|\^/| Maple 2025 (APPLE ARM64 MACOS) ._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2025 \ MAPLE / All rights reserved. Maple is a trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. # See file ../HowToRunThis.txt how to set up the code. > read "RC.m": > infolevel[LREtools] := 2; infolevel[LREtools] := 2 > interface(screenwidth = 150); _Env_LRE_tau := E; _Env_LRE_x := n; 80 _Env_LRE_tau := E _Env_LRE_x := n > with(LREtools); [AbsoluteFactorization, AnalyticityConditions, GCRD, GeneralizedExponents, GuessRecurrence, Homomorphisms, HypergeometricTerm, IntegralBasis, IsDesingularizable, IsIrreducible, LCLM, MinimalRecurrence, MultiplyOperators, NormalForm, OperatorToRecurrence, REcontent, REcreate, REplot, REprimpart, REreduceorder, REtoDE, REtodelta, REtoproc, RecurrenceToOperator, ReduceToOrderTwo, RightDivision, RightFactors, SearchTable, SolveLRE, SumDecompose, SymmetricPower, SymmetricProduct, ValuesAtPoint, autodispersion, constcoeffsol, dAlembertiansols, delta, dispersion, divconq, firstlin, hypergeomsols, mhypergeomsols, polysols, ratpolysols, riccati, shift] > for i in RC do > lprint(i[1]); > SolveLRE( i[2] ); > od; "A000023" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" memory used=48.3MB, alloc=108.3MB, time=0.23 LREtools/SearchTable: "SearchTable successful" n n {(-1) (n + 1) ((2 n + 3) BesselI(n, 2) - 4 BesselI(n - 1, 2)), (-1) (n + 1) ((2 n + 3) BesselK(n, -2) - 4 BesselK(n - 1, -2))} "A000085" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A000090" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / { 0 irem(n1, 3) = 0\ | { | | { 0 irem(n1, 3) = 1| | { | | { / n1 \ | | { |---- - 2/3| | |n - 1 { \ 3 / // n1 \ \2 2 n1 n1 n1 | |----- { (n1 - 1) n1 (-1/3) ||---- - 2/3|!| binomial(---- - 4/3, ---- - 2/3) binomial(n1 - 2, ---- - 2/3) irem(n1, 3) = 2| | \ { \\ 3 / / 3 3 3 | {n! | ) --------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 3) = 0\ | { | | { / n1 \ | | { |---- - 1/3| | | { \ 3 / n1 n1 | | { (-9) GAMMA(---- + 2/3) GAMMA(---- + 1) irem(n1, 3) = 1| |n - 1 { 3 3 | |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ---------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 3 / n1 n1 | | { (-9) GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 3) = 0| | { 3 3 | | { | |n - 1 { 0 irem(n1, 3) = 1| |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ---------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A000130" memory used=124.0MB, alloc=144.3MB, time=0.54 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 4) = 0 |----- { \n1 = 0 0 , irem(n1, 4) = 1 0 , irem(n1, 4) = 2 / n1 \ |---- - 3/4| \ 4 / n1 n1 2 // n1 \ \3 n1 / 1/2 (-1/4) binomial(---- - 3/2, ---- - 3/4) ||---- - 3/4|!| binomial(n1 - 3, ---- - 3/2) (n1 - 2) (n1 - 1) n1 , irem(n1, 4) = 3 2 4 \\ 4 / / 2 / { 0 irem(n1, 4) = 0\ | { | | { 0 irem(n1, 4) = 1| | { | | { / n1 \ | | { |---- - 1/2| | | { \ 4 / n1 n1 n1 | | { (-64) GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 3/4) irem(n1, 4) = 2| \ |n - 1 { 4 4 4 | | |----- { | | | \ { 0 irem(n1, 4) = 3| (n1 + 1)!|, n! | ) ----------------------------------------------------------------------------------------------|, | | / (n1 + 1)! | | |----- | / \n1 = 0 / / { 0 irem(n1, 4) = 0\ | { | | { / n1 \ | | { |---- - 1/4| | | { \ 4 / // n1 \ \3 3 n1 n1 n1 | | { n1 (-1) ||---- - 1/4|!| binomial(---- - 3/4, ---- - 1/4) binomial(n1 - 1, ---- - 1/4) irem(n1, 4) = 1| | { \\ 4 / / 4 4 4 | | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ---------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 4 / n1 n1 n1 | | { (-64) GAMMA(---- + 1) GAMMA(---- + 3/4) GAMMA(---- + 1/2) irem(n1, 4) = 0| | { 4 4 4 | | { | | { 0 irem(n1, 4) = 1| | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ----------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A000150" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n binomial(2 n, n) { { 2 4 {----------------, { 2 binomial(n - 1, n/2 - 1/2) , { -------------------------- n::even} n + 1 { ---------------------------- n::odd { n (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A000153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (-1) (n1 + 1) | n! (n + 3 n + 1) | ) -------------------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) + 3 n1 + 4) (n1 + 3 n1 + 1)| n! (n + 3 n + 1) \n1 = 0 / {-----------------, ----------------------------------------------------------------------------} n n "A000155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 1), (-1) BesselK(n, -1)} "A000159" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" memory used=203.0MB, alloc=176.3MB, time=0.92 LREtools/SearchTable: "SearchTable successful" n 3 2 2 {(-1) (n + 3) ((2 n + 9 n + 14 n + 9) BesselI(n, 2) + (-2 n - 9 n - 12) BesselI(n - 1, 2)), n 3 2 2 (-1) (n + 3) ((2 n + 9 n + 14 n + 9) BesselK(n, -2) + (-2 n - 9 n - 12) BesselK(n - 1, -2))} "A000166" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000167" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n n {1, (-1) , (-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A000172" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, -n, -n], [1, 1], -1)} "A000179" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A000180" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ (-1) 3 | {3 n!, 3 n! | ) -----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A000181" memory used=286.0MB, alloc=176.3MB, time=1.33 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 4 3 2 3 2 {(-1) (n + 4) ((4 n + 28 n + 71 n + 83 n + 43) BesselI(n, 2) + (-4 n - 28 n - 67 n - 59) BesselI(n - 1, 2)), n 4 3 2 3 2 (-1) (n + 4) ((4 n + 28 n + 71 n + 83 n + 43) BesselK(n, -2) + (-4 n - 28 n - 67 n - 59) BesselK(n - 1, -2))} "A000183" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 5, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000184" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 2), ----------------------------------------------} n + 1 "A000185" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" memory used=431.4MB, alloc=184.3MB, time=1.95 LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 4 3 2 {(-1) (n + 5) ((4 n + 40 n + 151 n + 276 n + 259 n + 109) BesselI(n, 2) + (-4 n - 40 n - 147 n - 240 n - 152) BesselI(n - 1, 2)), n 5 4 3 2 4 3 2 (-1) (n + 5) ((4 n + 40 n + 151 n + 276 n + 259 n + 109) BesselK(n, -2) + (-4 n - 40 n - 147 n - 240 n - 152) BesselK(n - 1, -2))} "A000186" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000207" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 binomial(n, n/2) { 3 4 { ------------------ n::even { -------------------------------- n::even (2 n + 1) binomial(2 n, n) { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {--------------------------, { , { , (n + 3) (n + 2) (n + 1) { 6 binomial(n + 1, n/2 + 1/2) { (2 n - 2) { ---------------------------- n::odd { 2 2 { n + 3 { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { { 0 irem(n, 3) = 1 { /2 n \ { { |--- - 2/3| { /2 n \ , { \ 3 / , { |--- - 4/3| { 2 GAMMA(n/3 + 1/2) { \ 3 / { ----------------------------- irem(n, 3) = 1 { 2 GAMMA(n/3 + 1/2) { GAMMA(n/3 + 2) { ----------------------------- irem(n, 3) = 2 { { GAMMA(n/3 + 2) { 0 irem(n, 3) = 2 { 2 n { 3 binomial(---, n/3) { 3 { -------------------- irem(n, 3) = 0 { n + 3 } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A000222" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 5) BesselI(n, 2) + (2 n - 7) BesselI(n - 1, 2)), (-1) ((2 n + 5) BesselK(n, -2) + (2 n - 7) BesselK(n - 1, -2))} "A000239" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A000240" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A000246" memory used=562.8MB, alloc=184.3MB, time=2.55 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 { 2 ((n/2)!) { 2 binomial(n, n/2) ((n/2)!) n::even { ------------ n::even { {{ n , { (-n - 1) 2 2 } { { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) { (n - 1) 2 { ----------------------------------------------------- n::odd { 2 ((n/2 - 1/2)!) n::odd { n "A000254" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000255" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! (n + 2), n! (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2)| |----- | \n1 = 0 / "A000256" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) | {(-1/4) (8 n + 5), (-1/4) (8 n + 5) | ) ----------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) (-1/4) (8 n1 + 13) (32 n1 + 20)| \n1 = 0 / "A000259" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\n / 1/2\n |----- | | 5 5 | | 5 5 | | 5 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|11/2 - ------| , |11/2 + ------| , |11/2 - ------| | ) |-2 (-1) (-11 + 5 5 ) (11 + 5 5 ) \ 2 / \ 2 / \ 2 / | / | |----- | \n1 = 0 \ / / 1/2\(-n2 - 1) \\\ |n1 - 1 | 5 5 | 2 ||| |----- |11/2 + ------| (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (5 n2 + 9) (19 n2 + 77 n2 + 70)||| | \ \ 2 / ||| | ) --------------------------------------------------------------------------------------------------|||} | / (n2 + 3) (n2 + 2) (n2 + 1) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) ||| |----- ||| \n2 = 0 /// "A000261" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 | \ (-1) (n1 + 1) | n! (n + 6 n + 8 n + 1) | ) -----------------------------------------------------------------------| | / 3 2 3 2 | 3 2 |----- (n1 + 1)! ((n1 + 1) + 6 (n1 + 1) + 8 n1 + 9) (n1 + 6 n1 + 8 n1 + 1)| n! (n + 6 n + 8 n + 1) \n1 = 0 / {------------------------, ---------------------------------------------------------------------------------------------------------} n n "A000262" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------} n "A000266" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |---- - 1/2| | | { (-2) |----|! n1::even| |n - 1 { \ 2 / / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 (-1/2) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) ----------------------------------------------------------------------------------|, n! | ) ------------------------------------|, n! | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / } "A000270" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n ((n + 2) BesselI(n, 2) - 2 BesselI(n - 1, 2)), (-1) n ((n + 2) BesselK(n, -2) - 2 BesselK(n - 1, -2))} "A000271" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n + 1) BesselI(n, 2) - BesselI(n - 1, 2)), (-1) ((n + 1) BesselK(n, -2) - BesselK(n - 1, -2))} "A000274" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! n, n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A000276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A000279" memory used=699.0MB, alloc=184.3MB, time=3.18 LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + 4 hypergeom([-n, -n, -n], [1, 1], -1)) {--------------------------------------------------------------------------------------------------} n + 2 "A000287" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n n 2 n 2 {(-1) , (-1/2) (27 n + 279 n + 718), (-1/2) (27 n + 279 n + 718) /n - 1 \ |----- | | \ (2 n1 + 11) (2 n1 + 9) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) | | ) ---------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) (27 (n1 + 1) + 279 n1 + 997) (54 n1 + 558 n1 + 1436)| \n1 = 0 / "A000305" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\n / 1/2\n |----- | | 5 5 | | 5 5 | | 5 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|11/2 - ------| , |11/2 + ------| , |11/2 - ------| | ) |-2 (-1) (-11 + 5 5 ) (11 + 5 5 ) \ 2 / \ 2 / \ 2 / | / | |----- | \n1 = 0 \ / / 1/2\(-n2 - 1) \\\ |n1 - 1 | 5 5 | 2 ||| |----- |11/2 + ------| (3 n2 + 4) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (95 n2 + 404 n2 + 405)||| | \ \ 2 / ||| | ) ----------------------------------------------------------------------------------------------------|||} | / (n2 + 3) (n2 + 2) (n2 + 1) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) ||| |----- ||| \n2 = 0 /// "A000313" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) (-1) | {n! (n - 1), n! (n - 1) | ) ---------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A000316" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {-(-2 2 ) n! (n + 1) ((2 n + 1) 2 BesselI(n + 1/2, 2 ) - 2 BesselI(n - 1/2, 2 )), 1/2 n 1/2 1/2 1/2 -(-2 2 ) n! (n + 1) ((2 n + 1) 2 BesselK(n + 1/2, -2 ) - 2 BesselK(n - 1/2, -2 ))} "A000321" LREtools/SearchTable: "SearchTable successful" n {(-1) HermiteH(n, 1/2)} "A000346" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} n + 1 "A000349" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000354" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ (-1) 2 | {2 n!, 2 n! | ) -----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A000365" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (4 n + 13) {(n + 3) (n + 2) 4 (8 n + 35), ---------------------------------------------------------} n + 1 "A000387" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (-1) | {n! | ) ------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A000399" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ n2! (n2 + 1) || | n1! (n1 + 1) | ) ------------------|| /n - 1 \ |n - 1 | / (n2 + 3) (n2 + 1)!|| |----- | |----- |----- || | \ n1! (n1 + 1) | | \ \n2 = 0 /| {n! (n + 1) (n + 2), n! (n + 1) (n + 2) | ) ------------------|, n! (n + 1) (n + 2) | ) ----------------------------------------|} | / (n1 + 3) (n1 + 1)!| | / (n1 + 1)! (n1 + 3) | |----- | |----- | \n1 = 0 / \n1 = 0 / "A000425" memory used=831.1MB, alloc=184.3MB, time=3.79 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n + 1) ((n + 1) BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n + 1) ((n + 1) BesselK(n, -2) - BesselK(n - 1, -2))} "A000426" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 3) BesselI(n, 2) - 4 BesselI(n - 1, 2)), (-1) ((2 n + 3) BesselK(n, -2) - 4 BesselK(n - 1, -2))} "A000449" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A000454" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 4) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 4) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 4) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 4) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|} | / (n1 + 4) (n1 + 1)! | |----- | \n1 = 0 / "A000459" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 ) n! BesselI(n + 1/2, 2 ), (-2 ) n! BesselK(n + 1/2, -2 )} "A000473" 2 n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (4 n + 46 n + 123) {(n + 4) (n + 3) (n + 2) 4 (8 n + 35), ----------------------------------------------------------------------------} n + 1 "A000475" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A000482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 5) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 5) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 5) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 5) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 5) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 5) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A000483" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 3) (n + 2) (n + 1) n! (n + 5) (n + 4), (n + 3) (n + 2) (n + 1) n! (n + 5) (n + 4) | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 4) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) n! (n + 5) (n + 4) | ) ----------------------------------------|} | / (n1 + 1)! (n1 + 5) | |----- | \n1 = 0 / "A000489" memory used=954.2MB, alloc=184.3MB, time=4.39 LREtools/SearchTable: "SearchTable successful" 3 2 {((9 n + 54 n + 117 n + 68) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) 4 3 2 2 + (-27 n - 234 n - 681 n - 738 n - 280) hypergeom([-n, -n, -n], [1, 1], -1)) (n + 1) /((n + 4) (n + 3) (n + 2))} "A000502" 2 n 2 (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {(n + 5) (n + 4) (n + 3) (n + 2) 4 (64 n + 1072 n + 3843), ------------------------------------------------------------------------------} n + 1 "A000522" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000531" n (2 n + 3) (2 n + 1) binomial(2 n, n) {4 , ------------------------------------} n + 1 "A000536" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 4) (n + 3) (n + 2) (n + 1) n! (n + 5 n + 5), /n - 1 \ |----- n1 | 2 | \ (-1) | (n + 4) (n + 3) (n + 2) (n + 1) n! (n + 5 n + 5) | ) --------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 5 n1 + 10) (n1 + 5 n1 + 5)| \n1 = 0 / "A000658" memory used=1079.8MB, alloc=186.9MB, time=4.90 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A000681" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) ((2 n + 2) LaguerreL(n + 1, -n - 1/2, 1/2) + LaguerreL(n, -n + 1/2, 1/2))} "A000704" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 1/2 (- n/2) 2 1/2 n 1/2 {2 HermiteH(n, ----), (-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} 2 "A000757" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- n2 ||| n n | \ n1 | n | \ | n1 | \ (-1) ||| {(-1) , (-1) | ) (-(-1) n1!)|, (-1) | ) |-(-1) n1! | ) ---------|||} | / | | / | | / (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A000774" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000775" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 3)| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A000776" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A000777" binomial(2 n, n) (n + 2) {1, ------------------------} n + 1 "A000778" binomial(2 n, n) (5 n + 4) {1, --------------------------} (n + 1) (n + 2) "A000779" n n {2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A000781" binomial(2 n, n) (11 n + 4) {1, ---------------------------} (n + 1) (n + 2) "A000806" LREtools/SearchTable: "SearchTable successful" {BesselI(n + 1/2, 1), BesselK(n + 1/2, -1)} "A000846" {binomial(2 n, n), binomial(3 n, n)} "A000898" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, I)} "A000900" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" memory used=1208.4MB, alloc=186.9MB, time=5.50 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SearchTable: "SearchTable successful" { (n/4) 1/2 n 1/2 { (-1) HermiteH(n/2 - 1/2, -I) n::even {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), { , { (n/2 + 1/2) (n/2 + 1/2) { (-I) (-1) HermiteH(n/2, -I) n::odd { (n/2) { (n/2) { (-I) HermiteH(n/2, I) n::even { (-I) HermiteH(n/2 - 1/2, I) n::even { , { , { (n/2 - 1/2) { (n/2 + 1/2) { (-I) HermiteH(n/2 - 1/2, I) n::odd { (-I) HermiteH(n/2, I) n::odd { (n/4) { (-1) HermiteH(n/2, -I) I n::even { } { (n/2 - 1/2) (n/2) { (-I) (-1) HermiteH(n/2 - 1/2, -I) n::odd "A000902" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, I)} "A000904" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 4 3 2 2 {(-1) ((n + 6 n + 14 n + 15 n + 7) BesselI(n, 2) - (n + 2) (n + 4 n + 5) BesselI(n - 1, 2)), n 4 3 2 2 (-1) ((n + 6 n + 14 n + 15 n + 7) BesselK(n, -2) - (n + 2) (n + 4 n + 5) BesselK(n - 1, -2))} "A000912" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n binomial(2 n, n) { { 2 4 {----------------, { 2 binomial(n - 1, n/2 - 1/2) , { -------------------------- n::even} n + 1 { ---------------------------- n::odd { n (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A000932" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A000957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-1/2) , (-1/2) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-1/2) | \n1 = 0 / "A000958" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (5 n1 + 9) | {(-1/2) , (-1/2) | ) -----------------------------------------|} | / (n1 + 1)| |----- (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A000985" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000986" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A000987" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001002" LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4) {------------------------------------------------------------} n + 1 "A001003" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {---------------------------------------, ---------------------------------------} n n "A001005" memory used=1337.5MB, alloc=186.9MB, time=6.10 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001006" LREtools/SearchTable: "SearchTable successful" n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------} n + 2 "A001040" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A001046" LREtools/SearchTable: "SearchTable not successful" {} "A001052" LREtools/SearchTable: "SearchTable not successful" {} "A001053" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A001120" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A001171" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (n1 + 1) \| n n | \ | 2 n1! || {(2 n + 1) (-1/2) n! binomial(2 n, n), (2 n + 1) (-1/2) n! binomial(2 n, n) | ) |- --------------------------------------------------------||, | / \ (n1 + 3) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ | 2 (-1) (2 n1 + 1) binomial(2 n1, n1) n1! || n (2 n + 1) (-1/2) n! binomial(2 n, n) | ) |- -----------------------------------------------||, (2 n + 1) (-1/2) n! binomial(2 n, n) | / \ (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / / / /n1 - 1 \ \\ | | |----- (n2 + 1) | || | | n1 | \ 2 n2! | || | | 2 (-1) (2 n1 + 1) | ) -----------------------------------------------------------------| binomial(2 n1, n1) n1!|| |n - 1 | | / (n2 + 3) (n2 + 4) (2 n2 + 3) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!| || |----- | |----- | || | \ | \n2 = 0 / || | ) |- ---------------------------------------------------------------------------------------------------------------------||} | / \ (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / "A001181" LREtools/SearchTable: "SearchTable successful" (3 n + 4) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) - 8 hypergeom([-n, -n, -n], [1, 1], -1) {-------------------------------------------------------------------------------------------------} 2 n (n + 3) (n + 2) "A001189" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {1, (-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A001205" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001233" memory used=1460.0MB, alloc=186.9MB, time=6.69 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 6) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 6) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, (n + 5) | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 1) n5! ||||| | | | | (n4 + 1) n4! | ) ------------------||||| | | | |n3 - 1 | / (n5 + 6) (n5 + 1)!||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------|||| | | |n2 - 1 | / (n4 + 6) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------|} | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / "A001234" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 1) n5! ||||| | | | | (n4 + 1) n4! | ) ------------------||||| | | | |n3 - 1 | / (n5 + 7) (n5 + 1)!||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------|, (n + 6) (n + 5) (n + 4) | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / (n + 3) (n + 2) (n + 1) n! / / / / / /n5 - 1 \\\\\\ | | | | | |----- |||||| | | | | | | \ (n6 + 1) n6! |||||| | | | | | (n5 + 1) n5! | ) ------------------|||||| | | | | |n4 - 1 | / (n6 + 7) (n6 + 1)!|||||| | | | | |----- |----- |||||| | | | | | \ \n6 = 0 /||||| | | | | (n4 + 1) n4! | ) ----------------------------------------||||| | | | |n3 - 1 | / (n5 + 7) (n5 + 1)! ||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) --------------------------------------------------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ------------------------------------------------------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) --------------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A001260" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (-1) | {n! (n - 2), n! (n - 2) | ) ------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A001261" memory used=1576.6MB, alloc=186.9MB, time=7.29 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (n1 - 1) (-1) | {n! (n - 3), n! (n - 3) | ) ---------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A001266" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001267" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001268" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001277" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- n2 | \ \ | \ (-1) | {1, ) (n1 + 1) (n1 + 2) (n1 + 3) n1!, ) (n1 + 1) (n1 + 2) (n1 + 3) n1! | ) ------------------------------------|} / / | / (n2 + 2) (n2 + 3) (n2 + 4) (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A001278" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! (2 n1 + 3)| {1, (2 n + 7) | ) -----------------------------------------------------------|, (2 n + 7) | / (2 n1 + 9) (2 n1 + 7) | |----- | \n1 = 0 / / /n1 - 1 \\ | |----- n2 || | | \ (-1) (2 n2 + 9) || | (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! (2 n1 + 3) | ) ----------------------------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 1)! (2 n2 + 5) (2 n2 + 3)|| |----- |----- || | \ \n2 = 0 /| | ) -------------------------------------------------------------------------------------------------------------------------------|, 2 n + 7 | / (2 n1 + 9) (2 n1 + 7) | |----- | \n1 = 0 / } "A001338" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- | ----- \ | \ 1 | \ {1, ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A001339" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A001340" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {n! (n + n + 1), n! (n + n + 1) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + n1 + 2) (n1 + n1 + 1)| \n1 = 0 / "A001341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 3 2 3 2 | \ 1 | {n! (n + 3 n + 5 n + 2), n! (n + 3 n + 5 n + 2) | ) -----------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 3 (n1 + 1) + 5 n1 + 7) (n1 + 3 n1 + 5 n1 + 2)| \n1 = 0 / "A001342" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 {n! (n + 6 n + 17 n + 20 n + 9), /n - 1 \ |----- | 4 3 2 | \ 1 | n! (n + 6 n + 17 n + 20 n + 9) | ) --------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 6 (n1 + 1) + 17 (n1 + 1) + 20 n1 + 29) (n1 + 6 n1 + 17 n1 + 20 n1 + 9)| \n1 = 0 / "A001405" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A001453" (2 n + 3) (2 n + 1) binomial(2 n, n) {1, ------------------------------------} (n + 3) (n + 2) (n + 1) "A001454" memory used=1701.0MB, alloc=186.9MB, time=7.88 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=1785.5MB, alloc=186.9MB, time=8.24 LREtools/SearchTable: "SearchTable successful" 7 6 5 4 3 2 {((2 n + 43 n - 3304 n - 41474 n - 192670 n - 431069 n - 468012 n - 198396) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 5 4 3 2 2 / 2 2 2 2 - 9 (2 n + 37 n - 140 n - 2611 n - 8556 n - 8532) (n + 1) hypergeom([1/2, -n, -n], [1, 1], 4)) / ((n + 5) (n + 4) (n + 3) (n + 2) ), / (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) ----------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A001455" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 memory used=1888.5MB, alloc=186.9MB, time=8.65 memory used=1965.5MB, alloc=186.9MB, time=8.91 memory used=2024.6MB, alloc=186.9MB, time=9.14 memory used=2091.0MB, alloc=186.9MB, time=9.40 LREtools/SearchTable: "SearchTable successful" memory used=2200.2MB, alloc=186.9MB, time=9.75 memory used=2276.1MB, alloc=186.9MB, time=9.99 memory used=2335.8MB, alloc=186.9MB, time=10.18 memory used=2380.4MB, alloc=186.9MB, time=10.36 memory used=2417.9MB, alloc=186.9MB, time=10.51 memory used=2458.3MB, alloc=186.9MB, time=10.67 memory used=2503.6MB, alloc=186.9MB, time=10.83 memory used=2552.0MB, alloc=186.9MB, time=11.00 memory used=2604.3MB, alloc=186.9MB, time=11.18 memory used=2646.9MB, alloc=186.9MB, time=11.35 memory used=2676.1MB, alloc=186.9MB, time=11.49 memory used=2706.6MB, alloc=186.9MB, time=11.63 memory used=2738.7MB, alloc=186.9MB, time=11.77 memory used=2770.8MB, alloc=186.9MB, time=11.91 memory used=2804.4MB, alloc=186.9MB, time=12.06 memory used=2840.4MB, alloc=186.9MB, time=12.22 memory used=2877.4MB, alloc=186.9MB, time=12.38 memory used=2916.0MB, alloc=186.9MB, time=12.55 memory used=2953.5MB, alloc=186.9MB, time=12.72 memory used=2992.0MB, alloc=186.9MB, time=12.88 memory used=3032.2MB, alloc=186.9MB, time=13.04 memory used=3075.8MB, alloc=186.9MB, time=13.24 memory used=3117.7MB, alloc=186.9MB, time=13.39 memory used=3159.6MB, alloc=186.9MB, time=13.56 memory used=3204.8MB, alloc=186.9MB, time=13.75 memory used=3248.5MB, alloc=186.9MB, time=13.92 memory used=3292.3MB, alloc=186.9MB, time=14.08 memory used=3340.0MB, alloc=186.9MB, time=14.28 memory used=3384.6MB, alloc=186.9MB, time=14.45 memory used=3432.9MB, alloc=186.9MB, time=14.65 memory used=3478.4MB, alloc=186.9MB, time=14.82 memory used=3526.6MB, alloc=186.9MB, time=15.02 memory used=3571.8MB, alloc=186.9MB, time=15.18 memory used=3621.0MB, alloc=186.9MB, time=15.39 memory used=3666.9MB, alloc=186.9MB, time=15.57 memory used=3715.3MB, alloc=186.9MB, time=15.77 memory used=3763.0MB, alloc=186.9MB, time=15.97 memory used=3817.0MB, alloc=186.9MB, time=16.16 memory used=3864.9MB, alloc=186.9MB, time=16.36 memory used=3910.8MB, alloc=186.9MB, time=16.53 memory used=3959.0MB, alloc=186.9MB, time=16.73 memory used=4010.7MB, alloc=186.9MB, time=16.93 memory used=4058.4MB, alloc=186.9MB, time=17.12 memory used=4113.0MB, alloc=186.9MB, time=17.36 memory used=4159.0MB, alloc=186.9MB, time=17.54 memory used=4213.4MB, alloc=186.9MB, time=17.76 memory used=4260.0MB, alloc=186.9MB, time=17.92 memory used=4312.7MB, alloc=186.9MB, time=18.13 memory used=4361.4MB, alloc=186.9MB, time=18.34 LREtools/SearchTable: "SearchTable successful" 9 8 7 6 5 4 3 2 {((2 n + 63 n - 32352 n - 649242 n - 5386926 n - 24163785 n - 63339428 n - 97012332 n - 80399232 n - 27851040) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 7 6 5 4 3 2 2 / - 9 (2 n + 57 n - 3004 n - 57279 n - 390676 n - 1290276 n - 2090304 n - 1334880) (n + 1) hypergeom([1/2, -n, -n], [1, 1], 4)) / ( / 2 2 2 2 2 13 12 11 10 9 8 7 (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) ), (-16 (32 n + 1936 n + 53488 n + 844600 n + 8139154 n + 46952145 n + 127006780 n 6 5 4 3 2 4 - 251678158 n - 3821742598 n - 16876026295 n - 42415244920 n - 64535844564 n - 55610508960 n - 20934056448) (n + 1) 16 15 14 13 12 11 hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1) + (2 n + 1) (64 n + 4160 n + 124880 n + 1905072 n + 11727340 n - 78730948 n 10 9 8 7 6 5 4 - 2301631841 n - 23571958707 n - 149730207266 n - 659111681226 n - 2089409714545 n - 4820891654795 n - 8037170711960 n 3 2 - 9437293200932 n - 7404393492720 n - 3484159664256 n - 743580739584) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1)) / 3 3 3 3 3 3 binomial(2 n, n) / ((n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8))} / "A001464" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 | 2 | 2 {|- ----| HermiteH(n, ----)} \ 2 / 2 "A001465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 1/2 (- n/2) 2 1/2 n 1/2 {2 HermiteH(n, ----), (-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} 2 "A001470" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001471" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A001472" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001473" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A001475" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {(-1/2 I 2 ) (2 (n + 1) HermiteH(n, 1/2 I 2 ) I + HermiteH(n + 1, 1/2 I 2 )) 2 I} "A001495" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001499" memory used=4479.2MB, alloc=186.9MB, time=18.85 LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) LaguerreL(n, -n - 1/2, -1/2)} "A001500" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001501" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001506" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 3 2 | \ (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 3) n1 binomial(2 n1 + 2, n1 + 1) | (27 n + 18 n - 5 n - 4) | ) ------------------------------------------------------------------------------| | / 3 2 3 2 | 3 2 |----- (n1 + 1) (27 (n1 + 1) + 18 (n1 + 1) - 5 n1 - 9) (27 n1 + 18 n1 - 5 n1 - 4)| 27 n + 18 n - 5 n - 4 \n1 = 0 / {----------------------------, -----------------------------------------------------------------------------------------------------------------} (2 n + 1) n binomial(2 n, n) (2 n + 1) n binomial(2 n, n) "A001507" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 27 n + 45 n + 4 n - 4 {----------------------------, (2 n + 1) n binomial(2 n, n) /n - 1 \ |----- 2 2 | 3 2 | \ n1 (n1 - 1) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 10) binomial(2 n1 + 2, n1 + 1)| (27 n + 45 n + 4 n - 4) | ) ---------------------------------------------------------------------------------------------| | / 2 3 2 3 2 | |----- (n1 + 1) (n1 + 2) (27 (n1 + 1) + 45 (n1 + 1) + 4 n1) (27 n1 + 45 n1 + 4 n1 - 4) | \n1 = 0 / --------------------------------------------------------------------------------------------------------------------------------} (2 n + 1) n binomial(2 n, n) "A001508" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 27 n + 126 n + 157 n + 22 n - 12 4 3 2 {--------------------------------------, (27 n + 126 n + 157 n + 22 n - 12) (2 n + 3) (2 n + 1) n binomial(2 n, n) /n - 1 \ |----- 2 2 2 2 | | \ n1 (n1 - 1) (n1 - 2) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 24 n1 + 21) binomial(2 n1 + 2, n1 + 1) | | ) ----------------------------------------------------------------------------------------------------------------------------------|/( | / 2 2 4 3 2 4 3 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1) (27 (n1 + 1) + 126 (n1 + 1) + 157 (n1 + 1) + 22 n1 + 10) (27 n1 + 126 n1 + 157 n1 + 22 n1 - 12)| \n1 = 0 / (2 n + 3) (2 n + 1) n binomial(2 n, n))} "A001514" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n - 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) ((n - 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A001515" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1), (-1) BesselK(n + 1/2, -1)} "A001516" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((n - 3 n + 4) BesselI(n + 1/2, 1) + 4 BesselI(n - 1/2, 1)), (-1) ((n - 3 n + 4) BesselK(n + 1/2, -1) + 4 BesselK(n - 1/2, -1))} "A001517" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/2), (-1) BesselK(n + 1/2, -1/2)} "A001518" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/3), (-1) BesselK(n + 1/2, -1/3)} "A001540" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- | | \ (-1) | | \ 2 n1 + 3 | {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A001558" memory used=4603.4MB, alloc=186.9MB, time=19.49 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (11 n1 + 35)| {(-1/2) , (-1/2) | ) -----------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A001559" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 27)| {(-1/2) , (-1/2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A001623" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001680" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001681" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001683" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 n::even { 2 binomial(n, n/2) binomial(2 n, n) { { ------------------ n::even {----------------, { (2 n - 2) , { n + 2 , (n + 1) (n + 2) { 2 { { ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { { 0 irem(n, 3) = 1 { 2 n { { 3 binomial(--- - 2/3, n/3 - 1/3) { /2 n \ , { 3 , { |--- - 4/3| { -------------------------------- irem(n, 3) = 1 { \ 3 / { n + 2 { 2 GAMMA(1/6 + n/3) { { ----------------------------- irem(n, 3) = 2 { 0 irem(n, 3) = 2 { GAMMA(5/3 + n/3) { /2 n\ { |---| { \ 3 / { 2 GAMMA(1/6 + n/3) { ----------------------- irem(n, 3) = 0} { GAMMA(5/3 + n/3) { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A001705" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A001706" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 1) (n + 2) (n + 3) n!, (n + 1) (n + 2) (n + 3) n! | ) ------------------|, | / (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 4) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 1) (n + 2) (n + 3) n! | ) ----------------------------------------|} | / (n1 + 4) (n1 + 1)! | |----- | \n1 = 0 / "A001707" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 5) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 5) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 5) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A001708" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 6) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 6) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|} | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / "A001709" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=4720.3MB, alloc=186.9MB, time=20.08 LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 1) n5! ||||| | | | | (n4 + 1) n4! | ) ------------------||||| | | | |n3 - 1 | / (n5 + 7) (n5 + 1)!||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A001711" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 1) (n + 2) (n + 3) n!, (n + 1) (n + 2) (n + 3) n! | ) ------------------|} | / (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / "A001712" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 5) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A001713" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 6) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|} | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / "A001714" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A001716" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / "A001717" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 6) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|} | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / "A001718" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A001719" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 8) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 8) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, (n + 7) | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|} | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / "A001721" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / "A001722" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A001723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 8) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 8) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|} | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / "A001724" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- | | \ (n1 + 1) n1! | (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 9) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 9) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 9) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 9) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 9) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 9) (n1 + 1)! | |----- | \n1 = 0 / (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 9) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 9) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 9) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ------------------------------------------------------------------------------------|} | / (n1 + 9) (n1 + 1)! | |----- | \n1 = 0 / "A001757" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 3 2 { 2 8 ((n/2)!) { 2 2 (n + 1) binomial(n, n/2) ((n/2)!) { ------------------------ n::even { -------------------------------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {{ , { } { (3 n + 3) 2 { (-n + 1) 3 3 2 { 2 ((n/2 + 1/2)!) { 4 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) { ------------------------------------------ n::odd { ---------------------------------------------------------- n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { n + 1 "A001819" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 2 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A001820" memory used=4831.4MB, alloc=186.9MB, time=20.69 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 2 2 2 2 2 2 | \ (n1 + 1) (n1!) | {(n + 2) (n + 1) (n!) , (n + 2) (n + 1) (n!) | ) ----------------------|, | / 2 2| |----- (n1 + 3) ((n1 + 1)!) | \n1 = 0 / / /n1 - 1 \\ | |----- 2 2 || | 2 2 | \ (n2 + 1) (n2!) || | (n1 + 1) (n1!) | ) ----------------------|| |n - 1 | / 2 2|| |----- |----- (n2 + 3) ((n2 + 1)!) || 2 2 2 | \ \n2 = 0 /| (n + 2) (n + 1) (n!) | ) ------------------------------------------------|} | / 2 2 | |----- (n1 + 3) ((n1 + 1)!) | \n1 = 0 / "A001821" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 2 2 2 2 2 2 2 2 | \ (n1 + 1) (n1!) | {(n + 3) (n + 2) (n + 1) (n!) , (n + 3) (n + 2) (n + 1) (n!) | ) ----------------------|, | / 2 2| |----- (n1 + 4) ((n1 + 1)!) | \n1 = 0 / / /n1 - 1 \\ | |----- 2 2 || | 2 2 | \ (n2 + 1) (n2!) || | (n1 + 1) (n1!) | ) ----------------------|| |n - 1 | / 2 2|| |----- |----- (n2 + 4) ((n2 + 1)!) || 2 2 2 2 | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) (n!) | ) ------------------------------------------------|, | / 2 2 | |----- (n1 + 4) ((n1 + 1)!) | \n1 = 0 / / / /n2 - 1 \\\ | | |----- 2 2 ||| | | 2 2 | \ (n3 + 1) (n3!) ||| | | (n2 + 1) (n2!) | ) ----------------------||| | |n1 - 1 | / 2 2||| | |----- |----- (n3 + 4) ((n3 + 1)!) ||| | 2 2 | \ \n3 = 0 /|| | (n1 + 1) (n1!) | ) ------------------------------------------------|| |n - 1 | / 2 2 || |----- |----- (n2 + 4) ((n2 + 1)!) || 2 2 2 2 | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) (n!) | ) --------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 4) ((n1 + 1)!) | \n1 = 0 / "A001824" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2 2 2 \| 2 n 2 2 2 n 2 2 | \ | 4 (2 n1 + 1) binomial(2 n1, n1) (n1!) || {(2 n + 1) (1/4) (n!) binomial(2 n, n) , (2 n + 1) (1/4) (n!) binomial(2 n, n) | ) |----------------------------------------------------|| | / | 2 2 2|| |----- \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / } "A001825" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 2 n 2 2 {(2 n + 1) (2 n + 3) (1/4) (n!) binomial(2 n, n) , /n - 1 \ |----- / 2 2 2 \| 2 2 n 2 2 | \ | 4 (2 n1 + 1) binomial(2 n1, n1) (n1!) || 2 2 (2 n + 1) (2 n + 3) (1/4) (n!) binomial(2 n, n) | ) |----------------------------------------------------||, (2 n + 1) (2 n + 3) | / | 2 2 2|| |----- \(2 n1 + 5) binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / n 2 2 (1/4) (n!) binomial(2 n, n) / / /n1 - 1 \\\ | | |----- / 2 2 2 \||| | | 2 2 2 | \ | 4 (2 n2 + 1) binomial(2 n2, n2) (n2!) |||| | |4 (2 n1 + 1) binomial(2 n1, n1) (n1!) | ) |----------------------------------------------------|||| |n - 1 | | / | 2 2 2|||| |----- | |----- \(2 n2 + 5) binomial(2 n2 + 2, n2 + 1) ((n2 + 1)!) /||| | \ | \n2 = 0 /|| | ) |--------------------------------------------------------------------------------------------------------||} | / | 2 2 2 || |----- \ (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A001850" LREtools/SearchTable: "SearchTable successful" {LegendreP(n, 3), LegendreQ(n, 3)} "A001883" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 5, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001887" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 5, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" memory used=4939.6MB, alloc=186.9MB, time=21.25 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A001895" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 binomial(n, n/2) { 3 4 { ------------------ n::even { -------------------------------- n::even (2 n + 3) (2 n + 1) binomial(2 n, n) { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {------------------------------------, { , { } (n + 3) (n + 2) (n + 1) { 6 binomial(n + 1, n/2 + 1/2) { (2 n - 2) { ---------------------------- n::odd { 2 2 { n + 3 { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A001900" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 ((n/2)!) n::even { (-n) 2 2 { { 2 binomial(n, n/2) ((n/2)!) (n + 1) n::even {{ (n + 1) 2 , { } { 2 ((n/2 + 1/2)!) { (-n + 1) 2 2 2 { ------------------------ n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { n + 1 "A001907" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2)| n n | \ (-1) 2 | {4 n!, 4 n! | ) -------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A001908" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ (-1) 5 | {5 n!, 5 n! | ) -----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A001909" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 n! (n + 10 n + 29 n + 24 n + 1) 4 3 2 {----------------------------------, n! (n + 10 n + 29 n + 24 n + 1) n /n - 1 \ |----- n1 | | \ (-1) (n1 + 1) | | ) ----------------------------------------------------------------------------------------------------|/n} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 10 (n1 + 1) + 29 (n1 + 1) + 24 n1 + 25) (n1 + 10 n1 + 29 n1 + 24 n1 + 1)| \n1 = 0 / "A001910" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 5 4 3 2 n! (n + 15 n + 75 n + 145 n + 89 n + 1) 5 4 3 2 {-------------------------------------------, n! (n + 15 n + 75 n + 145 n + 89 n + 1) n /n - 1 \ |----- n1 | | \ (-1) (n1 + 1) | | ) ------------------------------------------------------------------------------------------------------------------------------|/n} | / 5 4 3 2 5 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 15 (n1 + 1) + 75 (n1 + 1) + 145 (n1 + 1) + 89 n1 + 90) (n1 + 15 n1 + 75 n1 + 145 n1 + 89 n1 + 1)| \n1 = 0 / "A002002" LREtools/SearchTable: "SearchTable successful" (5 n + 3) LegendreP(n, 3) + (-n - 1) LegendreP(n + 1, 3) (5 n + 3) LegendreQ(n, 3) + (-n - 1) LegendreQ(n + 1, 3) {- --------------------------------------------------------, - --------------------------------------------------------} n n "A002003" LREtools/SearchTable: "SearchTable successful" (7 n + 3) LegendreP(n, 3) + (-n - 1) LegendreP(n + 1, 3) (7 n + 3) LegendreQ(n, 3) + (-n - 1) LegendreQ(n + 1, 3) {- --------------------------------------------------------, - --------------------------------------------------------} n n "A002005" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 2 8 binomial(---, n/2) { (n - 1) 3 n 3 n { 2 {{ 2 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) , { ----------------------- n::even} { 2 2 { (n + 1) (n + 2) { -------------------------------------------------------------------------------- n::odd { { n (n + 1) (n + 2) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A002018" LREtools/SearchTable: "SearchTable successful" n 2 (-1) (n!) ((n + 1) (2 n - 1) LaguerreL(n + 1, -n - 1/2, 1/2) + n LaguerreL(n, -n + 1/2, 1/2)) {-----------------------------------------------------------------------------------------------} n "A002019" LREtools/SearchTable: "SearchTable successful" n I (-(n + 2) hypergeom([-n - 1, 1 + 1/2 I], [2], 2) I + hypergeom([-n, 1 + 1/2 I], [2], 2)) n! {- ----------------------------------------------------------------------------------------------} n "A002020" LREtools/SearchTable: "SearchTable successful" n I (-(n + 2) hypergeom([-n - 1, 1 + 1/2 I], [2], 2) I + hypergeom([-n, 1 + 1/2 I], [2], 2)) n! {- ----------------------------------------------------------------------------------------------} n "A002026" memory used=5067.9MB, alloc=186.9MB, time=21.87 LREtools/SearchTable: "SearchTable successful" n (-1) ((2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------} (n + 3) (n + 2) "A002104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- | ----- \ | \ 1 | \ {1, ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A002119" LREtools/SearchTable: "SearchTable successful" {BesselI(n + 1/2, 1/2), BesselK(n + 1/2, -1/2)} "A002135" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002136" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002137" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002212" LREtools/SearchTable: "SearchTable successful" (4 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n "A002370" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) binomial(2 n, n) LaguerreL(n, - 1/4 - n, 1/4)} "A002426" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([1/2, -n], [1], 4)} "A002455" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2 \| n 2 n 2 | \ | (n1!) || {4 (n!) , 4 (n!) | ) |1/4 ------------||} | / | 2|| |----- \ ((n1 + 1)!) /| \n1 = 0 / "A002464" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002467" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A002468" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (-1) (3 n1 - 5) (n1 + 1) n1 | n! (n - 2 n - 1) | ) -------------------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) - 2 n1 - 3) (n1 - 2 n1 - 1)| n! n! (n - 2 n - 1) \n1 = 0 / {---------, -----------------, ----------------------------------------------------------------------------} n (n - 1) n (n - 1) n (n - 1) "A002469" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (-1) (3 n1 - 2) (n1 + 1) | n! (n - 2) | ) -----------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) - 2) (n1 - 2)| n! (n - 2) \n1 = 0 / {-----------, --------------------------------------------------------} n n "A002493" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" memory used=5188.0MB, alloc=186.9MB, time=22.45 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002538" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 2) (n1 + 1) n1! | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A002539" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 2) (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|, | / (n1 + 6) (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 3) (n2 + 1) n2! || | (n1 + 2) (n1 + 1) n1! | ) ---------------------------|| |n - 1 | / (n2 + 5) (n2 + 4) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------------------------|} | / (n1 + 6) (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A002627" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A002628" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002629" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002633" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002695" LREtools/SearchTable: "SearchTable successful" {(n + 1) (3 LegendreP(n + 1, 3) - LegendreP(n, 3)), (n + 1) (3 LegendreQ(n + 1, 3) - LegendreQ(n, 3))} "A002720" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -1)} "A002740" (n - 2) (n - 1) binomial(2 n, n) {--------------------------------} 2 n - 1 "A002741" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- n2 ||| n n | \ n1 | n | \ | n1 | \ (-1) ||| {(-1) , (-1) | ) (-(-1) n1!)|, (-1) | ) |-(-1) n1! | ) ---------|||} | / | | / | | / (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A002742" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ | n | \ n1 | {(-1) , (-1) | ) (-(n1 + 2) (n1 + 1) n1!)|, (-1) | ) (-(-1) (n1 + 2) (n1 + 1) n1!)|, | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / / / / /n2 - 1 \\\\ | | | |----- |||| | | | n2 | \ / 1 \|||| | | | (n2 + 1) (-1) n2! | ) |- ---------------------------||||| |n - 1 | |n1 - 1 | / \ (n3 + 2) (n3 + 3) (n3 + 1)!/|||| |----- | |----- |----- |||| n | \ | n1 | \ \n3 = 0 /||| (-1) | ) |-(-1) (n1 + 2) (n1 + 1) n1! | ) ------------------------------------------------------------|||} | / | | / (n2 + 1)! ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A002747" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ / 1 \|| | (-1) (n1 + 1) n1! | ) |- ------------------||| |n - 1 | / \ (n2 + 2) (n2 + 1)!/|| |----- |----- || n | \ \n2 = 0 /| {n! (n + 1), (n + 1) (-1) n!, n! (n + 1) | ) ---------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A002771" memory used=5302.5MB, alloc=218.9MB, time=23.07 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A002777" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { / 1/2\(n/2) {{ | 2 | 1/2 { 2 |- ----| binomial(n, n/2) (n/2)! BesselI(n/2 + 1, 2 ) , n::even { \ 2 / (- n/4 + 3/4) (n/2) 1/2 1/2 1/2 -I 2 (-1) (2 (n - 1) BesselI(n/2 + 1/2, 2 ) - 2 BesselI(n/2 - 1/2, 2 )) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! , n::odd { / 1/2\(n/2) , { | 2 | 1/2 { 2 |- ----| binomial(n, n/2) (n/2)! BesselK(n/2 + 1, -2 ) , n::even { \ 2 / (- n/4 + 3/4) (n/2) 1/2 1/2 1/2 -I 2 (-1) (2 (n - 1) BesselK(n/2 + 1/2, -2 ) - 2 BesselK(n/2 - 1/2, -2 )) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! , { 1/2 (n/2) 1/2 1/2 1/2 1/2 { 2 (-2 2 ) (n/2)! 2 (2 (n - 1) BesselI(n/2 + 1/2, 2 ) - 2 BesselI(n/2 - 1/2, 2 )) { ------------------------------------------------------------------------------------------------- n::even { n , { , n::odd { 1/2 (n/2 + 1/2) 1/2 { 4 (-2 2 ) (n/2 + 1/2)! BesselI(n/2 + 1, 2 ) { ---------------------------------------------------------- n::odd { n + 1 { 1/2 (n/2) 1/2 1/2 1/2 1/2 { 2 (-2 2 ) (n/2)! 2 (2 (n - 1) BesselK(n/2 + 1/2, -2 ) - 2 BesselK(n/2 - 1/2, -2 )) { --------------------------------------------------------------------------------------------------- n::even { n { } { 1/2 (n/2 + 1/2) 1/2 { 4 (-2 2 ) (n/2 + 1/2)! BesselK(n/2 + 1, -2 ) { ----------------------------------------------------------- n::odd { n + 1 "A002793" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -1)} "A002801" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, - 1/4 - n, 1/4)} "A002807" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- | ----- \ | \ (n2 + 1) n2| \ {1, ) n1! | ) -----------|, ) n1!} / | / (n2 + 1)! | / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A002816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A002829" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002867" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 2 { 2 4 ((n/2)!) n::even { (n + 1) ((n/2)!) binomial(n, n/2) n::even { {{ , { (2 n + 2) 2 } { 2 2 2 { 2 ((n/2 + 1/2)!) { 2 n ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) n::odd { -------------------------- n::odd { n + 1 "A002893" memory used=5456.2MB, alloc=218.9MB, time=23.83 LREtools/SearchTable: "SearchTable successful" {hypergeom([1/2, -n, -n], [1, 1], 4)} "A002895" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)} "A002896" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([1/2, -n, -n], [1, 1], 4)} "A002898" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002899" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A002909" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=5585.8MB, alloc=218.9MB, time=24.45 memory used=5730.2MB, alloc=223.3MB, time=25.16 memory used=5834.3MB, alloc=223.9MB, time=26.28 memory used=5942.2MB, alloc=223.9MB, time=26.74 memory used=6029.9MB, alloc=223.9MB, time=27.03 memory used=6145.5MB, alloc=223.9MB, time=27.42 memory used=6231.0MB, alloc=224.1MB, time=27.93 memory used=6302.4MB, alloc=224.1MB, time=28.42 memory used=6388.8MB, alloc=224.2MB, time=28.91 memory used=6537.7MB, alloc=256.2MB, time=29.59 memory used=6660.7MB, alloc=256.2MB, time=30.19 LREtools/SolveLRE: "Reduced the order of" (n+2)*(4*n^2+16*n+5)*(n+5)^2*E^4+(-24*n^5-340*n^4-1790*n^3-4223*n^2-4113*n-1010)*E^3+(20*n^4+200*n^3+ 659*n^2+795*n+290)*E^2+(24*n^5+260*n^4+990*n^3+1627*n^2+1133*n+230)*E-4*n^5-36*n^4-97*n^3-75*n^2 "to two: Half integer product u(n/2) * u(n/2+1/2)" (n+2)*(2*n+3)*E^2+(-12*n^2-30*n-19)*E+(n+1)*(2*n+1) LREtools/SearchTable: "SearchTable not successful" {1} "A002928" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A003011" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselJ(n + 1/2, -1), (-1) n! BesselY(n + 1/2, -1)} "A003048" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A003148" memory used=6797.3MB, alloc=240.2MB, time=30.89 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 1) n1 | n n | \ 2 (-1) (2 n1 + 1) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) -----------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A003149" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) n1!| {(n + 1) (1/2) n!, (n + 1) (1/2) n! | ) ----------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A003168" LREtools/SearchTable: "SearchTable successful" (2 n + 2) hypergeom([2 n + 3, -n - 1], [1], -1) + (-11 n - 8) hypergeom([-n, 2 n + 1], [1], -1) {-----------------------------------------------------------------------------------------------} (17 n + 11) n "A003169" LREtools/SearchTable: "SearchTable successful" 4 hypergeom([2 n + 3, -n - 1], [1], -1) - 5 hypergeom([-n, 2 n + 1], [1], -1) {-----------------------------------------------------------------------------} 17 n + 11 "A003221" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 + 2) (n1 - 1)| {n! | ) ------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A003422" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1!} / ----- n1 = 0 "A003435" LREtools/SearchTable: "SearchTable successful" n 2 {(-2) n! (n + 1) (n + 2) ((4 n + 6 n + 3) BesselI(n + 1/2, 1) + (-2 n - 2) BesselI(n - 1/2, 1)), n 2 (-2) n! (n + 1) (n + 2) ((4 n + 6 n + 3) BesselK(n + 1/2, -1) + (-2 n - 2) BesselK(n - 1/2, -1))} "A003436" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n + 1/2, 1) + BesselI(n - 1/2, 1)), (-1) (BesselK(n + 1/2, -1) + BesselK(n - 1/2, -1))} "A003440" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A003441" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { { 0 irem(n, 3) = 1 { /2 n \ (2 n + 1) binomial(2 n, n) { { |--- - 2/3| {--------------------------, { /2 n \ , { \ 3 / , (n + 3) (n + 2) { |--- - 4/3| { 2 GAMMA(n/3 + 1/2) { \ 3 / { ----------------------------- irem(n, 3) = 1 { 2 GAMMA(n/3 + 1/2) { GAMMA(n/3 + 2) { ----------------------------- irem(n, 3) = 2 { { GAMMA(n/3 + 2) { 0 irem(n, 3) = 2 { 2 n { 3 binomial(---, n/3) { 3 { -------------------- irem(n, 3) = 0 { n + 3 } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A003442" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=6950.4MB, alloc=240.2MB, time=31.68 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 n::even (n + 1) (2 n + 3) (2 n + 1) binomial(2 n, n) { {--------------------------------------------, { (2 n - 2) , (n + 4) (n + 3) (n + 2) { 2 (n + 1) { 1/2 -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { 0 irem(n, 4) = 0 { { 4 binomial(n, n/2) (n + 1) { 0 irem(n, 4) = 1 { -------------------------- n::even { { (n + 4) (n + 2) , { 0 irem(n, 4) = 2, { { { 0 n::odd { (n/2 - 3/2) { 2 GAMMA(n/4 + 1/2) { ----------------------------- irem(n, 4) = 3 { GAMMA(n/4 + 2) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1/2) { n , { ----------------------------- irem(n, 4) = 1, { 2 { GAMMA(n/4 + 2) { -------------------------------------- irem(n, 4) = 2 { { n (n + 4) binomial(n/2 - 1, n/4 - 1/2) { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { 4 binomial(n/2, n/4) { -------------------- irem(n, 4) = 0 { n + 4 { { 0 irem(n, 4) = 1} { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A003443" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { 0 irem(n, 5) = 2 (2 n + 3) (2 n + 1) (n + 2) (n + 1) binomial(2 n, n) { {----------------------------------------------------, { 0 irem(n, 5) = 3, (n + 5) (n + 4) (n + 3) { { /2 n \ { |--- - 8/5| { \ 5 / { 2 GAMMA(n/5 + 1/2) { ----------------------------- irem(n, 5) = 4 { GAMMA(n/5 + 2) { 0 irem(n, 5) = 0 { 0 irem(n, 5) = 0 { { { 0 irem(n, 5) = 1 { 0 irem(n, 5) = 1 { { { 0 irem(n, 5) = 2 { /2 n \ { { |--- - 4/5| { /2 n \ , { \ 5 / , { |--- - 6/5| { 2 GAMMA(n/5 + 1/2) { \ 5 / { ----------------------------- irem(n, 5) = 2 { 2 GAMMA(n/5 + 1/2) { GAMMA(n/5 + 2) { ----------------------------- irem(n, 5) = 3 { { GAMMA(n/5 + 2) { 0 irem(n, 5) = 3 { { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { 2 n { { 5 binomial(---, n/5) { /2 n \ { 5 { |--- - 2/5| { -------------------- irem(n, 5) = 0 { \ 5 / { n + 5 { 2 GAMMA(n/5 + 1/2) { { ----------------------------- irem(n, 5) = 1, { 0 irem(n, 5) = 1} { GAMMA(n/5 + 2) { { { 0 irem(n, 5) = 2 { 0 irem(n, 5) = 2 { { { 0 irem(n, 5) = 3 { 0 irem(n, 5) = 3 { { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 4 "A003444" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 n::even (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) { {----------------------------------------------, { 8 binomial(n - 1, n/2 - 1/2) n (n + 2) , (n + 5) (n + 4) (n + 3) (n + 1) { -------------------------------------- n::odd { (n + 1) (n + 3) (n + 5) { 0 irem(n, 4) = 0 { { n { 0 irem(n, 4) = 1 { 4 (n + 2) { { 1/2 ---------------------------------------- n::even, { 0 irem(n, 4) = 2, { (n + 1) (n + 3) (n + 5) binomial(n, n/2) { { { 8 binomial(n/2 - 3/2, n/4 - 3/4) (n - 1) { 0 n::odd { ---------------------------------------- irem(n, 4) = 3 { (n + 1) (n + 5) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { (n - 1) { { 4 2 { (n/2 - 1) , { ---------------------------------------------- irem(n, 4) = 1, { 2 GAMMA(n/4 + 3/4) { (n + 1) (n + 5) binomial(n/2 - 1/2, n/4 - 1/4) { --------------------------- irem(n, 4) = 2 { { GAMMA(n/4 + 9/4) { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { (n/2) { 2 GAMMA(n/4 + 3/4) { ----------------------- irem(n, 4) = 0 { GAMMA(n/4 + 9/4) { } { 0 irem(n, 4) = 1 { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A003446" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 binomial(n, n/2) { 2 4 { ------------------ n::even { -------------------------- n::even binomial(2 n, n) n { n + 2 { n (n + 1) binomial(n, n/2) {------------------, { , { , (n + 1) (n + 2) { 2 binomial(n - 1, n/2 - 1/2) { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { { 0 irem(n, 3) = 1 { 2 n { { 3 binomial(--- - 2/3, n/3 - 1/3) { /2 n \ , { 3 , { |--- - 4/3| { -------------------------------- irem(n, 3) = 1 { \ 3 / { n + 2 { 2 GAMMA(1/6 + n/3) { { ----------------------------- irem(n, 3) = 2 { 0 irem(n, 3) = 2 { GAMMA(5/3 + n/3) { /2 n\ { |---| { \ 3 / { 2 GAMMA(1/6 + n/3) { ----------------------- irem(n, 3) = 0} { GAMMA(5/3 + n/3) { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A003447" memory used=7097.8MB, alloc=240.2MB, time=32.47 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 4 { -------------------------------- n::even (n + 1) (2 n + 3) (2 n + 1) binomial(2 n, n) { (n + 1) (n + 3) binomial(n, n/2) {--------------------------------------------, { , (n + 4) (n + 3) (n + 2) { (2 n + 2) { 2 (8 n + 5) { -------------------------------------------------- n::odd { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { 0 irem(n, 4) = 0 { 4 binomial(n, n/2) (8 n + 5) { { ---------------------------- n::even { 0 irem(n, 4) = 1 { (n + 4) (n + 2) { { , { 0 irem(n, 4) = 2, { 32 binomial(n - 1, n/2 - 1/2) n { { ------------------------------- n::odd { (n/2 - 3/2) { (n + 1) (n + 3) { 2 GAMMA(n/4 + 1/2) { ----------------------------- irem(n, 4) = 3 { GAMMA(n/4 + 2) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1/2) { n , { ----------------------------- irem(n, 4) = 1, { 2 { GAMMA(n/4 + 2) { -------------------------------------- irem(n, 4) = 2 { { n (n + 4) binomial(n/2 - 1, n/4 - 1/2) { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { 4 binomial(n/2, n/4) { -------------------- irem(n, 4) = 0 { n + 4 { { 0 irem(n, 4) = 1} { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A003449" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 -------------------------------- n::even (2 n + 3) (2 n + 1) binomial(2 n, n) { (n + 1) (n + 3) binomial(n, n/2) {------------------------------------, { , (n + 4) (n + 3) (n + 2) { (2 n - 2) { 2 (n + 1) { 5/2 -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { 0 irem(n, 4) = 0 { 20 binomial(n, n/2) (n + 1) { { --------------------------- n::even { 0 irem(n, 4) = 1 { (n + 4) (n + 2) { { , { 0 irem(n, 4) = 2, { 4 binomial(n + 1, n/2 + 1/2) { { ---------------------------- n::odd { (n/2 - 3/2) { n + 3 { 2 GAMMA(n/4 + 1/2) { ----------------------------- irem(n, 4) = 3 { GAMMA(n/4 + 2) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1/2) { n , { ----------------------------- irem(n, 4) = 1, { 2 { GAMMA(n/4 + 2) { -------------------------------------- irem(n, 4) = 2 { { n (n + 4) binomial(n/2 - 1, n/4 - 1/2) { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { 4 binomial(n/2, n/4) { -------------------- irem(n, 4) = 0 { n + 4 { { 0 irem(n, 4) = 1} { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A003470" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselJ(n, -2) + BesselJ(n - 1, -2)), (-1) (n BesselY(n, -2) + BesselY(n - 1, -2))} "A003471" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" memory used=7241.8MB, alloc=240.2MB, time=33.20 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (- n/4 + 1/2) (n/2) 1/2 1/2 1/2 { 2 (-1) (2 BesselI(n/2 + 1, 2 ) - BesselI(n/2, 2 )) binomial(n, n/2) (n/2)! n::even { {{ / 1/2\(n/2 - 1/2) , { | 2 | 1/2 { 2 |- ----| binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n BesselI(n/2 + 1/2, 2 ) n::odd { \ 2 / { (- n/4 + 1/2) (n/2) 1/2 1/2 1/2 { 2 (-1) (2 BesselK(n/2 + 1, -2 ) - BesselK(n/2, -2 )) binomial(n, n/2) (n/2)! n::even { { / 1/2\(n/2 - 1/2) , { | 2 | 1/2 { 2 |- ----| binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n BesselK(n/2 + 1/2, -2 ) n::odd { \ 2 / { 1/2 (n/2) 1/2 { 2 (-2 2 ) (n/2)! BesselI(n/2 + 1/2, 2 ) n::even { { 1/2 (n/2 + 1/2) 1/2 1/2 1/2 1/2 , { (-2 2 ) (n/2 + 1/2)! 2 (2 BesselI(n/2 + 1, 2 ) - BesselI(n/2, 2 )) { ----------------------------------------------------------------------------------------- n::odd { n + 1 { 1/2 (n/2) 1/2 { 2 (-2 2 ) (n/2)! BesselK(n/2 + 1/2, -2 ) n::even { { 1/2 (n/2 + 1/2) 1/2 1/2 1/2 1/2 } { (-2 2 ) (n/2 + 1/2)! 2 (2 BesselK(n/2 + 1, -2 ) - BesselK(n/2, -2 )) { ------------------------------------------------------------------------------------------- n::odd { n + 1 "A003583" n {n binomial(2 n, n), 4 (n + 2)} "A003692" LREtools/SearchTable: "SearchTable successful" n n (-I) n! (n + 1) ((n + 1) LegendreP(n + 1, I) I + LegendreP(n, I)) (-I) n! (n + 1) ((n + 1) LegendreQ(n + 1, I) I + LegendreQ(n, I)) {- ------------------------------------------------------------------, - ------------------------------------------------------------------} n (n + 2) (n + 3) n (n + 2) (n + 3) "A003703" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n - I), (-1) GAMMA(n + I)} "A004040" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n 2 n 2 | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1) || {(-1) (27 n + 63 n + 34), (-1) (27 n + 63 n + 34) | ) |- -------------------------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (n1 + 2) (27 (n1 + 1) + 63 n1 + 97) (27 n1 + 63 n1 + 34)/| \n1 = 0 / "A004041" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 (2 n1 + 1) binomial(2 n1, n1) n1! \| {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A004148" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A004149" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A004303" n binomial(2 n, n) {2 , ----------------, n + 1} n + 1 "A004400" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) 2 n1!} / ----- n1 = 0 "A004527" memory used=7390.2MB, alloc=240.2MB, time=33.97 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) 3 2 n { 4 2 ((n/3)!) binomial(---, n/3) irem(n, 3) = 0 { 3 { { (n/3 + 2/3) 3 2 n { 9 2 ((n/3 + 2/3)!) binomial(--- + 4/3, n/3 + 2/3) { 3 {{ ------------------------------------------------------------- irem(n, 3) = 1, { 2 { (n + 2) { { (n/3 + 1/3) 3 2 n { 6 2 ((n/3 + 1/3)!) binomial(--- + 2/3, n/3 + 1/3) { 3 { ------------------------------------------------------------- irem(n, 3) = 2 { n + 1 { n 2 { n 2 { 6 2 GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) { 9 2 GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) { --------------------------------------- irem(n, 3) = 0 { --------------------------------------- irem(n, 3) = 0 { n + 1 { 2 { { (n + 2) { (n - 1) 2 { { 4 2 GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) irem(n, 3) = 1, { (n - 1) 2 } { { 6 2 GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) { (n + 1) 2 { --------------------------------------------- irem(n, 3) = 1 { 9 2 GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) { n + 1 { --------------------------------------------- irem(n, 3) = 2 { { 2 { n 2 { (n + 2) { 2 GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) irem(n, 3) = 2 "A004529" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n 3 { 2 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1/4) irem(n, 4) = 0 { { n 3 { 64 2 GAMMA(n/4 + 3/2) GAMMA(n/4 + 1) { -------------------------------------- irem(n, 4) = 1 { 3 { (n + 2) { {{ n 3 , { 16 2 GAMMA(n/4 + 5/4) GAMMA(n/4 + 3/4) { ---------------------------------------- irem(n, 4) = 2 { 2 { (n + 1) { { (n + 1) 3 { 2 2 GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { ------------------------------------------- irem(n, 4) = 3 { n { (- n/2) 3 4 { 2 binomial(n/2, n/4) ((n/4)!) irem(n, 4) = 0 { { (- n/2 + 1/2) 3 4 { 2 binomial(n/2 - 1/2, n/4 - 1/4) ((n/4 - 1/4)!) (n/2 + 1/2) irem(n, 4) = 1 { , { (- n/2 + 1) 2 3 4 { 1/4 2 n binomial(n/2 - 1, n/4 - 1/2) ((n/4 - 1/2)!) irem(n, 4) = 2 { { (- n/2 + 3/2) 3 3 4 { 1/8 2 (n - 1) binomial(n/2 - 3/2, n/4 - 3/4) ((n/4 - 3/4)!) irem(n, 4) = 3 { n 3 { 4 2 GAMMA(n/4 + 5/4) GAMMA(n/4 + 3/4) { --------------------------------------- irem(n, 4) = 0 { 2 { (n + 1) { { (n - 1) 3 { 2 2 GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { ------------------------------------------- irem(n, 4) = 1 { n , { { n 3 { 1/4 2 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1/4) irem(n, 4) = 2 { { (n + 1) 3 { 8 2 GAMMA(n/4 + 3/2) GAMMA(n/4 + 1) { ------------------------------------------- irem(n, 4) = 3 { 3 { (n + 2) { (n/2) 4 { 2 2 ((n/4)!) binomial(n/2, n/4) { ------------------------------------- irem(n, 4) = 0 { n { { (n/2 - 1/2) 4 { 2 ((n/4 - 1/4)!) binomial(n/2 - 1/2, n/4 - 1/4) irem(n, 4) = 1 { { (n/2 + 1) 4 { 8 2 ((n/4 + 1/2)!) binomial(n/2 + 1, n/4 + 1/2) } { --------------------------------------------------------- irem(n, 4) = 2 { 3 { (n + 2) { { (n/2 + 1/2) 4 { 4 2 ((n/4 + 1/4)!) binomial(n/2 + 1/2, n/4 + 1/4) { ------------------------------------------------------------- irem(n, 4) = 3 { 2 { (n + 1) "A004664" 2 {n , n!} "A005008" 2 {n , n!} "A005043" LREtools/SearchTable: "SearchTable successful" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4))} "A005095" {n, n!} "A005096" {n, n!} "A005165" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) , (-1) | ) (-(-1) (n1 + 1) n1!)|} | / | |----- | \n1 = 0 / "A005189" LREtools/SearchTable: "SearchTable successful" n! (n + 1) (LaguerreL(n + 1, -1) - 2 LaguerreL(n, -1)) {------------------------------------------------------} 2 n "A005190" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" memory used=7540.5MB, alloc=240.2MB, time=34.72 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 3 n 3 n 3 n {{ 128 binomial(3 n, ---) hypergeom([-n, - ---], [- --- + 1/2], 1/2) , n::even { 2 2 2 / 3 n 3 n 64 |3 (3 n + 8) (3 n + 4) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) \ 2 2 { { 2 3 n 3 n \ 3 n { + (-17/4 n - 17 n - 63/4) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| binomial(3 n + 3, --- + 3/2)/((5 n + 13) (n + 1)) , n::odd, { 32 2 2 / 2 { { { n / 3 n 3 n 64 (3 n + 2) |3 (3 n + 8) (3 n + 4) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) \ 2 2 2 3 n 3 n \ / / 2 3 n \ + (-17/4 n - 17 n - 63/4) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| / |(n + 1) (3 n + 1) (5 n + 13) binomial(3 n, ---)| , n::even 2 2 / / \ 2 / (6 n - 6) 3 n 3 n 64 2 (3 n - 1) hypergeom([-n, - ---], [- --- + 1/2], 1/2) 2 2 ------------------------------------------------------------------ , n::odd} 3 n n (3 n - 2) binomial(3 n - 3, --- - 3/2) 2 "A005191" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005213" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::even { {{ (n/2 + 1/2) / /5 n \ \ , { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) / /5 n \ \ { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------- n::even} { n + 1 { { (n/2 - 1/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::odd "A005218" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=7689.7MB, alloc=240.2MB, time=35.48 n {(-1) (hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)), { (n/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::even { { (n/2 + 1/2) / /5 n \ \ , { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) / /5 n \ \ { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------- n::even} { n + 1 { { (n/2 - 1/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::odd "A005258" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([-n, -n, -n], [1, -2 n], 1)} "A005259" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1)} "A005260" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, -n, -n, -n], [1, 1, 1], 1)} "A005261" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005317" n {2 , binomial(2 n, n)} "A005322" LREtools/SearchTable: "SearchTable successful" n (-1) ((13 n + 27) hypergeom([1/2, -n - 1], [1], 4) + (-15 n - 33) hypergeom([1/2, -n], [1], 4)) (n + 1) {--------------------------------------------------------------------------------------------------------} (n + 5) (n + 3) (n + 6) "A005323" memory used=7837.5MB, alloc=240.2MB, time=36.19 LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((20 n + 228 n + 796 n + 837) hypergeom([1/2, -n - 1], [1], 4) + (-21 n - 237 n - 816 n - 843) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) "A005324" LREtools/SearchTable: "SearchTable successful" n 4 3 2 {(-1) ((121 n + 2202 n + 13985 n + 36324 n + 32076) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 + (-123 n - 2202 n - 13647 n - 34140 n - 28404) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 10) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9))} "A005325" memory used=7951.0MB, alloc=240.2MB, time=36.77 LREtools/SearchTable: "SearchTable successful" n 4 3 2 {(-1) ((364 n + 7089 n + 46223 n + 120294 n + 104976) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 + (-366 n - 6981 n - 43941 n - 107850 n - 85104) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 6) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12))} "A005415" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005425" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 2 I)} "A005436" n n (n - 1) binomial(2 n, n) {4 (2 n + 7), --------------------------} 2 n - 1 "A005442" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|1/2 - ----| n!, |1/2 + ----| n!} \ 2 / \ 2 / "A005443" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|1/2 - ----| n!, |1/2 + ----| n!} \ 2 / \ 2 / "A005554" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n (n + 2) "A005558" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 { 16 { 16 binomial(n, n/2) (n + 1) { 1/2 ---------------------------------- n::even { ---------------------------- n::even { 2 2 { 2 { (n + 1) (n + 3) binomial(n, n/2) {{ (n + 2) , { } { { (4 n - 4) { 2 { 2 2 (n + 1) { 4 binomial(n + 1, n/2 + 1/2) { --------------------------------------- n::odd { ----------------------------- n::odd { 2 2 2 { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A005559" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { n { 32 binomial(n, n/2) (n + 1) { 16 (n + 2) { ----------------------------- n::even { 1/4 ------------------------------------------ n::even { 2 { 2 2 { (n + 2) (n + 4) { (n + 1) (n + 3) (n + 5) binomial(n, n/2) {{ , { } { 2 { (4 n - 4) 2 { 8 binomial(n + 1, n/2 + 1/2) (n + 1) (n + 2) { 2 (n + 1) { --------------------------------------------- n::odd { ----------------------------------------------- n::odd { 2 { 2 2 2 { (n + 3) (n + 5) { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A005560" memory used=8095.1MB, alloc=240.2MB, time=37.54 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { n 2 { 64 binomial(n, n/2) (n + 1) (n + 3) { 16 (n + 2) { ------------------------------------- n::even { 1/8 -------------------------------------------------- n::even { 2 2 { 2 2 { (n + 6) (n + 4) { (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) {{ , { } { 2 2 { (4 n - 4) 2 { 16 binomial(n + 1, n/2 + 1/2) (n + 1) (n + 2) { 2 (n + 1) (n + 3) { ----------------------------------------------- n::odd { 1/2 ------------------------------------------------ n::odd { 2 { 2 2 2 2 { (n + 3) (n + 5) (n + 7) { n (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A005561" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 2 { 128 binomial(n, n/2) (n + 1) (n + 3) { --------------------------------------- n::even { 2 2 { (n + 4) (n + 6) (n + 8) {{ , { 2 2 { 32 binomial(n + 1, n/2 + 1/2) (n + 2) (n + 1) (n + 4) { ------------------------------------------------------- n::odd { 2 2 { (n + 5) (n + 7) (n + 9) { n 2 { 16 (n + 2) (n + 4) { 1/16 --------------------------------------------------- n::even { 2 2 2 { (n + 1) (n + 5) (n + 7) (n + 9) binomial(n, n/2) { } { (4 n - 4) 2 2 { 2 (n + 1) (n + 3) { 1/4 -------------------------------------------------------- n::odd { 2 2 2 2 { n (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) "A005562" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 2 { 256 binomial(n, n/2) (n + 1) (n + 3) (n + 5) { ----------------------------------------------- n::even { 2 2 2 { (n + 6) (n + 8) (n + 10) {{ , { 2 2 2 { 64 binomial(n + 1, n/2 + 1/2) (n + 2) (n + 4) (n + 1) { -------------------------------------------------------- n::odd { 2 2 { (n + 5) (n + 7) (n + 9) (n + 11) { n 2 2 { 16 (n + 2) (n + 4) { 1/32 ------------------------------------------------------------ n::even { 2 2 2 { (n + 1) (n + 5) (n + 7) (n + 9) (n + 11) binomial(n, n/2) { } { (4 n - 4) 2 2 { 2 (n + 1) (n + 3) (n + 5) { 1/8 ---------------------------------------------------------- n::odd { 2 2 2 2 2 { n (n + 6) (n + 8) (n + 10) binomial(n - 1, n/2 - 1/2) "A005566" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 16 { 2 { -------------------------- n::even { 4 binomial(n, n/2) (n + 1) { 2 2 { --------------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {{ , { } { (4 n + 4) { 2 2 { 2 { 16 binomial(n - 1, n/2 - 1/2) n { ------------------------------------------- n::odd { --------------------------------- n::odd { 2 { 2 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { (n + 1) "A005572" LREtools/SearchTable: "SearchTable successful" n 2 (2 n hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 1) hypergeom([-1/2, -n], [1], -2)) {---------------------------------------------------------------------------------------} n + 2 "A005573" LREtools/SearchTable: "SearchTable successful" n {2 ((4 n + 1) hypergeom([-1/2, -n - 1], [1], -2) + (-4 n - 3) hypergeom([-1/2, -n], [1], -2))} "A005650" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" memory used=8238.8MB, alloc=240.2MB, time=38.31 LREtools/SearchTable: "SearchTable successful" 1/2 n {(2 ) ((2 n + 2) LaguerreL(n + 1, -n - 1/2, 1/2) + LaguerreL(n, -n + 1/2, 1/2)) n!, 1/2 n (-2 ) ((2 n + 2) LaguerreL(n + 1, -n - 1/2, 1/2) + LaguerreL(n, -n + 1/2, 1/2)) n!} "A005654" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {--------------------------, { , { } n + 1 { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A005656" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { -------------------------- n::even { 2 binomial(n, n/2) n::even binomial(2 n, n) (n - 1) { n (n + 1) binomial(n, n/2) { {------------------------, { , { binomial(n + 1, n/2 + 1/2) (n - 1) } n + 1 { (2 n - 2) { ---------------------------------- n::odd { 2 2 { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A005717" LREtools/SearchTable: "SearchTable successful" n (-1) (n + 1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------} n + 2 "A005721" LREtools/SearchTable: "SearchTable successful" {binomial(6 n, 3 n) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)} "A005723" LREtools/SearchTable: "SearchTable successful" {(6 (6 n + 5) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (3 n + 1) (3 n + 2) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n) (2 n + 1)/((3 n + 1) (3 n + 2) (5 n + 4))} "A005724" memory used=8383.2MB, alloc=240.2MB, time=39.13 LREtools/SearchTable: "SearchTable successful" {((6 n + 5) (17 n + 16) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - 2 (3 n + 2) (3 n + 1) (n + 1) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n) (2 n + 1)/((3 n + 1) (3 n + 2) (3 n + 4) (5 n + 4))} "A005725" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005726" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005768" n n n (n - 1) binomial(2 n, n) {2 , 4 (2 n + 13), --------------------------} 2 n - 1 "A005773" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n "A005774" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n + 2 "A005775" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((9 n + 50 n + 63) hypergeom([1/2, -n - 1], [1], 4) + (-9 n - 48 n - 57) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) "A005802" LREtools/SearchTable: "SearchTable successful" (2 n + 7) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-18 n - 45) hypergeom([1/2, -n, -n], [1, 1], 4) {--------------------------------------------------------------------------------------------------------} 2 (n + 2) "A005814" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A005817" memory used=8527.1MB, alloc=272.2MB, time=39.89 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 { 4 16 { 16 binomial(n, n/2) (n + 1) { ----------------------------------- n::even { ---------------------------- n::even { 2 2 2 { 2 { (n + 1) (n + 3) binomial(n, n/2) { (n + 4) (n + 2) {{ , { } { (4 n + 4) { 2 2 { 2 { 64 binomial(n - 1, n/2 - 1/2) n { ---------------------------------------------------- n::odd { --------------------------------- n::odd { 2 2 { 2 2 { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) "A005921" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|1/2 - ----| n!, |1/2 + ----| n!} \ 2 / \ 2 / "A005922" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {(n + 1) |1/2 - ----| n!, (n + 1) |1/2 + ----| n!} \ 2 / \ 2 / "A006026" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A006027" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A006040" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A006041" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1 + 1 | (n + 1) (n!) | ) ------------| | / 2| 2 |----- ((n1 + 1)!) | (n + 1) (n!) \n1 = 0 / {-------------, -----------------------------------} n n "A006077" LREtools/SearchTable: "SearchTable successful" n {3 hypergeom([- n/3, - n/3 + 2/3, 1/3 - n/3], [1, 1], 1)} "A006078" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 4 binomial(n, n/2) (n + 1) (2 n + 3) (2 n + 1) binomial(2 n, n) { { -------------------------- n::even {------------------------------------, { (2 n - 2) , { (n + 2) (n + 4) (n + 4) (n + 3) (n + 2) { 2 (n + 1) { { 1/2 -------------------------------------------- n::odd { 0 n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) } "A006079" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even binomial(2 n, n) { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {----------------, { , { } n + 1 { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A006081" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { 2 binomial(n, n/2) (n - 1) { ------------------------ n::even { -------------------------- n::even n (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) { n + 2 {2 , --------------------------, { , { } (n + 1) (n + 2) { (2 n + 2) { 4 binomial(n - 1, n/2 - 1/2) n { 2 (n - 1) { ------------------------------ n::odd { ------------------------------------------ n::odd { n + 1 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A006134" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / n1 + 1 ----- n1 = 0 "A006135" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 ----- ----- |----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) \ | \ 2 {1, ) ----------------------------------------, ) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) | ) (n2 + 1) (n2 + 2) / (n1 + 1) (n1 + 2) / | / ----- ----- |----- n1 = 0 n1 = 0 \n2 = 0 2 (15 n2 + 90 n2 + 131) /n2 - 1 \ |----- 3 3 2 2 3 2 | | \ (2 n3 + 5) (2 n3 + 3) (2 n3 + 1) binomial(2 n3, n3) (175 n3 + 1635 n3 + 5040 n3 + 5114) (2 n3 + 7) binomial(2 n3 + 2, n3 + 1)| / | ) -----------------------------------------------------------------------------------------------------------------------------------| / | / 3 3 2 2 2 | / |----- (n3 + 3) (n3 + 2) (n3 + 1) (15 (n3 + 1) + 90 n3 + 221) (15 n3 + 90 n3 + 131) | \n3 = 0 / \ | 2 2 | ((2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) binomial(2 n2 + 2, n2 + 1))|/((n1 + 1) (n1 + 2)), | | / n - 1 ----- \ (n1 + 1) (n1 + 2) ) ---------------------------------------------------} / (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) ----- n1 = 0 "A006139" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) LegendreP(n, I), (-2 I) LegendreQ(n, I)} "A006152" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! (n + 1) {---------------------------------------------------------------------} n "A006183" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | n! (n - 1) | ) ---------------------| | / (n1 + 1)! n1 (n1 - 1)| |----- | n! (n - 1) \n1 = 0 / {----------, -----------------------------------------} n n "A006184" memory used=8711.5MB, alloc=272.2MB, time=40.89 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- n2 | ----- \ | \ (-1) | \ {1, n (n - 3), ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A006198" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 2) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) ((2 n + 2) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A006199" LREtools/SearchTable: "SearchTable successful" {(n + 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1), (n + 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1)} "A006200" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((n + 1) (4 n + 14 n + 11) BesselI(n + 1/2, 1) - 2 (n + 2) BesselI(n - 1/2, 1)), n 2 2 (-1) ((n + 1) (4 n + 14 n + 11) BesselK(n + 1/2, -1) - 2 (n + 2) BesselK(n - 1/2, -1))} "A006228" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n - 1) 2 2 { 2 GAMMA(n/2 - 1/2 - RootOf(2 _Z + 2 _Z + 1)) GAMMA(n/2 + 1/2 + RootOf(2 _Z + 2 _Z + 1)) n::odd { n 2 2 { 2 GAMMA(n/2 - 1/2 RootOf(_Z + 1)) GAMMA(1/2 RootOf(_Z + 1) + n/2) n::even} { { 0 n::odd "A006231" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ 1 | {1, ) (n1 + 1) n1!, ) (n1 + 1) n1! | ) ---------|} / / | / (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A006251" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" /{ n1 \ |{ 4 | |{ ------------------------------------ n1::even| n - 1 n - 1 |{ n1 | ----- ----- |{ (n1 + 1) (n1 + 3) binomial(n1, ----) | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) \ |{ 2 | {1, ) ----------------------------------------, ) |{ |, / (n1 + 3) (n1 + 2) (n1 + 1) / |{ (2 n1 - 2) | ----- ----- |{ 2 2 | n1 = 0 n1 = 0 |{ - ---------------------------------------- n1::odd | |{ n1 | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | \{ 2 / /{ n1 \ |{ 4 binomial(n1, ----) | n - 1 |{ 2 | ----- |{ - -------------------- n1::even| \ |{ n1 + 2 | ) |{ |} / |{ n1 | ----- |{ 2 binomial(n1 + 1, ---- + 1/2) | n1 = 0 |{ 2 | |{ ------------------------------ n1::odd | \{ n1 + 3 / "A006256" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) binomial(3 n1, n1) | {(27/4) , (27/4) | ) ----------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (2 n1 + 1) (27/4) | \n1 = 0 / "A006295" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 2) {(n + 3) (n + 2) 4 (7 n + 10), ------------------------------------------------------} n + 1 "A006296" 2 n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (166 n + 1075 n + 1539) {(n + 4) (n + 3) (n + 2) 4 (7 n + 17), ---------------------------------------------------------------------------------} n + 1 "A006300" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n n | \ (2 n1 + 3) (2 n1 + 1) 3 binomial(2 n1, n1) (5 n1 + 12)| {(-4) , 12 , (-4) | ) --------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (n1 + 1) (-4) | \n1 = 0 / "A006301" memory used=8889.9MB, alloc=272.2MB, time=41.84 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 2 n {(-4) (16 n - 18 n + 1) n, 12 (50 n - 59) n, /n - 1 \ |----- n1 5 4 3 2 | n 2 | \ 3 binomial(2 n1, n1) (448 n1 + 1178 n1 - 1073 n1 - 449 n1 - 500 n1 + 36)| (-4) (16 n - 18 n + 1) n | ) ------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (-4) (16 (n1 + 1) - 18 n1 - 17) (n1 + 1) (16 n1 - 18 n1 + 1) | \n1 = 0 / "A006318" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {---------------------------------------, ---------------------------------------} n n "A006319" LREtools/SearchTable: "SearchTable successful" (n + 2) LegendreP(n + 1, 3) + (-7 n - 6) LegendreP(n, 3) (n + 2) LegendreQ(n + 1, 3) + (-7 n - 6) LegendreQ(n, 3) {--------------------------------------------------------, --------------------------------------------------------} n (n - 1) n (n - 1) "A006320" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (3 n + n + 2) LegendreP(n + 1, 3) + (-n - 3 n - 6) LegendreP(n, 3) (3 n + n + 2) LegendreQ(n + 1, 3) + (-n - 3 n - 6) LegendreQ(n, 3) {--------------------------------------------------------------------, --------------------------------------------------------------------} (n + 3) (n + 2) n (n + 3) (n + 2) n "A006321" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (7 n + 6 n + 5 n - 6) LegendreP(n + 1, 3) + (-n + 2 n + 9 n + 18) LegendreP(n, 3) {-------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) n 3 2 3 2 (7 n + 6 n + 5 n - 6) LegendreQ(n + 1, 3) + (-n + 2 n + 9 n + 18) LegendreQ(n, 3) -------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) n "A006343" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A006347" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A006348" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A006419" n (2 n + 1) binomial(2 n, n) (3 n + 4) {4 , ------------------------------------} (n + 1) (n + 2) "A006438" LREtools/SearchTable: "SearchTable successful" {n! LegendreP(n, 4), n! LegendreQ(n, 4)} "A006442" LREtools/SearchTable: "SearchTable successful" {LegendreP(n, 5), LegendreQ(n, 5)} "A006453" LREtools/SearchTable: "SearchTable successful" {n! LegendreP(n, 6), n! LegendreQ(n, 6)} "A006481" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) { { 2 (-1) binomial(n, n/2) {1, { (n/2 - 1/2) , { ---------------------------- n::even, { (-16) { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A006603" memory used=9073.7MB, alloc=272.2MB, time=42.77 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + _Z + 1, index = 1) , RootOf(_Z - _Z + _Z + 1, index = 2) , RootOf(_Z - _Z + _Z + 1, index = 3) } "A006604" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - _Z - _Z + 1, index = 1) , RootOf(2 _Z - _Z - _Z + 1, index = 2) , RootOf(2 _Z - _Z - _Z + 1, index = 3) } "A006605" LREtools/SearchTable: "SearchTable successful" (hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + 2 hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------} 13 n + 9 "A006675" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {n! n, n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A006678" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n n | \ -I (-I 2 ) n1! 2 (LegendreP(n1, 1/2 I 2 ) + 2 LegendreP(n1 + 1, 1/2 I 2 ) I) (n1 + 1)| {(-1) n!, (-1) n! | ) ---------------------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- n1 (-1) (n1 + 1)! | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n | \ -I (-I 2 ) n1! 2 (LegendreQ(n1, 1/2 I 2 ) + 2 LegendreQ(n1 + 1, 1/2 I 2 ) I) (n1 + 1)| (-1) n! | ) ---------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- n1 (-1) (n1 + 1)! | \n1 = 0 / "A006743" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n1 \\\ | | |{ 4 (4 n1 - 4) ||| | | |{ ------------------ n1::even||| | | |{ n1 ||| | | |{ binomial(n1, ----) ||| | | n1 |{ 2 ||| | | (-1) |{ ||| | | |{ (2 n1 + 2) ||| | | |{ 2 n1 (2 n1 - 1) ||| | | |{ ------------------------------------- n1::odd ||| |n - 1 | |{ n1 ||| |----- | |{ (n1 + 1) binomial(n1 + 1, ---- + 1/2) ||| n n n n | \ | \{ 2 /|| {(-1) , (-1) (n - 1), 2 (3 n + 17), (-1) (n - 1) | ) |- ----------------------------------------------------------------||, | / \ n1 (n1 - 1) /| |----- | \n1 = 0 / / / /{ n1 \\\ | | |{ 1/2 (2 n1 - 1) n1 binomial(n1, ----) n1::even||| | | n1 |{ 2 ||| | | (-1) |{ ||| |n - 1 | |{ n1 ||| |----- | |{ 2 binomial(n1 - 1, ---- - 1/2) n1 (n1 - 1) n1::odd ||| n | \ | \{ 2 /|| (-1) (n - 1) | ) |- ---------------------------------------------------------------------||} | / \ n1 (n1 - 1) /| |----- | \n1 = 0 / "A006847" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A006848" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A006849" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (n/2) (2 n + 1) 3 binomial(2 n, n) { { 2 3 binomial(n, n/2) {-----------------------------, { (n/2 - 1/2) , { ------------------------- n::even} (n + 1) (n + 2) { 48 { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A006882" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=9254.3MB, alloc=272.2MB, time=43.72 { 0 n::even { (n/2) {{ , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A006902" LREtools/SearchTable: "SearchTable successful" {n! binomial(2 n, n) hypergeom([-n], [n + 1], 1)} "A006947" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" (3 n + 11) (3 n + 8) {--------------------------------------------------------, (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) /n - 1 \ |----- 2 3 3 2 2 | | \ (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) binomial(2 n1 + 2, n1 + 1)| (3 n + 11) (3 n + 8) | ) ----------------------------------------------------------------------------------------------| | / 2 2 2 2 | |----- (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (3 n1 + 14) (3 n1 + 11) (3 n1 + 8) | \n1 = 0 / ----------------------------------------------------------------------------------------------------------------------------} (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) "A007007" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 4 binomial(n, n/2) (n + 1) { 1/2 ------------------------ n::even { -------------------------- n::even { (n + 3) binomial(n, n/2) { n + 2 {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A007043" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007060" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! BesselI(n + 1/2, 1), (-2) n! BesselK(n + 1/2, -1)} "A007107" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007122" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)), { (n/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::even { { (n/2 + 1/2) / /5 n \ \ , { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) / /5 n \ \ { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { - ------------------------------------------------------------------------------------------------------------------- n::even} { n + 1 { { (n/2 - 1/2) { -6 (-1) hypergeom([1/2, - n/2], [1], 4) n::odd "A007123" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even binomial(2 n, n) { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {----------------, { , { } n + 1 { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A007137" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n (2 n + 1) 3 binomial(2 n, n) {-----------------------------} (n + 1) (n + 2) "A007165" memory used=9428.9MB, alloc=272.2MB, time=44.63 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { / / 11 n \ \ { 32 |(n + 2) hypergeom([n + 3, - n/2 - 1], [1], -1) + |- ---- - 8| hypergeom([n + 1, - n/2], [1], -1)| {{ \ \ 2 / / { - ----------------------------------------------------------------------------------------------------- , n::even { n (17 n + 22) 2 8 (10 (n + 2) (n + 3) hypergeom([n + 4, - n/2 - 3/2], [1], -1) + (-339/4 n - 343 n - 1353/4) hypergeom([n + 2, - n/2 - 1/2], [1], -1)) - --------------------------------------------------------------------------------------------------------------------------------------- , n (n + 1) (17 n + 39) { { , { n::odd { { 2 2 (10 (n + 2) (n + 3) hypergeom([n + 4, - n/2 - 3/2], [1], -1) + (-339/4 n - 343 n - 1353/4) hypergeom([n + 2, - n/2 - 1/2], [1], -1)) - --------------------------------------------------------------------------------------------------------------------------------------- , n (n + 1) (17 n + 39) n::even / / 11 n \ \ 8 |(n + 2) hypergeom([n + 3, - n/2 - 1], [1], -1) + |- ---- - 8| hypergeom([n + 1, - n/2], [1], -1)| \ \ 2 / / } - ---------------------------------------------------------------------------------------------------- , n::odd n (17 n + 22) "A007207" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007297" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 binomial(---, n/2) { 2 { --------------------- n::even { n + 1 {{ , { (2 n - 2) 3 n { 4 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - ----------------------------------------------------- n::odd { n (n + 1) { 3 n 3 n { 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { - --------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { 3 n 3 n { binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ----------------------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) (3 n + 2) "A007317" LREtools/SearchTable: "SearchTable successful" {hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4)} "A007339" 3 {n , n!} "A007345" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {hypergeom([-1/2, -n - 1], [1], -8) - hypergeom([-1/2, -n], [1], -8), hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4)} "A007403" n n {8 (n + 2), n 2 binomial(2 n, n)} "A007440" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n | 1/2 5 5 | 1/2 n | 1/2 5 5 | (-5 ) |5 LegendreP(n + 1, ----) - 5 LegendreP(n, ----)| (-5 ) |-5 LegendreQ(n + 1, ----) + 5 LegendreQ(n, ----)| \ 5 5 / \ 5 5 / {-------------------------------------------------------------, - --------------------------------------------------------------} n + 2 n + 2 "A007477" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007489" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1!} / ----- n1 = 0 "A007526" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A007564" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 2) - 2 LegendreP(n, 2)) 2 (LegendreQ(n + 1, 2) - 2 LegendreQ(n, 2)) {--------------------------------------------, --------------------------------------------} n n "A007566" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) 2| n n | \ 2 (2 n1 + 1) | {2 n!, 2 n! | ) ----------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A007578" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007579" memory used=9643.9MB, alloc=304.2MB, time=45.72 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=9774.9MB, alloc=611.9MB, time=46.51 memory used=9777.4MB, alloc=643.9MB, time=47.16 memory used=9777.5MB, alloc=899.9MB, time=47.79 memory used=9794.0MB, alloc=1155.9MB, time=48.66 memory used=9953.5MB, alloc=1187.9MB, time=49.62 { // 5 63 4 3 2 \ {{ - 24576 ||1/8 n + -- n + 50 n + 855/2 n + 3727/2 n + 2790| hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4) { \\ 16 / 4 3 2 \ / 3 - 9/16 (n + 2) (2 n + 55 n + 574 n + 2924 n + 5904) hypergeom([1/2, - n/2, - n/2], [1, 1], 4)| binomial(n, n/2) / ((n + 2) (n + 4) / / 2 2 (n + 6) (n + 8) ) , n::even // 5 69 4 3 2 46233\ - 12288 ||1/8 n + -- n + 239/4 n + 3633/8 n + 14641/8 n + -----| hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) \\ 16 16 / 2 2 \ / - 9/16 (n + 3) (2 n + 35 n + 141) (n + 12 n + 47) hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)| binomial(n + 1, n/2 + 1/2) / ( / / 3 2 { n / (n + 1) (n + 3) (n + 5) (n + 7) (n + 9)) , n::odd, { - 48 4 | { \ / 5 69 4 3 2 46233\ |1/8 n + -- n + 239/4 n + 3633/8 n + 14641/8 n + -----| hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) \ 16 16 / 2 2 \ / 2 3 - 9/16 (n + 3) (2 n + 35 n + 141) (n + 12 n + 47) hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)| / ((n + 1) (n + 3) (n + 5) / / 2 (n + 7) (n + 9) binomial(n, n/2)) , n::even (2 n - 2) // 5 63 4 3 2 \ - 96 2 ||1/8 n + -- n + 50 n + 855/2 n + 3727/2 n + 2790| hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4) \\ 16 / 4 3 2 \ / 3 2 2 - 9/16 (n + 2) (2 n + 55 n + 574 n + 2924 n + 5904) hypergeom([1/2, - n/2, - n/2], [1, 1], 4)| / (n (n + 2) (n + 4) (n + 6) (n + 8) / / binomial(n - 1, n/2 - 1/2)) , n::odd} "A007580" memory used=10139.4MB, alloc=1187.9MB, time=50.67 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" memory used=10326.1MB, alloc=1187.9MB, time=51.79 memory used=10430.2MB, alloc=1187.9MB, time=52.57 memory used=10500.7MB, alloc=1187.9MB, time=53.29 memory used=10583.7MB, alloc=1187.9MB, time=54.01 memory used=10672.0MB, alloc=1187.9MB, time=54.74 memory used=10725.3MB, alloc=1187.9MB, time=55.41 memory used=10781.8MB, alloc=1187.9MB, time=56.08 memory used=10844.3MB, alloc=1187.9MB, time=56.80 memory used=10907.7MB, alloc=1187.9MB, time=57.48 memory used=10976.1MB, alloc=1187.9MB, time=58.22 memory used=11046.6MB, alloc=1187.9MB, time=58.96 memory used=11118.8MB, alloc=1187.9MB, time=59.71 memory used=11193.5MB, alloc=1187.9MB, time=60.49 memory used=11268.1MB, alloc=1187.9MB, time=61.26 memory used=11352.1MB, alloc=1187.9MB, time=62.07 memory used=11433.0MB, alloc=1187.9MB, time=62.86 memory used=11508.5MB, alloc=1187.9MB, time=63.65 memory used=11588.0MB, alloc=1187.9MB, time=64.43 memory used=11670.1MB, alloc=1187.9MB, time=65.24 memory used=11753.3MB, alloc=1187.9MB, time=66.05 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=11925.7MB, alloc=1187.9MB, time=67.27 {} "A007595" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 binomial(n, n/2) (2 n + 1) binomial(2 n, n) { { ------------------ n::even {--------------------------, { (2 n - 2) , { n + 2 } (n + 1) (n + 2) { 2 { { ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A007611" n {2 , n!} "A007661" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { (n/3) { { { 3 (n/3)! irem(n, 3) = 0 {{ 0 irem(n, 3) = 1, { (n/3 - 1/3) , { } { { 3 GAMMA(n/3 + 1) irem(n, 3) = 1 { 0 irem(n, 3) = 1 { (n/3 - 2/3) { { { 3 GAMMA(n/3 + 1) irem(n, 3) = 2 { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A007662" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 {{ , { , { 0 irem(n, 4) = 2 { 1/2 n (n/4 - 1/2)! binomial(n/2 - 1, n/4 - 1/2) irem(n, 4) = 2 { { { (n/2 - 3/2) { 0 irem(n, 4) = 3 { 2 GAMMA(n/4 + 1) irem(n, 4) = 3 { 0 irem(n, 4) = 0 { (n/2) { { 2 (n/4)! irem(n, 4) = 0 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1) irem(n, 4) = 1, { 0 irem(n, 4) = 1} { { { 0 irem(n, 4) = 2 { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 "A007676" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/3) { (n/3) { (n/3) { (-1) %6 irem(n, 3) = 0 { (-1) %3 irem(n, 3) = 0 { (-1) %5 irem(n, 3) = 0 { { { {{ (n/3 + 2/3) , { (n/3 + 2/3) , { (n/3 - 1/3) , { (-1) %4 irem(n, 3) = 1 { (-1) %1 irem(n, 3) = 1 { (-1) %6 irem(n, 3) = 1 { { { { (n/3 + 1/3) { (n/3 + 1/3) { (n/3 + 1/3) { (-1) %5 irem(n, 3) = 2 { (-1) %2 irem(n, 3) = 2 { (-1) %4 irem(n, 3) = 2 { (n/3) { (n/3) { (n/3) { (-1) %2 irem(n, 3) = 0 { (-1) %4 irem(n, 3) = 0 { (-1) %1 irem(n, 3) = 0 { { { { (n/3 - 1/3) , { (n/3 - 1/3) , { (n/3 - 1/3) } { (-1) %3 irem(n, 3) = 1 { (-1) %5 irem(n, 3) = 1 { (-1) %2 irem(n, 3) = 1 { { { { (n/3 + 1/3) { (n/3 - 2/3) { (n/3 - 2/3) { (-1) %1 irem(n, 3) = 2 { (-1) %6 irem(n, 3) = 2 { (-1) %3 irem(n, 3) = 2 %1 := (-12 n - 21) BesselK(n/3 + 5/6, -1/2) + 9 BesselK(n/3 - 1/6, -1/2) %2 := (-12 n - 27) BesselK(n/3 + 1/2, -1/2) + 9 BesselK(n/3 - 1/2, -1/2) %3 := (-24 n - 12) BesselK(1/6 + n/3, -1/2) + 18 BesselK(n/3 - 5/6, -1/2) %4 := (-12 n - 21) BesselI(n/3 + 5/6, 1/2) + 9 BesselI(n/3 - 1/6, 1/2) %5 := (-12 n - 27) BesselI(n/3 + 1/2, 1/2) + 9 BesselI(n/3 - 1/2, 1/2) %6 := (-24 n - 12) BesselI(1/6 + n/3, 1/2) + 18 BesselI(n/3 - 5/6, 1/2) "A007677" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/3) { (n/3) { (n/3) { (-1) %6 irem(n, 3) = 0 { (-1) %3 irem(n, 3) = 0 { (-1) %5 irem(n, 3) = 0 { { { {{ (n/3 + 2/3) , { (n/3 + 2/3) , { (n/3 - 1/3) , { (-1) %4 irem(n, 3) = 1 { (-1) %1 irem(n, 3) = 1 { (-1) %6 irem(n, 3) = 1 { { { { (n/3 + 1/3) { (n/3 + 1/3) { (n/3 + 1/3) { (-1) %5 irem(n, 3) = 2 { (-1) %2 irem(n, 3) = 2 { (-1) %4 irem(n, 3) = 2 { (n/3) { (n/3) { (n/3) { (-1) %2 irem(n, 3) = 0 { (-1) %4 irem(n, 3) = 0 { (-1) %1 irem(n, 3) = 0 { { { { (n/3 - 1/3) , { (n/3 - 1/3) , { (n/3 - 1/3) } { (-1) %3 irem(n, 3) = 1 { (-1) %5 irem(n, 3) = 1 { (-1) %2 irem(n, 3) = 1 { { { { (n/3 + 1/3) { (n/3 - 2/3) { (n/3 - 2/3) { (-1) %1 irem(n, 3) = 2 { (-1) %6 irem(n, 3) = 2 { (-1) %3 irem(n, 3) = 2 %1 := (-12 n - 21) BesselK(n/3 + 5/6, -1/2) + 9 BesselK(n/3 - 1/6, -1/2) %2 := (-12 n - 27) BesselK(n/3 + 1/2, -1/2) + 9 BesselK(n/3 - 1/2, -1/2) %3 := (-24 n - 12) BesselK(1/6 + n/3, -1/2) + 18 BesselK(n/3 - 5/6, -1/2) %4 := (-12 n - 21) BesselI(n/3 + 5/6, 1/2) + 9 BesselI(n/3 - 1/6, 1/2) %5 := (-12 n - 27) BesselI(n/3 + 1/2, 1/2) + 9 BesselI(n/3 - 1/2, 1/2) %6 := (-24 n - 12) BesselI(1/6 + n/3, 1/2) + 18 BesselI(n/3 - 5/6, 1/2) "A007808" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A007817" 3 2 n (2 n + 1) (2 n + 3) (2 n + 5) binomial(2 n, n) (29 n + 358 n + 1511 n + 2142) {4 , -------------------------------------------------------------------------------} (n + 7) (n + 6) (n + 4) (n + 3) (n + 2) (n + 1) "A007851" (2 n + 1) binomial(2 n, n) (n + 4) {1, ----------------------------------} (n + 1) (n + 2) "A007852" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) ((8 n + 4) hypergeom([1/2, -n - 1], [n + 2], -4) + (-5 n - 5) hypergeom([1/2, -n], [n + 1], -4)) binomial(2 n, n) {----------------, -----------------------------------------------------------------------------------------------------------------} n + 1 (n + 1) (3 n + 1) "A007854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(9/2) , (9/2) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (9/2) | \n1 = 0 / "A007856" memory used=12132.0MB, alloc=1187.9MB, time=68.81 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 binomial(2 n, n) n (4 (2 n + 1) hypergeom([1/2, -n - 1], [n + 2], -4) - 5 (5 n + 2) (n + 1) hypergeom([1/2, -n], [n + 1], -4)) binomial(2 n, n) {------------------, -----------------------------------------------------------------------------------------------------------------------------} n + 1 (n + 1) (3 n + 1) "A007857" binomial(2 n, n) binomial(3 n, n) {----------------, ----------------} n + 1 n + 1 "A007858" LREtools/SearchTable: "SearchTable successful" {(24 (11 n + 5) (3 n + 2) (3 n + 4) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) 3 2 / 2 + (-665 n - 1737 n - 1402 n - 336) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1) / ((n + 1) (2 n + 1) (4 n + 1) / (17 n + 24))} "A007863" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007868" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), n!} "A007901" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 7 _Z + 14 _Z - 9, index = 1) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 2) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 3) } "A007911" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) { 2 (n/2)! (- n/2 - 1) n::even { 2 (n + 1) binomial(n, n/2) (n/2)! n::even {{ , { } { (n/2 + 1/2) { (- n/2 + 1/2) { 1/2 2 (n/2 + 1/2)! n::odd { -2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A007971" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} (n - 1) n "A007985" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A007987" memory used=12338.1MB, alloc=1187.9MB, time=70.32 LREtools/SolveLRE: "Reduced the order of" (8*n^2+15*n+4)*(n+3)^2*E^3-(n+2)*(56*n^3+265*n^2+345*n+90)*E^2-3*(n+2)*(56*n^3+233*n^2+266*n+57)*E+27*( 8*n^2+31*n+27)*n^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" {- ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) 2 ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4))/n } "A008269" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 ) n! BesselJ(n + 1/2, -2 ), (-2 ) n! BesselY(n + 1/2, -2 )} "A008271" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {-(-2 ) n! (n - 1) (n + 1) ((2 n + 1) 2 BesselJ(n + 1/2, -2 ) + 2 BesselJ(n - 1/2, -2 )), 1/2 n 1/2 1/2 1/2 -(-2 ) n! (n - 1) (n + 1) ((2 n + 1) 2 BesselY(n + 1/2, -2 ) + 2 BesselY(n - 1/2, -2 ))} "A008549" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} n + 1 "A008909" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A009024" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n - 1 - I) n, (-1) GAMMA(n - 1 + I) n} "A009027" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" {((n + 1) LaguerreL(n + 1, -n - I, 1) + LaguerreL(n, -n + 1 - I, 1)) n!, ((n + 1) LaguerreL(n + 1, -n + I, 1) + LaguerreL(n, -n + 1 + I, 1)) n!} "A009053" memory used=12543.9MB, alloc=1187.9MB, time=71.81 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" RightFactors: "Input is not over Q" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A009084" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/ReduceToOrderTwo: "Only implemented for absolutely irreducible operators of order 3 or 4 whose coefficients are of type ratpoly(rational)" LREtools/SolveLRE: "Cannot reduce the operator to order two" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/ReduceToOrderTwo: "Only implemented for absolutely irreducible operators of order 3 or 4 whose coefficients are of type ratpoly(rational)" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A009097" memory used=12736.4MB, alloc=1187.9MB, time=73.36 memory used=12868.5MB, alloc=1187.9MB, time=74.61 RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -n - I, -I), n! LaguerreL(n, -n + I, -I), -n! LaguerreL(n, -n - I, I), -n! LaguerreL(n, -n + I, I)} "A009102" memory used=13073.2MB, alloc=1187.9MB, time=76.15 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ (-I) | n | \ I | {(-1) n!, (-1) n! | ) ----------------------|, (-1) n! | ) ----------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-1) (n1 + 1)!| |----- (-1) (n1 + 1)!| \n1 = 0 / \n1 = 0 / "A009108" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) n!, (-1/2 + 1/2 I) n!} "A009128" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 2 | |----- n1 2 | n n | \ (-I) (n1 + (1 - 2 I) n1 - 2 - 2 I)| n | \ I (n1 + (1 + 2 I) n1 - 2 + 2 I)| {(-1) n!, (-1) n! | ) -------------------------------------|, (-1) n! | ) ----------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-1) (n1 + 1)! | |----- (-1) (n1 + 1)! | \n1 = 0 / \n1 = 0 / "A009132" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / n1 (n1 - 1)\| {(-1) n!, (-1) n! | ) |- -----------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A009179" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- / n1 \| n n | \ / 1 \| n | \ | (-1) || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- ---------||} | / \ (n1 + 1)!/| | / \ (n1 + 1)!/| |----- | |----- | \n1 = 0 / \n1 = 0 / "A009218" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A009280" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" {((n + 1) LaguerreL(n + 1, -n - 2 - I, -1) + (n + 1 + I) LaguerreL(n, -n - 1 - I, -1)) n!, ((n + 1) LaguerreL(n + 1, -n - 2 + I, -1) + (n + 1 - I) LaguerreL(n, -1 + I - n, -1)) n!} "A009281" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 \| n n | \ | (-1) (n1 + 3 n1 + 4)|| {(-1) n!, (-1) n! | ) |- -----------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A009294" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) n!, (-1/2 + 1/2 I) n!} "A009383" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n n (-1) n! (-1/2 - 1/2 I) n! (-1/2 + 1/2 I) n! {--------, ------------------, ------------------} n n n "A009410" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=13274.3MB, alloc=1187.9MB, time=77.75 /n - 1 /n1 - 1 \ \ /n - 1 /n1 - 1 \ \ |----- |----- n2 | | |----- |----- n2 | | n n n | \ n1 | \ (-1) n2!| | n | \ n1 | \ (-1) n2! (n2 + 1)| | {(-I) , I , (-I) | ) (-1) | ) ----------| I|, (-I) | ) (-1) | ) -------------------| I|, | / | / (n2 + 1) | | | / | / (n2 + 1) | | |----- |----- I | | |----- |----- I | | \n1 = 0 \n2 = 0 / / \n1 = 0 \n2 = 0 / / / / / / /n3 - 1 \ \\\ \ | | | | |----- n4 | ||| | | | | | | \ (-I) (n4 + 3/2 - I) | ||| | | | | | (n3 + 1) | ) -------------------------------| n3!||| | | | |n2 - 1 | | / (n4 + 1) | ||| | | | |----- | |----- (-1) (n4 + 1)! (n4 + 2)| ||| | | | n2 | \ | \n4 = 0 / ||| | | | (-1) n2! | ) |- -----------------------------------------------------||| | |n - 1 |n1 - 1 | / \ (n3 + 1)! /|| | |----- |----- |----- || | n | \ n1 | \ \n3 = 0 /| | (-I) | ) (-1) | ) -----------------------------------------------------------------------------| I|, | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / / / / / /n3 - 1 \ \\\ \ | | | | |----- n4 | ||| | | | | | | \ I (n4 + 3/2 + I) | ||| | | | | | (n3 + 1) | ) -------------------------------| n3!||| | | | |n2 - 1 | | / (n4 + 1) | ||| | | | |----- | |----- (-1) (n4 + 1)! (n4 + 2)| ||| | | | n2 | \ | \n4 = 0 / ||| | | | (-1) n2! | ) |- -----------------------------------------------------||| | |n - 1 |n1 - 1 | / \ (n3 + 1)! /|| | |----- |----- |----- || | n | \ n1 | \ \n3 = 0 /| | (-I) | ) (-1) | ) -----------------------------------------------------------------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A009411" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2 \| n n | \ | (n1 - n1 + 2) n1! || {(-1) n!, (-1) n! | ) |- ---------------------||} | / \ n1 (n1 - 1) (n1 + 1)!/| |----- | \n1 = 0 / "A009430" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) n!, (-1/2 + 1/2 I) n!} "A009454" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n - I), (-1) GAMMA(n + I)} "A009457" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" {((n + 1) LaguerreL(n + 1, -n - 2 - I, -1) + (n + 1 + I) LaguerreL(n, -n - 1 - I, -1)) n!, ((n + 1) LaguerreL(n + 1, -n - 2 + I, -1) + (n + 1 - I) LaguerreL(n, -1 + I - n, -1)) n!} "A009461" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" {((n + 1) LaguerreL(n + 1, -n - I, 1) + LaguerreL(n, -n + 1 - I, 1)) n!, ((n + 1) LaguerreL(n + 1, -n + I, 1) + LaguerreL(n, -n + 1 + I, 1)) n!} "A009537" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 2 | |----- n1 2 | n n | \ (-I) (n1 + (1 - 2 I) n1 - 2 - 2 I)| n | \ I (n1 + (1 + 2 I) n1 - 2 + 2 I)| {(-1) n!, (-1) n! | ) -------------------------------------|, (-1) n! | ) ----------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-1) (n1 + 1)! | |----- (-1) (n1 + 1)! | \n1 = 0 / \n1 = 0 / "A009551" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-I) | | \ I | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A009557" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) n!, (-1/2 + 1/2 I) n!} "A009572" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ (-I) (n1 - 2 I) (n1 + 1)| n | \ I (n1 + 2 I) (n1 + 1)| {(-1) n!, (-1) n! | ) --------------------------|, (-1) n! | ) -----------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-1) (n1 + 1)! | |----- (-1) (n1 + 1)! | \n1 = 0 / \n1 = 0 / "A009573" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- / n1 \| n n | \ / (n1 - 2) (n1 + 1)\| n | \ | (-1) (n1 + 1) (n1 + 2)|| {(-1) n!, (-1) n! | ) |- -----------------||, (-1) n! | ) |- ------------------------||} | / \ (n1 + 1)! /| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A009574" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 1) (n1 + 2)|| {(-1) n!, (-1) n! | ) |- ------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A009575" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (n1 - 2) (n1 + 1) n1!\| {(-1) n!, (-1) n! | ) |- ---------------------||} | / \ n1 (n1 - 1) (n1 + 1)!/| |----- | \n1 = 0 / "A009578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (n1 - 2) (n1 + 1)\| {(-1) n!, (-1) n! | ) |- -----------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A009597" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A009628" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- / n1 \| n n | \ / 1 \| n | \ | (-1) || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- ---------||} | / \ (n1 + 1)!/| | / \ (n1 + 1)!/| |----- | |----- | \n1 = 0 / \n1 = 0 / "A009778" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) n!, (-1/2 + 1/2 I) n!} "A009940" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, 1)} "A010551" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 binomial(n, n/2) ((n/2)!) (2 n + 2) n::even {{ 2 , { } { 2 ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { ----------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { n + 1 "A010683" memory used=13461.9MB, alloc=1187.9MB, time=79.31 LREtools/SearchTable: "SearchTable successful" (2 n + 1) LegendreP(n + 1, 3) + (-2 n - 3) LegendreP(n, 3) (2 n + 1) LegendreQ(n + 1, 3) + (-2 n - 3) LegendreQ(n, 3) {----------------------------------------------------------, ----------------------------------------------------------} n (n + 2) n (n + 2) "A010736" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (3 n + 5 n + 1) LegendreP(n + 1, 3) + (-n - 3 n - 3) LegendreP(n, 3) (3 n + 5 n + 1) LegendreQ(n + 1, 3) + (-n - 3 n - 3) LegendreQ(n, 3) {----------------------------------------------------------------------, ----------------------------------------------------------------------} (n + 3) (n + 2) n (n + 3) (n + 2) n "A010752" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / | | | | | | | / 1/2\n / 1/2 \n / 1/2\n | | 5 | |5 | | 5 | | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | \ 2 / \ 2 / \ 2 / | | | | | | | | | \ / / /{ 0 irem(n2, 3) = 0\\\ | | |{ ||| | | |{ /2 n2 \ ||| | | |{ |---- - 2/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 2 GAMMA(---- + 1/2) ||| | | |{ 3 ||| n - 1 | |n1 - 1 |{ - --------------------------------- irem(n2, 3) = 1||| ----- | |----- / 1/2 \(-n2 - 1) |{ n2 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ GAMMA(---- + 1) ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 3 ||| / | | / \ 2 / |{ ||| ----- | |----- |{ /2 n2 \ ||| n1 = 0 | |n2 = 0 |{ |---- + 2/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------------- irem(n2, 3) = 2||| | | |{ n2 ||| | | |{ GAMMA(5/3 + ----) ||| \ \ \{ 3 /// \ / | | | | | | | | | | | | | | | / 1/2\n | | | 5 | | |, |1/2 - ----| | | \ 2 / | | | | | | | | | | | | | | | | | / \ / / /{ /2 n2\ \\\ | | |{ |----| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------- irem(n2, 3) = 0||| | | |{ n2 ||| n - 1 | |n1 - 1 |{ GAMMA(5/3 + ----) ||| ----- | |----- / 1/2 \(-n2 - 1) |{ 3 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 0 irem(n2, 3) = 1||| / | | / \ 2 / |{ ||| ----- | |----- |{ /2 n2 \ ||| n1 = 0 | |n2 = 0 |{ |---- - 4/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 2 GAMMA(---- + 1/2) ||| | | |{ 3 ||| | | |{ - --------------------------------- irem(n2, 3) = 2||| | | |{ n2 ||| | | |{ GAMMA(---- + 1) ||| \ \ \{ 3 /// \ | | | | | | | | / 1/2\n | | 5 | |, |1/2 - ----| | \ 2 / | | | | | | | | / / / / /{ 2 n2 n2 \\\\ |n - 1 | |n1 - 1 |{ -2 binomial(----, ----) irem(n2, 3) = 0|||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 2 n2 n2 ||||} | / | | / \ 2 / |{ binomial(---- + 4/3, ---- + 2/3) irem(n2, 3) = 1|||| |----- | |----- |{ 3 3 |||| |n1 = 0 | |n2 = 0 |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// "A010753" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / / | | | | |n - 1 | / 1/2\n / 1/2 \n / 1/2\n |----- | | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) \ 2 / \ 2 / \ 2 / | / | |----- | |n1 = 0 | | | | | \ \ / | | | / /{ 0 irem(n2, 3) = 0\\\\ | | |{ |||| | | |{ 2 n2 n2 |||| | |n1 - 1 |{ binomial(---- - 2/3, ---- - 1/3) (2 n2 + 1) |||| | |----- / 1/2 \(-n2 - 1) |{ 3 3 |||| / 1/2\n | | \ |5 | |{ ------------------------------------------- irem(n2, 3) = 1|||| | 5 | | | ) |---- + 1/2| |{ n2 + 2 ||||, |1/2 - ----| | | / \ 2 / |{ |||| \ 2 / | |----- |{ 2 n2 n2 |||| | |n2 = 0 |{ 2 binomial(---- - 4/3, ---- - 2/3) (2 n2 - 1) |||| | | |{ 3 3 |||| | | |{ - --------------------------------------------- irem(n2, 3) = 2|||| | \ \{ n2 + 1 //// | | | | \ / / /{ /2 n2\ \\\ | | |{ |----| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------- irem(n2, 3) = 0||| | | |{ n2 ||| n - 1 | |n1 - 1 |{ GAMMA(5/3 + ----) ||| ----- | |----- / 1/2 \(-n2 - 1) |{ 3 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ /2 n2 \ ||| / | | / \ 2 / |{ |---- - 2/3| ||| ----- | |----- |{ \ 3 / n2 ||| n1 = 0 | |n2 = 0 |{ 2 2 GAMMA(---- + 5/6) ||| | | |{ 3 ||| | | |{ - --------------------------------- irem(n2, 3) = 1||| | | |{ n2 ||| | | |{ GAMMA(---- + 4/3) ||| | | |{ 3 ||| | | |{ ||| \ \ \{ 0 irem(n2, 3) = 2/// \ | | | | | | | | / 1/2\n | | 5 | |, |1/2 - ----| | \ 2 / | | | | | | | | / / / / /{ /2 n2\ \\\\ | | | |{ |----| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 2 GAMMA(---- + 5/6) |||| | | | |{ 3 |||| | | | |{ - --------------------------- irem(n2, 3) = 0|||| | | | |{ n2 |||| |n - 1 | |n1 - 1 |{ GAMMA(---- + 4/3) |||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 0 irem(n2, 3) = 1||||} | / | | / \ 2 / |{ |||| |----- | |----- |{ /2 n2 \ |||| |n1 = 0 | |n2 = 0 |{ |---- + 2/3| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 GAMMA(---- + 7/6) |||| | | | |{ 3 |||| | | | |{ ------------------------------- irem(n2, 3) = 2|||| | | | |{ n2 |||| | | | |{ GAMMA(5/3 + ----) |||| \ \ \ \{ 3 //// "A010756" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / | | | | | | | / 1/2\n / 1/2 \n / 1/2\n | | 5 | |5 | | 5 | | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | \ 2 / \ 2 / \ 2 / | | | | | | | | | \ / / /{ 0 irem(n2, 3) = 0\\\ | | |{ ||| | | |{ /2 n2 \ ||| | | |{ |---- - 2/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 2 GAMMA(---- + 1/2) ||| | | |{ 3 ||| n - 1 | |n1 - 1 |{ - --------------------------------- irem(n2, 3) = 1||| ----- | |----- / 1/2 \(-n2 - 1) |{ n2 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ GAMMA(---- + 1) ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 3 ||| / | | / \ 2 / |{ ||| ----- | |----- |{ /2 n2 \ ||| n1 = 0 | |n2 = 0 |{ |---- + 2/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------------- irem(n2, 3) = 2||| | | |{ n2 ||| | | |{ GAMMA(5/3 + ----) ||| \ \ \{ 3 /// \ / | | | | | | | | | | | | | | | / 1/2\n | | | 5 | | |, |1/2 - ----| | | \ 2 / | | | | | | | | | | | | | | | | | / \ / / /{ /2 n2\ \\\ | | |{ |----| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------- irem(n2, 3) = 0||| | | |{ n2 ||| n - 1 | |n1 - 1 |{ GAMMA(5/3 + ----) ||| ----- | |----- / 1/2 \(-n2 - 1) |{ 3 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 0 irem(n2, 3) = 1||| / | | / \ 2 / |{ ||| ----- | |----- |{ /2 n2 \ ||| n1 = 0 | |n2 = 0 |{ |---- - 4/3| ||| | | |{ \ 3 / n2 ||| | | |{ 2 2 GAMMA(---- + 1/2) ||| | | |{ 3 ||| | | |{ - --------------------------------- irem(n2, 3) = 2||| | | |{ n2 ||| | | |{ GAMMA(---- + 1) ||| \ \ \{ 3 /// \ | | | | | | | | / 1/2\n | | 5 | |, |1/2 - ----| | \ 2 / | | | | | | | | / / / / /{ 2 n2 n2 \\\\ |n - 1 | |n1 - 1 |{ -2 binomial(----, ----) irem(n2, 3) = 0|||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 2 n2 n2 ||||} | / | | / \ 2 / |{ binomial(---- + 4/3, ---- + 2/3) irem(n2, 3) = 1|||| |----- | |----- |{ 3 3 |||| |n1 = 0 | |n2 = 0 |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// "A010757" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / / | | | | |n - 1 | / 1/2\n / 1/2 \n / 1/2\n |----- | | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) \ 2 / \ 2 / \ 2 / | / | |----- | |n1 = 0 | | | | | \ \ / | | | / /{ 0 irem(n2, 3) = 0\\\\ | | |{ |||| | | |{ 2 n2 n2 |||| | |n1 - 1 |{ binomial(---- - 2/3, ---- - 1/3) (2 n2 + 1) |||| | |----- / 1/2 \(-n2 - 1) |{ 3 3 |||| / 1/2\n | | \ |5 | |{ ------------------------------------------- irem(n2, 3) = 1|||| | 5 | | | ) |---- + 1/2| |{ n2 + 2 ||||, |1/2 - ----| | | / \ 2 / |{ |||| \ 2 / | |----- |{ 2 n2 n2 |||| | |n2 = 0 |{ 2 binomial(---- - 4/3, ---- - 2/3) (2 n2 - 1) |||| | | |{ 3 3 |||| | | |{ - --------------------------------------------- irem(n2, 3) = 2|||| | \ \{ n2 + 1 //// | | | | \ / / /{ /2 n2\ \\\ | | |{ |----| ||| | | |{ \ 3 / n2 ||| | | |{ 2 GAMMA(---- + 7/6) ||| | | |{ 3 ||| | | |{ ------------------------- irem(n2, 3) = 0||| | | |{ n2 ||| n - 1 | |n1 - 1 |{ GAMMA(5/3 + ----) ||| ----- | |----- / 1/2 \(-n2 - 1) |{ 3 ||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ /2 n2 \ ||| / | | / \ 2 / |{ |---- - 2/3| ||| ----- | |----- |{ \ 3 / n2 ||| n1 = 0 | |n2 = 0 |{ 2 2 GAMMA(---- + 5/6) ||| | | |{ 3 ||| | | |{ - --------------------------------- irem(n2, 3) = 1||| | | |{ n2 ||| | | |{ GAMMA(---- + 4/3) ||| | | |{ 3 ||| | | |{ ||| \ \ \{ 0 irem(n2, 3) = 2/// \ | | | | | | | | / 1/2\n | | 5 | |, |1/2 - ----| | \ 2 / | | | | | | | | / / / / /{ /2 n2\ \\\\ | | | |{ |----| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 2 GAMMA(---- + 5/6) |||| | | | |{ 3 |||| | | | |{ - --------------------------- irem(n2, 3) = 0|||| | | | |{ n2 |||| |n - 1 | |n1 - 1 |{ GAMMA(---- + 4/3) |||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 0 irem(n2, 3) = 1||||} | / | | / \ 2 / |{ |||| |----- | |----- |{ /2 n2 \ |||| |n1 = 0 | |n2 = 0 |{ |---- + 2/3| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 GAMMA(---- + 7/6) |||| | | | |{ 3 |||| | | | |{ ------------------------------- irem(n2, 3) = 2|||| | | | |{ n2 |||| | | | |{ GAMMA(5/3 + ----) |||| \ \ \ \{ 3 //// "A010763" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 1 "A010842" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A010843" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-3) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A010844" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=13647.4MB, alloc=1219.9MB, time=80.81 /n - 1 \ |----- (-n1 - 1)| n n | \ 2 | {2 n!, 2 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A010845" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1)| n n | \ 3 | {3 n!, 3 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A010849" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (10 n + 36 n + 32 n + 3) LegendreP(n + 1, 3) + (-2 n - 8 n - 12 n - 9) LegendreP(n, 3) {------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) n 3 2 3 2 (10 n + 36 n + 32 n + 3) LegendreQ(n + 1, 3) + (-2 n - 8 n - 12 n - 9) LegendreQ(n, 3) ------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) n "A011270" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A011365" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A011553" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n (2 n + 1) (-1/4) binomial(2 n, n) (8 n + 15) n {---------------------------------------------, (2 n + 1) (-1/4) binomial(2 n, n) (8 n + 15) (n + 3) (n + 2) (n + 1) /n - 1 \ |----- | | \ (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 12) | | ) --------------------------------------------------------------------------------------------------------|/((n + 3) (n + 2) (n + 1))} | / 2 (n1 + 1) | |----- (n1 + 3) (n1 + 2) (n1 + 1) (2 n1 + 3) (-1/4) binomial(2 n1 + 2, n1 + 1) (8 n1 + 23) (8 n1 + 15)| \n1 = 0 / "A011800" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A012000" LREtools/SearchTable: "SearchTable successful" n n {4 LegendreP(n, 1/2), 4 LegendreQ(n, 1/2)} "A012022" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-n1) 2 2 2\| n 2 n 2 | \ | 8 (2 n1 + 1) binomial(2 n1, n1) (n1!) || {(-2) (n!) binomial(2 n, n), (-2) (n!) binomial(2 n, n) | ) |-1/2 ---------------------------------------------||} | / | 2 || |----- \ binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A012125" LREtools/SearchTable: "SearchTable successful" n n {-4 (n + 1) (LegendreP(n, 1/2) - 2 LegendreP(n + 1, 1/2)), -4 (n + 1) (LegendreQ(n, 1/2) - 2 LegendreQ(n + 1, 1/2))} "A012244" LREtools/SearchTable: "SearchTable successful" n n {(-I) n! LegendreP(n, I), (-I) n! LegendreQ(n, I)} "A012899" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 GAMMA(n/2 - 1/2 %2) GAMMA(1/2 %2 + n/2) n::even { {{ (n + 1) , { 2 n GAMMA(n/2 + 1/2 - 1/2 %2) GAMMA(1/2 %2 + n/2 + 1/2) { -------------------------------------------------------------- n::odd { 2 { n - 2 n + 2 { n { 2 n GAMMA(n/2 - %1) GAMMA(n/2 + 1 + %1) { ---------------------------------------- n::even { 2 } { n - 2 n + 2 { { (n - 1) { 2 GAMMA(n/2 - 1/2 - %1) GAMMA(n/2 + 1/2 + %1) n::odd 2 %1 := RootOf(2 _Z + 2 _Z + 1) 2 %2 := RootOf(_Z + 1) "A012960" LREtools/SearchTable: "SearchTable successful" n I ((n + 2) hypergeom([-n - 1, 1 + 1/2 I], [2], 2) I + n hypergeom([-n, 1 + 1/2 I], [2], 2)) n! {-----------------------------------------------------------------------------------------------} n - 1 "A013069" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 2 2 { (n/2) 2 { (-1/4) ((n/2)!) binomial(n, n/2) { 4 (-4) ((n/2)!) { --------------------------------------- n::even { --------------------- n::even { n - 1 { (n - 2) n {{ , { } { (n/2 - 1/2) 2 2 { (n/2 + 1/2) 2 { n (-1/4) ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) { 4 (-4) ((n/2 + 1/2)!) { --------------------------------------------------------------- n::odd { --------------------------------- n::odd { n - 2 { 2 { (n + 1) (n - 1) "A013076" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=13888.2MB, alloc=1219.9MB, time=82.51 3 2 n 3 2 n {(1/2 RootOf(%1, index = 1) + 1/2 RootOf(%1, index = 1) - 1) n!, (1/2 RootOf(%1, index = 2) + 1/2 RootOf(%1, index = 2) - 1) n!, 3 2 n 3 2 n (1/2 RootOf(%1, index = 3) + 1/2 RootOf(%1, index = 3) - 1) n!, (1/2 RootOf(%1, index = 4) + 1/2 RootOf(%1, index = 4) - 1) n!} 4 2 %1 := _Z - 2 _Z + 2 "A013155" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 { 4 2 ((n/2)!) { 2 binomial(n, n/2) ((n/2)!) (2 n - 1) { -------------- n::even { ------------------------------------------- n::even { n {{ n - 1 , { } { { (n + 1) 2 { (-n + 1) 2 2 { 2 2 ((n/2 + 1/2)!) (2 n - 1) { 2 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { ------------------------------------ n::odd { 2 { (n - 1) (n + 1) "A013170" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 2 GAMMA(n/2 - %2) GAMMA(n/2 + 1 + %2) n::even { 2 n GAMMA(n/2 - %1) GAMMA(n/2 + 2 + %1) { { ---------------------------------------- n::even {{ (n + 1) , { 2 } { 2 n GAMMA(n/2 + 1/2 - %2) GAMMA(n/2 + 3/2 + %2) { n + 1 { ------------------------------------------------------ n::odd { { 2 { (n - 1) { n + 1 { 2 GAMMA(n/2 - 1/2 - %1) GAMMA(n/2 + 3/2 + %1) n::odd 2 %1 := RootOf(4 _Z + 8 _Z + 5) 2 %2 := RootOf(2 _Z + 2 _Z + 1) "A013277" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n {(1/2 RootOf(%1, index = 1) + 1/2 RootOf(%1, index = 1) - 1) n!, (1/2 RootOf(%1, index = 2) + 1/2 RootOf(%1, index = 2) - 1) n!, 3 2 n 3 2 n (1/2 RootOf(%1, index = 3) + 1/2 RootOf(%1, index = 3) - 1) n!, (1/2 RootOf(%1, index = 4) + 1/2 RootOf(%1, index = 4) - 1) n!} 4 2 %1 := _Z - 2 _Z + 2 "A013397" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n {(-1) n!, (-1) n! / / /{ 0 n1::even\\\ | | n1 |{ ||| |n - 1 | (-1) |{ (n1 - 1) n1 2 n1 2 ||| |----- | |{ 2 GAMMA(---- - 1/2 - RootOf(4 _Z + 8 _Z + 5)) GAMMA(---- + 3/2 + RootOf(4 _Z + 8 _Z + 5)) n1::odd ||| | \ | \{ 2 2 /|| | ) |- ------------------------------------------------------------------------------------------------------------------------------||, | / \ (n1 + 1)! /| |----- | \n1 = 0 / / / /{ n1 n1 2 n1 2 \\\ | | n1 |{ 2 GAMMA(---- - RootOf(2 _Z + 2 _Z + 1)) GAMMA(---- + 1 + RootOf(2 _Z + 2 _Z + 1)) n1::even||| |n - 1 | (-1) |{ 2 2 ||| |----- | |{ ||| n | \ | \{ 0 n1::odd /|| (-1) n! | ) |- ----------------------------------------------------------------------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A013430" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 2 n GAMMA(n/2 - %2) GAMMA(n/2 + 2 + %2) { -2 GAMMA(n/2 - %1) GAMMA(n/2 + 1 + %1) n::even { ---------------------------------------- n::even { {{ 2 , { (n + 1) } { n + 1 { 2 n GAMMA(n/2 + 1/2 - %1) GAMMA(n/2 + 3/2 + %1) { { ------------------------------------------------------ n::odd { (n - 1) { 2 { -2 GAMMA(n/2 - 1/2 - %2) GAMMA(n/2 + 3/2 + %2) n::odd { n + 1 2 %1 := RootOf(2 _Z + 2 _Z + 1) 2 %2 := RootOf(4 _Z + 8 _Z + 5) "A013436" n 2 2 n 2 2 {4 GAMMA(n - RootOf(2 _Z + 2 _Z + 1)) GAMMA(n + 1 + RootOf(2 _Z + 2 _Z + 1)), 4 GAMMA(n - 1/2 RootOf(_Z + 1)) GAMMA(1/2 RootOf(_Z + 1) + n) n} "A013459" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (n1 + 1) I n1! hypergeom([-n1, 1 + 1/2 I], [2], 2)| {(-1) n!, (-1) n! | ) ----------------------------------------------------|} | / (n1 + 1) | |----- (-1) (n1 + 1)! | \n1 = 0 / "A013463" memory used=14111.2MB, alloc=1219.9MB, time=84.07 RightFactors: "Input is not over Q" memory used=14319.0MB, alloc=1226.1MB, time=85.58 LREtools/ReduceToOrderTwo: "This step is only implemented over the rationals" RightFactors: "Input is not over Q" memory used=14524.5MB, alloc=1226.1MB, time=87.07 LREtools/ReduceToOrderTwo: "This step is only implemented over the rationals" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A013492" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ 0 n1::even\\\ | | |{ ||| | | n1 |{ / n1 \ ||| | | (-1) |{ |---- - 1/2| ||| |n - 1 | |{ \ 2 / // n1 \ \2 n1 2 ||| |----- | |{ n1 (-1/4) ||---- - 1/2|!| binomial(n1 - 1, ---- - 1/2) n1::odd ||| n n | \ | \{ \\ 2 / / 2 /|| {(-1) n!, (-1) n! | ) |- -----------------------------------------------------------------------------------------------||, | / \ (n1 + 1)! /| |----- | \n1 = 0 / / / /{ / n1 \ \\\ | | |{ |----| ||| | | |{ \ 2 / // n1 \ \2 ||| | | n1 |{ 2 (-4) ||----|!| ||| | | (-1) |{ \\ 2 / / ||| | | |{ ----------------------- n1::even||| |n - 1 | |{ n1 ||| |----- | |{ ||| n | \ | \{ 0 n1::odd /|| (-1) n! | ) |- --------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A013494" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 { 2 2 ((n/2)!) { 2 n binomial(n, n/2) ((n/2)!) { - -------------- n::even { ----------------------------------- n::even { n {{ n - 1 , { , { { (n + 1) 2 { (-n + 1) 2 2 { 2 2 n ((n/2 + 1/2)!) { -2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { ---------------------------- n::odd { 2 { (n - 1) (n + 1) { n 2 { 4 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { ---------------------------------------------------- irem(n, 4) = 0 { n - 1 { { (2 n - 2) 2 { -2 GAMMA(n/4 + 1/2) GAMMA(n/4 + 1/4) GAMMA(n/4 + 3/4) irem(n, 4) = 1 { { n 2 , { 16 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) { ------------------------------------------------------- irem(n, 4) = 2 { 2 { (n + 1) (n + 2) { { (2 n + 2) 2 { 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { - ------------------------------------------------------------ irem(n, 4) = 3 { n (n + 1) { n 2 { 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) { ---------------------------------------------------- irem(n, 4) = 0 { 2 { (n + 1) (n + 2) { { (2 n - 2) 2 { 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { - ------------------------------------------------------------ irem(n, 4) = 1, { n (n + 1) { { n 2 { 4 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { 1/16 ---------------------------------------------------- irem(n, 4) = 2 { n - 1 { { n 2 { -1/64 4 GAMMA(n/4 + 1/2) GAMMA(n/4 + 1/4) GAMMA(n/4 + 3/4) irem(n, 4) = 3 { (- n/2) 3 4 { -2 binomial(n/2, n/4) ((n/4)!) binomial(n, n/2) irem(n, 4) = 0 { { (- n/2 - 3/2) 3 4 { 2 binomial(n/2 + 3/2, n/4 + 3/4) ((n/4 + 3/4)!) binomial(n + 3, n/2 + 3/2) { ----------------------------------------------------------------------------------------- irem(n, 4) = 1 { 2 { (n + 1) (n + 2) { , { (- n/2 - 1) 3 4 { 2 binomial(n/2 + 1, n/4 + 1/2) ((n/4 + 1/2)!) binomial(n + 2, n/2 + 1) { - ----------------------------------------------------------------------------------- irem(n, 4) = 2 { n (n + 1) { { (- n/2 - 1/2) 3 4 { 2 binomial(n/2 + 1/2, n/4 + 1/4) ((n/4 + 1/4)!) binomial(n + 1, n/2 + 1/2) { ----------------------------------------------------------------------------------------- irem(n, 4) = 3 { n - 1 { (n/2) 4 3 n { 2 ((n/4)!) binomial(n, n/4) binomial(---, n/4) { 4 { - ---------------------------------------------------- irem(n, 4) = 0 { n { { (n/2 - 1/2) 4 3 n { 2 n ((n/4 - 1/4)!) binomial(n - 1, n/4 - 1/4) binomial(--- - 3/4, n/4 - 1/4) { 4 { ---------------------------------------------------------------------------------------- irem(n, 4) = 1 { n - 1 } { { (n/2 - 1) 4 3 n { 2 ((n/4 - 1/2)!) (-n + 1) binomial(n - 2, n/4 - 1/2) binomial(--- - 3/2, n/4 - 1/2) irem(n, 4) = 2 { 4 { { (n/2 + 1/2) 4 3 n { 2 ((n/4 + 1/4)!) binomial(n + 1, n/4 + 1/4) binomial(--- + 3/4, n/4 + 1/4) { 4 { -------------------------------------------------------------------------------------- irem(n, 4) = 3 { 2 { (n + 1) "A013498" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 (n + 3) { 2 { ------------------ n::even { binomial(n, n/2) (n + 2 n - 4) { n binomial(n, n/2) {{ 1/2 ------------------------------- n::even, { } { n - 1 { (2 n + 2) 2 { { 2 2 (n + 2 n - 4) { binomial(n - 1, n/2 - 1/2) (n + 3) n::odd { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A013981" n (2 n + 1) binomial(2 n, n) {2 , --------------------------, n + 3} n + 1 "A013989" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) (n + 1) HermiteH(n, 1/2 I 2 )} "A013999" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A014137" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A014138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A014140" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {1, (3 n + 4) | ) ----------------------------------------|, 3 n + 4} | / (n1 + 1) (n1 + 2) (3 n1 + 7) (3 n1 + 4) | |----- | \n1 = 0 / "A014143" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {1, (3 n + 7) | ) ---------------------------------------------------|, 3 n + 7} | / (n1 + 3) (n1 + 2) (n1 + 1) (3 n1 + 10) (3 n1 + 7) | |----- | \n1 = 0 / "A014144" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1!| {1, n, n | ) ---|} | / n1 | |----- | \n1 = 0 / "A014145" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {1, (n + 1) | ) n1!|, n + 1} | / | |----- | \n1 = 0 / "A014151" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) | 2 {(n + 2) (n + 3), (9 n + 33 n + 26) | ) -----------------------------------------------------------------|, 9 n + 33 n + 26} | / 2 2 | |----- (n1 + 1) (n1 + 2) (9 (n1 + 1) + 33 n1 + 59) (9 n1 + 33 n1 + 26)| \n1 = 0 / "A014178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A014180" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A014288" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1!} / ----- n1 = 0 "A014300" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=14760.4MB, alloc=1226.1MB, time=88.77 /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 5)| {(-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) | \n1 = 0 / "A014301" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 4)| {(-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) | \n1 = 0 / "A014314" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 1/2 ------------------------ n::even n { binomial(n, n/2) (2 n - 2) n::even { (n + 1) binomial(n, n/2) {2 , { , { } { binomial(n + 1, n/2 + 1/2) (n - 1) n::odd { (2 n - 2) { 2 (n - 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A014316" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ (2 n1 + 1) binomial(2 n1, n1) (5 n1 + 4) | 2 {(9 n + 6 n - 1) | ) ----------------------------------------------------|, 6 n + 5, 9 n + 6 n - 1} | / 2 2 | |----- (n1 + 1) (9 (n1 + 1) + 6 n1 + 5) (9 n1 + 6 n1 - 1)| \n1 = 0 / "A014318" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 1) binomial(2 n1, n1)| {2 , 2 | ) ----------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A014330" LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1) {----------------------------------------------------------------} n + 1 "A014333" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A014431" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (LegendreP(n, 1/7 I 7 ) + 7 LegendreP(n + 1, 1/7 I 7 ) I) {-----------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (LegendreQ(n, 1/7 I 7 ) + 7 LegendreQ(n + 1, 1/7 I 7 ) I) -----------------------------------------------------------------------------------} n "A014432" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 11 ) 11 (LegendreP(n, 1/11 I 11 ) + 11 LegendreP(n + 1, 1/11 I 11 ) I) {------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 11 ) 11 (LegendreQ(n, 1/11 I 11 ) + 11 LegendreQ(n + 1, 1/11 I 11 ) I) ------------------------------------------------------------------------------------------} n "A014433" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 (LegendreP(n, 1/15 I 15 ) + 15 LegendreP(n + 1, 1/15 I 15 ) I) {------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 (LegendreQ(n, 1/15 I 15 ) + 15 LegendreQ(n + 1, 1/15 I 15 ) I) ------------------------------------------------------------------------------------------} n "A014434" LREtools/SearchTable: "SearchTable successful" memory used=14974.2MB, alloc=1226.1MB, time=90.40 1/2 n 1/2 1/2 1/2 1/2 -I (-I 19 ) 19 (LegendreP(n, 1/19 I 19 ) + 19 LegendreP(n + 1, 1/19 I 19 ) I) {------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 19 ) 19 (LegendreQ(n, 1/19 I 19 ) + 19 LegendreQ(n + 1, 1/19 I 19 ) I) ------------------------------------------------------------------------------------------} n "A014435" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 23 ) 23 (LegendreP(n, 1/23 I 23 ) + 23 LegendreP(n + 1, 1/23 I 23 ) I) {------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 23 ) 23 (LegendreQ(n, 1/23 I 23 ) + 23 LegendreQ(n + 1, 1/23 I 23 ) I) ------------------------------------------------------------------------------------------} n "A014495" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A014508" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 + 1) | {1, (n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A014531" LREtools/SearchTable: "SearchTable successful" n (-1) ((4 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 15) hypergeom([1/2, -n], [1], 4)) (n + 1) {-----------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A014532" LREtools/SearchTable: "SearchTable successful" n (-1) ((13 n + 27) hypergeom([1/2, -n - 1], [1], 4) + (-15 n - 33) hypergeom([1/2, -n], [1], 4)) (n + 1) {--------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) "A014533" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((20 n + 228 n + 796 n + 837) hypergeom([1/2, -n - 1], [1], 4) + (-21 n - 237 n - 816 n - 843) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 7) (n + 8) "A014775" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A015735" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A016036" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A016065" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 2 {1, ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A018191" LREtools/SearchTable: "SearchTable successful" n {(-I) (HermiteH(n + 1, 1/2 I) + HermiteH(n, 1/2 I) I)} "A018217" n {4 (n + 1), (2 n + 1) binomial(2 n, n)} "A018218" n {4 (n + 1), (2 n + 1) binomial(2 n, n)} "A018224" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { -------------------------- n::even { 2 2 { 2 { (n + 1) binomial(n, n/2) { 4 binomial(n, n/2) n::even {{ , { } { (4 n - 4) { 2 { 4 2 { binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------ n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) "A018901" memory used=15193.0MB, alloc=1226.1MB, time=92.04 memory used=15383.6MB, alloc=1258.1MB, time=93.51 memory used=15598.4MB, alloc=1270.8MB, time=94.82 memory used=15789.6MB, alloc=1271.5MB, time=96.03 memory used=15987.5MB, alloc=1271.5MB, time=97.31 memory used=16179.5MB, alloc=1271.5MB, time=98.51 memory used=16377.1MB, alloc=1271.5MB, time=99.77 memory used=16580.7MB, alloc=1271.5MB, time=101.04 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A018934" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (-1) (n1 - 1) n1 (n1 + 1) | n! (n - n - 1) | ) ---------------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) - n1 - 2) (n1 - n1 - 1)| n! (n - n - 1) \n1 = 0 / {---------------, ----------------------------------------------------------------------} (n - 1) n (n - 1) n "A019460" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A019461" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A019462" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A019463" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A019464" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A019465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A019466" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A019497" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A019514" 3 {1, (n!) } "A019577" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-I) n! LegendreP(n + 1, I), (-I) n! LegendreQ(n + 1, I), n!} "A019581" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(n + 1) n!, (-I) n! (n + 1) LegendreP(n + 1, I), (-I) n! (n + 1) LegendreQ(n + 1, I)} "A020543" {1, n!} "A020549" 2 {1, (n!) } "A022558" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n 2 n 2 | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1) || {(-1) (27 n + 9 n - 2), (-1) (27 n + 9 n - 2) | ) |- ------------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (27 (n1 + 1) + 9 n1 + 7) (27 n1 + 9 n1 - 2)/| \n1 = 0 / "A022916" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n { 3 (n + 2) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { ------------------------------------------ irem(n, 3) = 0 { 2 { (n + 1) GAMMA(5/3 + n/3) { { (n - 1) { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) {{ ------------------------------------------ irem(n, 3) = 1, { 2 { GAMMA(n/3 + 4/3) { { n { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { ------------------------------------ irem(n, 3) = 2 { 2 { GAMMA(n/3 + 1) { 2 n { 9 binomial(---, n/3) binomial(n, n/3) irem(n, 3) = 0 { 3 { { 2 n { (n + 2) binomial(--- + 4/3, n/3 + 2/3) binomial(n + 2, n/3 + 2/3) { 3 , { ----------------------------------------------------------------- irem(n, 3) = 1 { n + 1 { { 2 n { 3 binomial(--- + 2/3, n/3 + 1/3) binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 { 3 { n { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { ------------------------------------ irem(n, 3) = 0 { 2 { GAMMA(n/3 + 4/3) { { (n - 1) { 9 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { -------------------------------------------- irem(n, 3) = 1} { 2 { GAMMA(n/3 + 1) { { (n + 1) { 3 (n + 2) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { ------------------------------------------------ irem(n, 3) = 2 { 2 { (n + 1) GAMMA(5/3 + n/3) "A023053" memory used=16836.0MB, alloc=1271.5MB, time=102.91 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n / 1/2\n | 5 | |5 | | 5 | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| \ 2 / \ 2 / \ 2 / / / / / 1/2 \(-n2 - 1) \\\ |n - 1 | |n1 - 1 |5 | ||| |----- | |----- |---- + 1/2| (3 n2 + 4) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2)||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 2 / ||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) -------------------------------------------------------------------------|||} | / | | / (n2 + 2) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A023421" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A023422" memory used=17092.4MB, alloc=1271.5MB, time=104.50 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A023425" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A023426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A023427" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A023428" memory used=17371.7MB, alloc=1271.5MB, time=106.13 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A023431" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A023432" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A023433" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A024167" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {(n + 1) n!, (n + 1) n! | ) -------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A024168" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {(n + 1) n!, (n + 1) n! | ) -------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A024187" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / "A024188" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) -------------------|} | / (n1 + 4) (n1 + 1)! | |----- | \n1 = 0 / "A024189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) -------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A024199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) binomial(2 n1, n1) n1! || {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A024200" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) binomial(2 n1, n1) n1! || {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A024216" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 4/3)\| {3 GAMMA(n + 4/3), 3 GAMMA(n + 4/3) | ) |1/3 ---------------||} | / \ GAMMA(n1 + 7/3)/| |----- | \n1 = 0 / "A024217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) GAMMA(n1 + 4/3)|| {3 GAMMA(n + 4/3), 3 GAMMA(n + 4/3) | ) |1/3 ----------------------||} | / \ GAMMA(n1 + 7/3) /| |----- | \n1 = 0 / "A024382" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 5/4)\| {4 GAMMA(n + 5/4), 4 GAMMA(n + 5/4) | ) |1/4 ---------------||} | / \ GAMMA(n1 + 9/4)/| |----- | \n1 = 0 / "A024395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 5/3)\| {3 GAMMA(n + 5/3), 3 GAMMA(n + 5/3) | ) |1/3 ---------------||} | / \ GAMMA(n1 + 8/3)/| |----- | \n1 = 0 / "A024396" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) GAMMA(n1 + 5/3)|| {3 GAMMA(n + 5/3), 3 GAMMA(n + 5/3) | ) |1/3 ----------------------||} | / \ GAMMA(n1 + 8/3) /| |----- | \n1 = 0 / "A024419" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) n1!| { 0 n::even {(n + 1) (1/2) n!, (n + 1) (1/2) n! | ) ----------------------|, { , | / (n1 + 2) (n1 + 1)! | { (-2 n + 2) 2 2 2 |----- | { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd \n1 = 0 / { 2 { ((n/2)!) n::even} { { 0 n::odd "A024718" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / n1 + 1 ----- n1 = 0 "A024719" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) {1, ) ----------------------------------------} / (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A024720" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1) {1, ) ---------------------------------------------------} / (n1 + 1) (3 n1 + 2) (3 n1 + 1) ----- n1 = 0 "A024721" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /3125\n1 ----- |----| GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) \ \256 / {1, ) ------------------------------------------------------------------------} / GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 5/4) ----- n1 = 0 "A024997" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((4 n + 15 n + 15) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A024998" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((2 n + 6 n + 1) hypergeom([1/2, -n - 1], [1], 4) + (10 n + 16 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------------} n (n + 2) "A025012" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025014" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A025163" memory used=17640.5MB, alloc=1271.5MB, time=108.04 LREtools/SearchTable: "SearchTable successful" /3 n \ /3 n \ |--- + 1/2| / 1/2 1/2 \ |--- + 1/2| / 1/2 1/2 \ \ 2 / | 1/2 2 2 | \ 2 / | 1/2 2 2 | {-2 (n + 1) |-2 LegendreP(n + 1, ----) + LegendreP(n, ----)|, -2 (n + 1) |-2 LegendreQ(n + 1, ----) + LegendreQ(n, ----)| \ 2 2 / \ 2 2 / } "A025164" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1), (-1) BesselK(n + 1/2, -1)} "A025165" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n | 2 | |2 | {|- ----| HermiteH(n, 1), |----| HermiteH(n, 1)} \ 2 / \ 2 / "A025166" LREtools/SearchTable: "SearchTable successful" n {2 n! LaguerreL(n, 1/2)} "A025167" LREtools/SearchTable: "SearchTable successful" n {2 n! LaguerreL(n, -1/2)} "A025168" LREtools/SearchTable: "SearchTable successful" n 2 ((-2 n - 3) LaguerreL(n, -1/2) + (2 n + 2) LaguerreL(n + 1, -1/2)) n! {------------------------------------------------------------------------} n "A025175" LREtools/SearchTable: "SearchTable successful" n n 4 (n + 1) (LegendreP(n + 1, 1/2) - 2 LegendreP(n, 1/2)) 4 (n + 1) (LegendreQ(n + 1, 1/2) - 2 LegendreQ(n, 1/2)) {--------------------------------------------------------, --------------------------------------------------------} n + 2 n + 2 "A025178" LREtools/SearchTable: "SearchTable successful" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4))} "A025179" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((4 n + 15 n + 15) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A025180" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((n + 3 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 15 n - 21) hypergeom([1/2, -n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A025181" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((8 n + 43 n + 72) hypergeom([1/2, -n - 1], [1], 4) + (-12 n - 81 n - 168) hypergeom([1/2, -n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A025182" LREtools/SearchTable: "SearchTable successful" memory used=17887.4MB, alloc=1271.5MB, time=109.86 n 3 2 3 2 (-1) ((13 n + 129 n + 464 n + 540) hypergeom([1/2, -n - 1], [1], 4) + (-15 n - 159 n - 624 n - 804) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------------} (n + 8) (n + 7) (n + 6) (n + 4) "A025191" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {3 , -----------------------------------------------------------------------------------------} n "A025227" LREtools/SearchTable: "SearchTable successful" n n (-2 I) ((3 n + 1) LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (-2 I) ((3 n + 1) LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- -------------------------------------------------------------------, - -------------------------------------------------------------------} (n - 1) n (n - 1) n "A025228" LREtools/SearchTable: "SearchTable successful" n 2 ((2 n - 1) hypergeom([-1/2, -n - 1], [1], -2) - 2 n hypergeom([-1/2, -n], [1], -2)) {--------------------------------------------------------------------------------------} n "A025229" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (LegendreP(n, 1/2 I 2 ) + 2 LegendreP(n + 1, 1/2 I 2 ) I) {-------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (LegendreQ(n, 1/2 I 2 ) + 2 LegendreQ(n + 1, 1/2 I 2 ) I) -------------------------------------------------------------------------------------} n "A025230" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n - 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n + 1) hypergeom([-1/2, -n], [1], -1)) {---------------------------------------------------------------------------------------------} n "A025231" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 2) - 2 LegendreP(n, 2)) 2 (LegendreQ(n + 1, 2) - 2 LegendreQ(n, 2)) {--------------------------------------------, --------------------------------------------} n n "A025232" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n 1/2 | 1/2 3 7 3 7 | (2 7 ) 7 |-7 LegendreP(n + 1, ------) + 3 LegendreP(n, ------)| \ 7 7 / {- ------------------------------------------------------------------------, n / 1/2 1/2 \ 1/2 n 1/2 | 1/2 3 7 3 7 | (2 7 ) 7 |-7 LegendreQ(n + 1, ------) + 3 LegendreQ(n, ------)| \ 7 7 / - ------------------------------------------------------------------------} n "A025235" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 7 ) (7 LegendreP(n + 1, 1/7 I 7 ) I - 7 LegendreP(n, 1/7 I 7 )) {-----------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 7 ) (7 LegendreQ(n + 1, 1/7 I 7 ) I - 7 LegendreQ(n, 1/7 I 7 )) -----------------------------------------------------------------------------} n + 2 "A025237" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 11 ) (11 LegendreQ(n + 1, 1/11 I 11 ) I - 11 LegendreQ(n, 1/11 I 11 )) {------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 11 ) (-11 LegendreP(n + 1, 1/11 I 11 ) I + 11 LegendreP(n, 1/11 I 11 )) - -------------------------------------------------------------------------------------} n + 2 "A025238" LREtools/SearchTable: "SearchTable successful" (4 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n "A025239" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (LegendreP(n, 1/2 I 2 ) + 2 LegendreP(n + 1, 1/2 I 2 ) I) {-------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (LegendreQ(n, 1/2 I 2 ) + 2 LegendreQ(n + 1, 1/2 I 2 ) I) -------------------------------------------------------------------------------------} n "A025240" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {---------------------------------------, ---------------------------------------} n n "A025241" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025242" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025243" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025244" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025245" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025246" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025247" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025248" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025249" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025250" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025251" memory used=18133.1MB, alloc=1271.5MB, time=111.71 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025252" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025253" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025254" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025255" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025256" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025257" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025258" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025259" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025260" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025261" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025262" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025263" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025264" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025265" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025266" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025267" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025268" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025269" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025270" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025271" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025272" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025273" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025274" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025275" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025276" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025277" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025278" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025279" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A025565" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n "A025566" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((2 n - 1) (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-2 n + 9 n - 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------------} (n - 1) n "A025567" LREtools/SearchTable: "SearchTable successful" n (-1) ((3 n + 5) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------} (n + 4) (n + 2) "A025568" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((18 n + 115 n + 144) hypergeom([1/2, -n - 1], [1], 4) + (-18 n - 105 n - 96) hypergeom([1/2, -n], [1], 4)) (n + 1) {---------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A025577" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=18374.6MB, alloc=1271.5MB, time=113.56 n - 1 ----- \ n1 {1, ) (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - hypergeom([1/2, -n1], [1], 4))} / ----- n1 = 0 "A025578" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {3 , ----------------------------------------------------------------------------------------} n "A025756" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- n1 ----- n1 \ 9 GAMMA(n1 + 1/3) \ 9 GAMMA(n1 + 2/3) {1, ) -------------------, ) -------------------} / GAMMA(n1 + 2) / GAMMA(n1 + 2) ----- ----- n1 = 0 n1 = 0 "A025757" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| n n | \ | 5 (-1) 20 binomial(2 n1, n1)|| n | \ | 5 (-1) 80 GAMMA(n1 + 1/4)|| {(-1/5) , (-1/5) | ) |- --------------------------------||, (-1/5) | ) |- -----------------------------||, | / \ n1 + 1 /| | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ |----- / n1 n1 \| n | \ | 5 (-1) 80 GAMMA(n1 + 3/4)|| (-1/5) | ) |- -----------------------------||} | / \ GAMMA(n1 + 2) /| |----- | \n1 = 0 / "A025758" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| |----- / n1 \| n n | \ |41 1025 GAMMA(n1 + 1/5)|| n | \ |41 1025 GAMMA(n1 + 2/5)|| n | \ |41 1025 GAMMA(n1 + 3/5)|| {(1/41) , (1/41) | ) |-------------------------||, (1/41) | ) |-------------------------||, (1/41) | ) |-------------------------||, | / \ GAMMA(n1 + 2) /| | / \ GAMMA(n1 + 2) /| | / \ GAMMA(n1 + 2) /| |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ |41 1025 GAMMA(n1 + 4/5)|| (1/41) | ) |-------------------------||} | / \ GAMMA(n1 + 2) /| |----- | \n1 = 0 / "A025759" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| /-1 \n /-1 \n | \ | 434 (-1) 3906 binomial(2 n1, n1)|| /-1 \n | \ | 434 (-1) 15624 GAMMA(1/6 + n1)|| {|---| , |---| | ) |- ------------------------------------||, |---| | ) |- ----------------------------------||, \434/ \434/ | / \ n1 + 1 /| \434/ | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| /-1 \n | \ | 434 (-1) 15624 GAMMA(n1 + 1/3)|| /-1 \n | \ | 434 (-1) 15624 GAMMA(n1 + 2/3)|| |---| | ) |- ----------------------------------||, |---| | ) |- ----------------------------------||, \434/ | / \ GAMMA(n1 + 2) /| \434/ | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ |----- / n1 n1 \| /-1 \n | \ | 434 (-1) 15624 GAMMA(n1 + 5/6)|| |---| | ) |- ----------------------------------||} \434/ | / \ GAMMA(n1 + 2) /| |----- | \n1 = 0 / "A025760" memory used=18629.0MB, alloc=1271.5MB, time=115.39 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| n n | \ |5713 279937 GAMMA(n1 + 1/7)|| n | \ |5713 279937 GAMMA(n1 + 2/7)|| {(1/5713) , (1/5713) | ) |-----------------------------||, (1/5713) | ) |-----------------------------||, | / \ GAMMA(n1 + 2) /| | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| n | \ |5713 279937 GAMMA(n1 + 3/7)|| n | \ |5713 279937 GAMMA(n1 + 4/7)|| (1/5713) | ) |-----------------------------||, (1/5713) | ) |-----------------------------||, | / \ GAMMA(n1 + 2) /| | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| n | \ |5713 279937 GAMMA(n1 + 5/7)|| n | \ |5713 279937 GAMMA(n1 + 6/7)|| (1/5713) | ) |-----------------------------||, (1/5713) | ) |-----------------------------||} | / \ GAMMA(n1 + 2) /| | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A025761" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| / -1 \n / -1 \n | \ | 90075 (-1) 1441200 binomial(2 n1, n1)|| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 1/4)|| {|-----| , |-----| | ) |- -----------------------------------------||, |-----| | ) |- --------------------------------------||, \90075/ \90075/ | / \ n1 + 1 /| \90075/ | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 1/8)|| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 3/4)|| |-----| | ) |- --------------------------------------||, |-----| | ) |- --------------------------------------||, \90075/ | / \ GAMMA(n1 + 2) /| \90075/ | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 3/8)|| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 5/8)|| |-----| | ) |- --------------------------------------||, |-----| | ) |- --------------------------------------||, \90075/ | / \ GAMMA(n1 + 2) /| \90075/ | / \ GAMMA(n1 + 2) /| |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ |----- / n1 n1 \| / -1 \n | \ | 90075 (-1) 5764800 GAMMA(n1 + 7/8)|| |-----| | ) |- --------------------------------------||} \90075/ | / \ GAMMA(n1 + 2) /| |----- | \n1 = 0 / "A026000" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([-n, 2 n + 1], [n + 1], -1)} "A026002" memory used=18874.8MB, alloc=1271.5MB, time=117.17 LREtools/SearchTable: "SearchTable successful" (2 n + 1) LegendreP(n + 1, 3) + LegendreP(n, 3) (2 n + 1) LegendreQ(n + 1, 3) + LegendreQ(n, 3) {-----------------------------------------------, -----------------------------------------------} n + 2 n + 2 "A026003" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n + 3) LegendreP(n/2 + 3/2, 3) + (-5 n - 11) LegendreP(n/2 + 1/2, 3) { --------------------------------------------------------------------- n::even {{ n + 1 , { { 2 LegendreP(n/2, 3) n::odd { (n + 3) LegendreQ(n/2 + 3/2, 3) + (-5 n - 11) LegendreQ(n/2 + 1/2, 3) { --------------------------------------------------------------------- n::even { n + 1 , { { 2 LegendreQ(n/2, 3) n::odd { 2 LegendreP(n/2, 3) n::even { { (n + 3) LegendreP(n/2 + 3/2, 3) + (-5 n - 11) LegendreP(n/2 + 1/2, 3) , { --------------------------------------------------------------------- n::odd { n + 1 { 2 LegendreQ(n/2, 3) n::even { { (n + 3) LegendreQ(n/2 + 3/2, 3) + (-5 n - 11) LegendreQ(n/2 + 1/2, 3) } { --------------------------------------------------------------------- n::odd { n + 1 "A026008" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 16 binomial(n, n/2) (n + 1) { 3/2 -------------------------------- n::even { --------------------------- n::even { (n + 3) (n + 5) binomial(n, n/2) { (n + 4) (n + 2) {{ , { } { (2 n - 2) { 12 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 2 (n + 1) { ------------------------------------- n::odd { -------------------------------------------- n::odd { (n + 5) (n + 3) { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A026010" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (3 n + 5) { 2 binomial(n, n/2) (3 n + 2) { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (3 n + 5) { 2 2 (3 n + 2) { ------------------------------------ n::odd { ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A026021" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { n 2 { 2 binomial(n, n/2) (3 n + 6 n + 8) { 2 4 (n + 2 n + 5) { ----------------------------------- n::even { ------------------------------------------------ n::even { (n + 2) (n + 6) (n + 4) { (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) {{ , { } { 2 { (2 n - 2) 2 { 4 binomial(n + 1, n/2 + 1/2) (n + 2 n + 5) { 2 (3 n + 6 n + 8) { ------------------------------------------- n::odd { ---------------------------------------------------- n::odd { (n + 3) (n + 5) (n + 7) { n (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A026023" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 4 binomial(n, n/2) (n + 1) { 1/2 ------------------------ n::even { -------------------------- n::even { (n + 3) binomial(n, n/2) { n + 4 {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------ n::odd { n + 3 { n (n + 4) binomial(n - 1, n/2 - 1/2) "A026034" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n + 2 n + 21) { 5/2 ------------------------------------------------ n::even { (n + 3) (n + 5) (n + 7) (n + 9) binomial(n, n/2) {{ , { (2 n - 2) 2 { 4 2 (n + 1) (n + 2 n + 12) { ------------------------------------------------------------ n::odd { n (n + 2) (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) { 2 { 32 binomial(n, n/2) (n + 2 n + 12) (n + 1) { ------------------------------------------- n::even { (n + 2) (n + 4) (n + 6) (n + 8) { } { 2 { 20 binomial(n + 1, n/2 + 1/2) (n + 1) (n + 2 n + 21) { ----------------------------------------------------- n::odd { (n + 3) (n + 5) (n + 7) (n + 9) "A026070" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((2 n + 12 n + 27 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 66 n - 267 n - 351) hypergeom([1/2, -n], [1], 4)) (n + 1) {---------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 5) (n + 3) (n + 6) "A026071" memory used=19111.0MB, alloc=1271.5MB, time=118.90 LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 {(-1) ((8 n + 135 n + 958 n + 3459 n + 6240 n + 4644) hypergeom([1/2, -n - 1], [1], 4) 5 4 3 2 + (-12 n - 237 n - 1986 n - 8493 n - 18144 n - 15516) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 8) (n + 7) (n + 4) (n + 5) (n + 3) (n + 6) )} "A026079" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (-1) (hypergeom([1/2, -n - 1], [1], 4) + (4 n + 5) hypergeom([1/2, -n], [1], 4)) {1, ---------------------------------------------------------------------------------} n + 2 "A026080" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((n + 1) (8 n - 7) hypergeom([1/2, -n - 1], [1], 4) + (28 n - 15 n - 7) hypergeom([1/2, -n], [1], 4)) {3 , ------------------------------------------------------------------------------------------------------------} n (n - 1) "A026083" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((2 n + 10 n + 15) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 36 n - 57) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A026084" LREtools/SearchTable: "SearchTable successful" n 4 3 2 2 (-1) ((8 n + 92 n + 394 n + 721 n + 483) hypergeom([1/2, -n - 1], [1], 4) + 3 (4 n + 24 n + 41) (n + 1) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A026085" LREtools/SearchTable: "SearchTable successful" n 4 3 2 4 3 2 {(-1) ((n + 12 n + 59 n + 132 n + 117) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 45 n - 258 n - 648 n - 603) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A026086" memory used=19327.4MB, alloc=1303.5MB, time=120.65 LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 {(-1) ((8 n + 140 n + 1030 n + 3745 n + 6654 n + 4527) hypergeom([1/2, -n - 1], [1], 4) 3 2 + 9 (4 n + 44 n + 173 n + 223) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 7) (n + 6) (n + 5) (n + 4) (n + 3))} "A026087" LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {(-1) ((2 n + 30 n + 185 n + 504 n + 425 n - 426 n - 828) hypergeom([1/2, -n - 1], [1], 4) 6 5 4 3 2 + (-6 n - 144 n - 1491 n - 8118 n - 24063 n - 36234 n - 20988) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 2) (n + 8) (n + 7) (n + 4) (n + 5) (n + 3) (n + 6))} "A026095" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((2 n + 6 n + 1) hypergeom([1/2, -n - 1], [1], 4) + (10 n + 16 n + 1) hypergeom([1/2, -n], [1], 4)) {1, -----------------------------------------------------------------------------------------------------------} n (n + 2) "A026096" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 2 n (-1) ((10 n - 4 n - 5) hypergeom([1/2, -n - 1], [1], 4) + (26 n - 18 n - 5) hypergeom([1/2, -n], [1], 4)) {3 , ------------------------------------------------------------------------------------------------------------} n (n - 1) "A026107" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((5 n + 13) (2 n + 8 n + 9) hypergeom([1/2, -n - 1], [1], 4) - 3 (2 n + 6 n + 1) (n + 1) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A026109" memory used=19577.0MB, alloc=1303.5MB, time=122.51 LREtools/SearchTable: "SearchTable successful" n 4 3 2 {(-1) ((13 n + 174 n + 881 n + 2064 n + 1863) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 + (-15 n - 213 n - 1182 n - 3108 n - 3177) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 8) (n + 7) (n + 4) (n + 5) (n + 3) (n + 6))} "A026110" LREtools/SearchTable: "SearchTable successful" n 4 3 2 {(-1) ((40 n + 573 n + 3239 n + 8334 n + 7776) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 + (-42 n - 609 n - 3549 n - 9570 n - 9504) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 10) (n + 9) (n + 8) (n + 6) (n + 5) (n + 4))} "A026122" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((8 n + 43 n + 72) hypergeom([1/2, -n - 1], [1], 4) + (-12 n - 81 n - 168) hypergeom([1/2, -n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 2) "A026123" LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 {(-1) ((10 n + 140 n + 825 n + 2478 n + 3649 n + 2106) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 - 3 (2 n + 16 n + 33 n - 33 n - 142) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A026124" memory used=19811.9MB, alloc=1303.5MB, time=124.23 LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 {(-1) ((8 n + 109 n + 610 n + 1697 n + 2112 n + 918) hypergeom([1/2, -n - 1], [1], 4) 5 4 3 2 + (-12 n - 207 n - 1560 n - 6129 n - 11928 n - 9162) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 2) (n + 8) (n + 7) (n + 4) (n + 5) (n + 3) (n + 6))} "A026125" LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {(-1) ((26 n + 602 n + 5959 n + 32032 n + 96651 n + 154602 n + 104976) hypergeom([1/2, -n - 1], [1], 4) 6 5 4 3 2 + (-30 n - 744 n - 8109 n - 49206 n - 171681 n - 323622 n - 257904) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 3))} "A026126" memory used=20010.9MB, alloc=1303.5MB, time=125.77 LREtools/SearchTable: "SearchTable successful" n 7 6 5 4 3 2 {(-1) ((80 n + 2699 n + 39595 n + 326321 n + 1615737 n + 4769060 n + 7747308 n + 5307120) hypergeom([1/2, -n - 1], [1], 4) 7 6 5 4 3 2 + (-84 n - 2877 n - 43281 n - 370071 n - 1926183 n - 6063852 n - 10657812 n - 7998480) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 12) (n + 11) (n + 8) (n + 5) (n + 7) (n + 6) (n + 10) (n + 9) (n + 4))} "A026135" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) (2 n + 1) {--------------------------------------------------------------------------------------------------} (n + 2) n "A026152" LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {(-1) ((5 n + 103 n + 1001 n + 5549 n + 17357 n + 28548 n + 19197) hypergeom([1/2, -n - 1], [1], 4) 5 4 3 2 - 3 (n + 11 n + 31 n - 107 n - 857 n - 1359) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3)) } "A026153" memory used=20214.9MB, alloc=1303.5MB, time=127.33 LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {(-1) ((4 n + 77 n + 703 n + 3769 n + 11389 n + 18810 n + 15552) hypergeom([1/2, -n - 1], [1], 4) 6 5 4 3 2 + (-6 n - 153 n - 1899 n - 13521 n - 54543 n - 117078 n - 105408) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 3))} "A026154" memory used=20397.9MB, alloc=1303.5MB, time=128.84 LREtools/SearchTable: "SearchTable successful" n 7 6 5 4 3 2 {(-1) ((78 n + 2406 n + 34945 n + 303932 n + 1637381 n + 5360002 n + 9921816 n + 7931520) hypergeom([1/2, -n - 1], [1], 4) 7 6 5 4 3 2 + (-90 n - 3024 n - 48867 n - 475542 n - 2871879 n - 10528254 n - 21546024 n - 18679680) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 12) (n + 11) (n + 8) (n + 5) (n + 7) (n + 6) (n + 10) (n + 9) (n + 4))} "A026155" memory used=20570.9MB, alloc=1303.5MB, time=130.17 LREtools/SearchTable: "SearchTable successful" n 8 7 6 5 4 3 2 {(-1) ((40 n + 1725 n + 34768 n + 423110 n + 3312260 n + 16732345 n + 52675652 n + 93248100 n + 69167520) hypergeom([1/2, -n - 1], [1], 4) 8 7 6 5 4 3 2 + (-42 n - 1851 n - 38508 n - 487170 n - 3985638 n - 21143079 n - 70176132 n - 131412540 n - 103805280) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 14) (n + 13) (n + 12) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5))} "A026163" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((2 n + 10 n + 44 n + 81) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 27 n - 144 n - 255) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 4) (n + 8) (n + 7) (n + 3) "A026164" LREtools/SearchTable: "SearchTable successful" n 2 2 4 3 2 (-1) ((n + 5 n + 12) (2 n + n - 7) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 25 n - 166 n - 419 n - 324) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A026165" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((2 n + 3) (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 7 n - 9) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A026243" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 2 n1 + 3 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A026269" memory used=20777.3MB, alloc=1303.5MB, time=131.84 LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((2 n + 7 n + 8 n + 6) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 3 n + 8 n + 6) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) n "A026270" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((4 n + 10 n + 5 n - 10) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 14 n - 35 n - 10) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 2) n "A026288" LREtools/SearchTable: "SearchTable successful" n 4 3 2 4 3 2 {(-1) ((2 n + 17 n + 63 n + 130 n + 102) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 13 n - 19 n + 34 n + 102) hypergeom([1/2, -n], [1], 4))/( (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A026290" LREtools/SearchTable: "SearchTable successful" n 7 6 5 4 3 2 {(-1) ((6 n + 157 n + 1776 n + 11325 n + 43554 n + 99998 n + 127584 n + 70632) hypergeom([1/2, -n - 1], [1], 4) 7 6 5 4 3 2 + (-6 n - 153 n - 1668 n - 10101 n - 36222 n - 76362 n - 90624 n - 50328) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 2) (n + 7) (n + 4) (n + 6) (n + 5) (n + 8) (n + 9) (n + 3) (n + 10))} "A026299" memory used=20989.4MB, alloc=1335.5MB, time=133.55 LREtools/SearchTable: "SearchTable successful" n 3 2 2 (-1) ((8 n + 2 n - 14 n + 1) hypergeom([1/2, -n - 1], [1], 4) - (4 n - 1) (2 n + 1) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------------------------------------} n (n + 2) (n - 1) "A026302" LREtools/SearchTable: "SearchTable successful" (hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + 2 hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) (n + 1) {----------------------------------------------------------------------------------------------------------------------} 13 n + 9 "A026325" LREtools/SearchTable: "SearchTable successful" n 2 3 2 (-1) (n (3 n + 3 n - 22) hypergeom([1/2, -n - 1], [1], 4) + (-9 n - 81 n - 318 n - 480) hypergeom([1/2, -n], [1], 4)) (n + 1) {---------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) "A026327" LREtools/SearchTable: "SearchTable successful" n 5 4 3 2 {(-1) ((13 n + 242 n + 2069 n + 10588 n + 29028 n + 29808) hypergeom([1/2, -n - 1], [1], 4) 5 4 3 2 + (-15 n - 306 n - 2931 n - 16308 n - 47004 n - 50832) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 8) (n + 10) (n + 9) (n + 7) (n + 6) (n + 5) (n + 4))} "A026328" memory used=21248.6MB, alloc=1335.5MB, time=135.41 LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {(-1) ((120 n + 3387 n + 43463 n + 329643 n + 1473913 n + 3401586 n + 2974320) hypergeom([1/2, -n - 1], [1], 4) 6 5 4 3 2 + (-126 n - 3627 n - 47991 n - 375951 n - 1726755 n - 4082334 n - 3678480) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 12) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5))} "A026329" LREtools/SearchTable: "SearchTable successful" n 7 6 5 4 3 2 {(-1) ((121 n + 4735 n + 84057 n + 884181 n + 5765174 n + 22145900 n + 44308008 n + 34362144) hypergeom([1/2, -n - 1], [1], 4) 7 6 5 4 3 2 + (-123 n - 4809 n - 85587 n - 904419 n - 5917962 n - 22734996 n - 45294168 n - 34826976) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 14) (n + 13) (n + 12) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6))} "A026375" LREtools/SearchTable: "SearchTable successful" {(2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)} "A026376" LREtools/SearchTable: "SearchTable successful" ((4 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3) hypergeom([-1/2, -n], [1], -4)) (n + 1) {--------------------------------------------------------------------------------------------------} n + 2 "A026377" LREtools/SearchTable: "SearchTable successful" memory used=21477.5MB, alloc=1335.5MB, time=137.09 2 2 ((20 n + 20 n + 9) hypergeom([-1/2, -n - 1], [1], -4) + (-20 n - 30 n - 15) hypergeom([-1/2, -n], [1], -4)) (n + 1) {---------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A026378" LREtools/SearchTable: "SearchTable successful" {(8 n + 3) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7) hypergeom([-1/2, -n], [1], -4)} "A026379" LREtools/SearchTable: "SearchTable successful" 2 2 ((8 n + 11 n + 4) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 15 n - 8) hypergeom([-1/2, -n], [1], -4)) (n + 1) {------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A026380" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (4 n + 12) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-4 n - 16) hypergeom([-1/2, - n/2 - 1/2], [1], -4) n::even {{ , { (8 n + 6) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-8 n - 14) hypergeom([-1/2, - n/2], [1], -4) n::odd { (8 n + 6) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-8 n - 14) hypergeom([-1/2, - n/2], [1], -4) n::even { , { (4 n + 12) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-4 n - 16) hypergeom([-1/2, - n/2 - 1/2], [1], -4) n::odd { (n/2 + 1) (n/2) { 2 5 (4 n + 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 2 5 (4 n + 7) hypergeom([3/2, - n/2], [1], 4/5) n::even { , { (n/2 + 3/2) (n/2 + 1/2) { 4 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 4 5 (n + 4) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) n::odd { (n/2 + 3/2) (n/2 + 1/2) { 4 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 4 5 (n + 4) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) n::even { } { (n/2 + 1) (n/2) { 2 5 (4 n + 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 2 5 (4 n + 7) hypergeom([3/2, - n/2], [1], 4/5) n::odd "A026387" LREtools/SearchTable: "SearchTable successful" {(8 n + 3) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7) hypergeom([-1/2, -n], [1], -4)} "A026388" LREtools/SearchTable: "SearchTable successful" 2 2 (16 n + 18 n + 7) hypergeom([-1/2, -n - 1], [1], -4) + (-16 n - 26 n - 13) hypergeom([-1/2, -n], [1], -4) {-----------------------------------------------------------------------------------------------------------} n + 2 "A026389" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 ((8 n + 23 n + 28 n + 7) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 27 n - 38 n - 17) hypergeom([-1/2, -n], [1], -4)) (n + 1) {-----------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A026392" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 { 2 ((n + 3) (4 n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-4 n - 19 n - 17) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) {{ - --------------------------------------------------------------------------------------------------------------------------- n::even, { { n + 1 { - { { (-20 n - 15) hypergeom([-1/2, - n/2 - 1], [1], -4) + (20 n + 35) hypergeom([-1/2, - n/2], [1], -4) n::odd 2 ( (n/2 + 3/2) (n/2 + 1/2) 2 5 (n + 3) (4 n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (4 n + 19 n + 17) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) /(n + 1) , n::even (n/2 + 1) (n/2) , -5 5 (4 n + 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) + 5 5 (4 n + 7) hypergeom([3/2, - n/2], [1], 4/5) , n::odd { (-20 n - 15) hypergeom([-1/2, - n/2 - 1], [1], -4) + (20 n + 35) hypergeom([-1/2, - n/2], [1], -4) n::even { { 2 , { { 2 ((n + 3) (4 n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-4 n - 19 n - 17) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) { { - --------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 (n/2 + 1) (n/2) -5 5 (4 n + 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) + 5 5 (4 n + 7) hypergeom([3/2, - n/2], [1], 4/5) , n::even - 2 ( (n/2 + 3/2) (n/2 + 1/2) 2 5 (n + 3) (4 n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (4 n + 19 n + 17) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) } /(n + 1) , n::odd "A026418" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026569" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026570" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026571" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026572" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026573" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026580" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 17 | | 17 | {|1/2 - -----| , |1/2 + -----| } \ 2 / \ 2 / "A026582" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 17 | | 17 | {1, |1/2 - -----| , |1/2 + -----| } \ 2 / \ 2 / "A026585" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026587" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026589" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026596" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 17 | | 17 | {|1/2 - -----| , |1/2 + -----| } \ 2 / \ 2 / "A026598" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 17 | | 17 | {1, |1/2 - -----| , |1/2 + -----| } \ 2 / \ 2 / "A026621" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { binomial(n, n/2) (7 n - 8) { 4 4 (7 n - 1) { -------------------------- n::even { -------------------------- n::even { n - 1 { n (n + 1) binomial(n, n/2) {{ , { } { 2 binomial(n - 1, n/2 - 1/2) (7 n - 1) { (2 n + 2) { -------------------------------------- n::odd { 2 2 (7 n - 8) { n + 1 { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A026623" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { binomial(n, n/2) (7 n - 8) n { { 1/2 -------------------------- n::even {1, 2 , { (2 n - 2) , { n - 1 } { 2 2 (7 n - 8) { { ------------------------------------ n::odd { 0 n::odd { n (n - 1) binomial(n - 1, n/2 - 1/2) "A026627" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(2 n1, n1) (51 n1 + n1 - 14)| {(-1/2) , (-1/2) | ) -------------------------------------|} | / (n1 + 1) | |----- (2 n1 - 1) (n1 + 1) (-1/2) | \n1 = 0 / "A026628" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | n n | \ binomial(2 n1, n1) (51 n1 + 205 n1 + 242 n1 + 78)| {(-1/2) , (-1/2) | ) ---------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A026629" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 3 2 | n n | \ (2 n1 + 1) binomial(2 n1, n1) (51 n1 + 460 n1 + 1515 n1 + 2174 n1 + 1140)| {(-1/2) , (-1/2) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A026630" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(2 n1, n1) (51 n1 + 103 n1 + 38)| {(-1/2) , (-1/2) | ) -----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) | \n1 = 0 / "A026631" memory used=21793.1MB, alloc=1367.5MB, time=139.37 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | n n | \ (2 n1 + 1) binomial(2 n1, n1) (51 n1 + 307 n1 + 598 n1 + 384)| {(-1/2) , (-1/2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A026632" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { n2 2 \ \ | | { binomial(n2, ----) (51 n2 + 2 n2 - 56) | | | | { 2 | | | | { --------------------------------------- n2::even| | | | { (n2 + 2) (n2 - 1) | | | | { | | | | { n2 2 | | | | { 2 binomial(n2 - 1, ---- - 1/2) (51 n2 + 104 n2 - 3) | | |n - 1 |n1 - 1 { 2 | | |----- |----- { ---------------------------------------------------- n2::odd | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { (n2 + 1) (n2 + 3) | | {(-1/2 I 2 ) , (1/2 I 2 ) , (-1/2 I 2 ) | ) (-1) 2 | ) ----------------------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 2 \ \ | | { 4 4 (51 n2 + 104 n2 - 3) | | | | { --------------------------------------- n2::even| | | | { n2 | | | | { n2 (n2 + 1) (n2 + 3) binomial(n2, ----) | | | | { 2 | | | | { | | | | { (2 n2 + 2) 2 | | | | { 2 2 (51 n2 + 2 n2 - 56) | | | | { ------------------------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { (n2 - 1) (n2 + 1) (n2 + 2) binomial(n2 + 1, ---- + 1/2) | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) -------------------------------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A026638" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A026639" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A026640" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) ---------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 5) (n1 + 4) (n1 + 1) (-1/2) | \n1 = 0 / "A026641" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A026642" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) ---------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 4) (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A026643" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { n2 \ \ | | { 4 binomial(n2, ----) (n2 + 1) | | | | { 2 | | | | { ----------------------------- n2::even| | | | { n2 + 2 | | | | { | | | | { n2 | | | | { 2 binomial(n2 + 1, ---- + 1/2) (n2 + 2) | | |n - 1 |n1 - 1 { 2 | | |----- |----- { --------------------------------------- n2::odd | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { n2 + 3 | | {(-1/2 I 2 ) , (1/2 I 2 ) , (-1/2 I 2 ) | ) (-1) 2 | ) ---------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { 4 (n2 + 2) | | | | { 1/2 ------------------------------------ n2::even| | | | { n2 | | | | { (n2 + 1) (n2 + 3) binomial(n2, ----) | | | | { 2 | | | | { | | | | { (2 n2 - 2) | | | | { 2 (n2 + 1) | | | | { ---------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) ----------------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A026645" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n 1/2 n 1/2 n {1, (-1) , 2 , (-1/2 I 2 ) , (1/2 I 2 ) , / / { 0 n2::even\ \ | | { | | | | { (2 n2 - 2) | | | | { 2 (n2 + 1) | | | | { 1/2 ---------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) --------------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { 2 binomial(n2, ----) (n2 + 1) | | | | { 2 | | | | { ----------------------------- n2::even| | |n - 1 |n1 - 1 { n2 + 2 | | |----- |----- { | | 1/2 n | \ n1 1/2 | \ { 0 n2::odd | | (-1/2 I 2 ) | ) (-1) 2 | ) -----------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A026652" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" (- n/2) 1/2 n 1/2 1/2 1/2 1/2 {3 (6 ) (3 (-3 + 6 ) LegendreP(n, 3 ) + 3 LegendreP(n + 1, 3 )), (- n/2) 1/2 n 1/2 1/2 1/2 1/2 3 (6 ) (3 (-3 + 6 ) LegendreQ(n, 3 ) + 3 LegendreQ(n + 1, 3 )), (- n/2) 1/2 n 1/2 1/2 1/2 1/2 3 (-6 ) (3 (-6 - 3) LegendreP(n, 3 ) + 3 LegendreP(n + 1, 3 )), (- n/2) 1/2 n 1/2 1/2 1/2 1/2 3 (-6 ) (3 (-6 - 3) LegendreQ(n, 3 ) + 3 LegendreQ(n + 1, 3 ))} "A026671" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A026672" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=22097.6MB, alloc=1367.5MB, time=141.63 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 - 2)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ---------------------------------------------------------------------||} | / | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A026673" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (3 n2 + 8)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------------------------------|| | / | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // } "A026674" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A026675" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 - 15 n2 - 40)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------------------------------||} | / | / (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A026676" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026677" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026678" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026679" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {1, RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026704" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 1) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 2) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 3) } "A026705" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 1) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 2) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 3) } "A026707" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 1) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 2) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 3) } "A026708" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 1) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 2) , RootOf(2 _Z - 11 _Z + 6 _Z - 1, index = 3) } "A026715" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 2 %1 := _Z - 9 _Z + 25 _Z - 30 _Z + 10 _Z - 1 "A026726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (5 n2 + 4)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A026731" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026732" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026733" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026737" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2) (n2 + 9)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A026742" memory used=22407.1MB, alloc=1367.5MB, time=143.92 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026743" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026759" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (4 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3) hypergeom([-1/2, -n], [1], -4) binomial(2 n, n) n {----------------------------------------------------------------------------------------, ------------------} n + 2 (n + 1) (n + 2) "A026765" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A026767" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A026770" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 LegendreP(n + 1, 3) - LegendreP(n, 3) 3 LegendreQ(n + 1, 3) - LegendreQ(n, 3) binomial(2 n, n) n {---------------------------------------, ---------------------------------------, ------------------} n + 2 n + 2 (n + 1) (n + 2) "A026773" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 3 LegendreP(n + 1, 3) - LegendreP(n, 3) 3 LegendreQ(n + 1, 3) - LegendreQ(n, 3) (2 n + 1) binomial(2 n, n) {---------------------------------------, ---------------------------------------, --------------------------} n + 2 n + 2 (n + 1) (n + 2) "A026842" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 2 || 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 48 n2 + 152)|| (2 + 5 ) (2 - 5 ) | ) ------------------------------------------------------------------------------------------------------|| | / (n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 2) (n2 + 1) || |----- || \n2 = 0 // } "A026843" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 63 n2 + 182)|| | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------------------------------------------||} | / | / (n2 + 4) (n2 + 5) (n2 + 6) (n2 + 7) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A026846" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 2 || 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 48 n2 + 152)|| (2 + 5 ) (2 - 5 ) | ) ------------------------------------------------------------------------------------------------------|| | / (n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 2) (n2 + 1) || |----- || \n2 = 0 // } "A026847" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 4 _Z - 1 "A026849" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 2 || 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 48 n2 + 152)|| (2 + 5 ) (2 - 5 ) | ) ------------------------------------------------------------------------------------------------------|| | / (n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 2) (n2 + 1) || |----- || \n2 = 0 // } "A026850" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 20)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------------------||} | / | / (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A026851" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 3 2 || 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 30 n2 + 23 n2 - 126)|| (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------------------------------------------------||} | / (n2 + 4) (n2 + 5) (n2 + 6) (n2 + 7) (n2 + 3) (n2 + 2) (n2 + 1) || |----- || \n2 = 0 // "A026854" memory used=22704.6MB, alloc=1367.5MB, time=146.12 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2) (n2 + 16 n2 + 18)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) --------------------------------------------------------------------||} | / | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A026855" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 27 n2 + 56) n2|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------------------------------|| | / | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // } "A026908" 2 binomial(2 n, n) (49 n + 99 n + 38) {1, ------------------------------------} (n + 1) (n + 2) "A026909" 2 (2 n + 1) binomial(2 n, n) (49 n + 197 n + 186) {1, ------------------------------------------------} (n + 3) (n + 2) (n + 1) "A026910" 2 (2 n + 1) binomial(2 n, n) (49 n + 197 n + 184) {1, ------------------------------------------------} (n + 4) (n + 2) (n + 1) "A026911" 4 3 2 (2 n + 3) (2 n + 1) binomial(2 n, n) (49 n + 638 n + 3077 n + 6508 n + 5100) {1, -------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A026912" 2 (2 n + 1) binomial(2 n, n) (49 n + 197 n + 186) {1, ------------------------------------------------} (n + 3) (n + 2) (n + 1) "A026913" 3 2 (2 n + 3) (2 n + 1) binomial(2 n, n) (49 n + 442 n + 1313 n + 1280) {1, ---------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A026914" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 { 4 (49 n + 296 n + 399) { 2 binomial(n, n/2) (49 n + 198 n + 152) { ---------------------------------------- n::even { ---------------------------------------- n::even { (n + 1) (n + 3) (n + 5) binomial(n, n/2) { (n + 2) (n + 4) {1, { , { } { (2 n - 2) 2 { 2 { 2 2 (49 n + 198 n + 152) { binomial(n + 1, n/2 + 1/2) (49 n + 296 n + 399) { -------------------------------------------- n::odd { ------------------------------------------------ n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { (n + 5) (n + 3) "A026933" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ n1 | n | \ n1 | {(-1) , (-1) | ) (-(-1) LegendreP(n1 + 1, 3))|, (-1) | ) (-(-1) LegendreQ(n1 + 1, 3))|} | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / "A026940" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 2 ((56 n + 142 n + 81) hypergeom([1/2, -2 n - 2], [1], 4) - 9 (2 n + 1) (4 n + 7) hypergeom([1/2, -2 n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------------} (4 n + 3) (2 n + 3) (2 n + 5) (n + 2) "A026943" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) (n + 2) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A026945" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" hypergeom([1/2, -2 n - 2], [1], 4) + 3 hypergeom([1/2, -2 n], [1], 4) {---------------------------------------------------------------------} 4 n + 3 "A026956" 2 binomial(2 n, n) (49 n - 105 n + 48) {1, -------------------------------------} (2 n - 3) (2 n - 1) "A027027" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027044" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n 3 2 n 3 2 n 3 2 n {1, 3 , RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027050" memory used=23049.9MB, alloc=1399.5MB, time=148.58 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027051" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027056" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027057" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027058" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027059" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027076" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n 3 2 n 3 2 n 3 2 n {1, 3 , RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027118" memory used=23447.5MB, alloc=1431.5MB, time=151.00 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A027171" binomial(2 n, n) (9 n + 5) {1, --------------------------} n + 1 "A027172" (2 n + 1) binomial(2 n, n) (9 n + 14) {1, -------------------------------------} (n + 1) (n + 2) "A027173" 2 (2 n + 1) binomial(2 n, n) (9 n + 32 n + 27) {1, ---------------------------------------------} (n + 3) (n + 2) (n + 1) "A027174" 2 (2 n + 3) (2 n + 1) binomial(2 n, n) (9 n + 50 n + 65) {1, -------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 1) "A027175" (2 n + 1) binomial(2 n, n) (9 n + 14) {1, -------------------------------------} (n + 1) (n + 2) "A027176" 2 (2 n + 3) (2 n + 1) binomial(2 n, n) (9 n + 41 n + 44) {1, -------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A027177" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (9 n + 19) { 2 binomial(n, n/2) (9 n + 10) { -------------------------------- n::even { ----------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (9 n + 19) { 2 2 (9 n + 10) { ------------------------------------- n::odd { ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A027277" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A027291" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { binomial(n, n/2) (3 n + 2) (n + 5) { 2 4 (n + 5) (3 n + 5) { ---------------------------------- n::even { -------------------------------- n::even n { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {2 , { , { } { 2 binomial(n - 1, n/2 - 1/2) (n + 5) (3 n + 5) n { (2 n + 2) { ------------------------------------------------ n::odd { 2 (n + 5) (3 n + 2) { (n + 1) (n + 3) { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A027292" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { binomial(n, n/2) (3 n + 2) n { { -------------------------- n::even {2 , { (2 n - 2) , { n + 2 } { 2 (3 n + 2) { { ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A027298" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 6) { 2 binomial(n, n/2) (n + 1) (n + 6) { 1/2 ------------------------ n::even { ---------------------------------- n::even n { (n + 3) binomial(n, n/2) { n + 4 {2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n + 1) (n + 6) { 2 (n + 1) (n + 6) { ------------------------------------------ n::odd { ------------------------------------ n::odd { n + 3 { n (n + 4) binomial(n - 1, n/2 - 1/2) "A027299" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n n { { 4 {2 , { 2 binomial(n - 1, n/2 - 1/2) n , { ------------------------ n::even} { ------------------------------ n::odd { (n + 3) binomial(n, n/2) { n + 3 { { 0 n::odd "A027305" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 4) { 1/2 ------------------------ n::even { 2 binomial(n, n/2) (n + 1) (n + 4) n { (n + 1) binomial(n, n/2) { ---------------------------------- n::even {2 , { , { n + 2 } { (2 n - 2) { { 2 (n + 1) (n + 4) { (n + 4) binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A027306" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {2 , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A027307" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-8 n - 5) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------} n (10 n + 7) "A027312" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) (n + 1) (n + 4) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4)) {3 , ---------------------------------------------------------------------------------------} n + 3 "A027390" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A027391" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A027392" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A027393" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A027394" memory used=23783.2MB, alloc=1431.5MB, time=153.44 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A027395" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A027412" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-1/2 I 2 ) (HermiteH(n + 1, 2 I) - 2 I 2 HermiteH(n, 2 I)) {-----------------------------------------------------------------------} n "A027432" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A027616" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |---- - 1/2| | | { (-2) |----|! n1::even| |n - 1 { \ 2 / / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 (-1/2) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) ----------------------------------------------------------------------------------|, n! | ) ------------------------------------|, n! | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / } "A027617" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / { 0 irem(n1, 3) = 0\ | { | | { 0 irem(n1, 3) = 1| | { | | { / n1 \ | | { |---- - 2/3| | |n - 1 { \ 3 / // n1 \ \2 2 n1 n1 n1 | |----- { (n1 - 1) n1 (-1/3) ||---- - 2/3|!| binomial(---- - 4/3, ---- - 2/3) binomial(n1 - 2, ---- - 2/3) irem(n1, 3) = 2| | \ { \\ 3 / / 3 3 3 | {n! | ) --------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 3) = 0\ | { | | { / n1 \ | | { |---- - 1/3| | | { \ 3 / n1 n1 | | { (-9) GAMMA(---- + 2/3) GAMMA(---- + 1) irem(n1, 3) = 1| |n - 1 { 3 3 | |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ---------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 3 / n1 n1 | | { (-9) GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 3) = 0| | { 3 3 | | { | |n - 1 { 0 irem(n1, 3) = 1| |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ---------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A027618" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, LegendreP(n, 3), LegendreQ(n, 3)} "A027908" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)} "A027909" LREtools/SearchTable: "SearchTable successful" ((9 n + 9) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4) + (11 n + 9) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4)) binomial(2 n, n) (2 n + 1) {-------------------------------------------------------------------------------------------------------------------------------------------} (3 n + 4) (13 n + 9) "A027910" LREtools/SearchTable: "SearchTable successful" 2 {(18 (n + 1) (14 n + 52 n + 47) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4) 3 2 + (737 n + 3162 n + 4183 n + 1674) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4)) binomial(2 n, n) (2 n + 1) (2 n + 3)/((3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (13 n + 9))} "A027911" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) (hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + 2 hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) {------------------------------------------------------------------------------------------------------------------------} 13 n + 9 "A027913" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" {{ { 12 binomial(n, n/2) hypergeom([- n/4, - n/4 + 1/2], [n/2 + 1], 4) , n::even 8 (3/2 (3 n + 7) (3 n + 5) hypergeom([- n/4 - 3/4, - n/4 - 1/4], [n/2 + 5/2], 4) { 2 , { + (-61/4 n - 61 n - 231/4) hypergeom([- n/4 + 1/4, - n/4 - 1/4], [n/2 + 3/2], 4)) binomial(n + 1, n/2 + 1/2)/((13 n + 31) (n + 1)) , n::odd { { n 8 4 (3/2 (3 n + 7) (3 n + 5) hypergeom([- n/4 - 3/4, - n/4 - 1/4], [n/2 + 5/2], 4) 2 / 2 + (-61/4 n - 61 n - 231/4) hypergeom([- n/4 + 1/4, - n/4 - 1/4], [n/2 + 3/2], 4)) / ((n + 1) (13 n + 31) binomial(n, n/2)) , n::even / (2 n - 2) 12 2 hypergeom([- n/4, - n/4 + 1/2], [n/2 + 1], 4) } ----------------------------------------------------------- , n::odd n binomial(n - 1, n/2 - 1/2) "A027914" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) hypergeom([1/2, -n], [1], 4)} "A027915" memory used=24133.1MB, alloc=1431.5MB, time=155.89 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n \ n1 {1, 3 , ) (-1) hypergeom([1/2, -n1 - 1], [1], 4)} / ----- n1 = 0 "A027995" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {1, (2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ------------------------------------------------------------||} | / | / (n2 + 3) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A028339" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), /n - 1 \ |----- | n | \ / 2 (2 n1 + 1) binomial(2 n1, n1) n1! \| n (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||, (2 n + 3) (2 n + 1) (1/2) n! | / \(2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / / / /n1 - 1 \ \\ | | |----- | || | | | \ / 2 (2 n2 + 1) binomial(2 n2, n2) n2! \| || | |2 (2 n1 + 1) | ) |-----------------------------------------------|| binomial(2 n1, n1) n1!|| |n - 1 | | / \(2 n2 + 5) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!/| || |----- | |----- | || | \ | \n2 = 0 / || binomial(2 n, n) | ) |----------------------------------------------------------------------------------------------||} | / \ (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / "A028340" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 1) (2 n + 3) (2 n + 5) (1/2) n! binomial(2 n, n), /n - 1 \ |----- | n | \ / 2 (2 n1 + 1) binomial(2 n1, n1) n1! \| (2 n + 1) (2 n + 3) (2 n + 5) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||, (2 n + 1) (2 n + 3) | / \(2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / / / /n1 - 1 \ \\ | | |----- | || | | | \ / 2 (2 n2 + 1) binomial(2 n2, n2) n2! \| || | |2 (2 n1 + 1) | ) |-----------------------------------------------|| binomial(2 n1, n1) n1!|| |n - 1 | | / \(2 n2 + 7) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!/| || |----- | |----- | || n | \ | \n2 = 0 / || (2 n + 5) (1/2) n! binomial(2 n, n) | ) |----------------------------------------------------------------------------------------------||, | / \ (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / / | | | | | | | |n - 1 |----- n | \ (2 n + 1) (2 n + 3) (2 n + 5) (1/2) n! binomial(2 n, n) | ) | / |----- \n1 = 0 / / / /n2 - 1 \ \\ \\ | | | |----- | || || | | | | \ / 2 (2 n3 + 1) binomial(2 n3, n3) n3! \| || || | | |2 (2 n2 + 1) | ) |-----------------------------------------------|| binomial(2 n2, n2) n2!|| || | |n1 - 1 | | / \(2 n3 + 7) binomial(2 n3 + 2, n3 + 1) (n3 + 1)!/| || || | |----- | |----- | || || | | \ | \n3 = 0 / || || |2 (2 n1 + 1) | ) |----------------------------------------------------------------------------------------------|| binomial(2 n1, n1) n1!|| | | / \ (2 n2 + 7) binomial(2 n2 + 2, n2 + 1) (n2 + 1)! /| || | |----- | || | \n2 = 0 / || |---------------------------------------------------------------------------------------------------------------------------------------------||} \ (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| | / "A028341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 1) (2 n + 3) (2 n + 5) (2 n + 7) (1/2) n! binomial(2 n, n), /n - 1 \ |----- | n | \ / 2 (2 n1 + 1) binomial(2 n1, n1) n1! \| (2 n + 1) (2 n + 3) (2 n + 5) (2 n + 7) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||, (2 n + 1) | / \(2 n1 + 9) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / n (2 n + 3) (2 n + 5) (2 n + 7) (1/2) n! binomial(2 n, n) / / /n1 - 1 \ \\ | | |----- | || | | | \ / 2 (2 n2 + 1) binomial(2 n2, n2) n2! \| || | |2 (2 n1 + 1) | ) |-----------------------------------------------|| binomial(2 n1, n1) n1!|| |n - 1 | | / \(2 n2 + 9) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!/| || |----- | |----- | || | \ | \n2 = 0 / || | ) |----------------------------------------------------------------------------------------------||, (2 n + 1) (2 n + 3) (2 n + 5) | / \ (2 n1 + 9) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / / | | | | | | | |n - 1 |----- n | \ (2 n + 7) (1/2) n! binomial(2 n, n) | ) | / |----- \n1 = 0 / / / /n2 - 1 \ \\ \\ | | | |----- | || || | | | | \ / 2 (2 n3 + 1) binomial(2 n3, n3) n3! \| || || | | |2 (2 n2 + 1) | ) |-----------------------------------------------|| binomial(2 n2, n2) n2!|| || | |n1 - 1 | | / \(2 n3 + 9) binomial(2 n3 + 2, n3 + 1) (n3 + 1)!/| || || | |----- | |----- | || || | | \ | \n3 = 0 / || || |2 (2 n1 + 1) | ) |----------------------------------------------------------------------------------------------|| binomial(2 n1, n1) n1!|| | | / \ (2 n2 + 9) binomial(2 n2 + 2, n2 + 1) (n2 + 1)! /| || | |----- | || | \n2 = 0 / || |---------------------------------------------------------------------------------------------------------------------------------------------||, \ (2 n1 + 9) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| | / / / / | | | | | | | | | | | | | | | | | | | | | |n - 1 | |n1 - 1 |----- | |----- n | \ | | \ (2 n + 1) (2 n + 3) (2 n + 5) (2 n + 7) (1/2) n! binomial(2 n, n) | ) |2 (2 n1 + 1) | ) | / | | / |----- | |----- \n1 = 0 \ \n2 = 0 / / / /n3 - 1 \ \\ \\ | | | |----- | || || | | | | \ / 2 (2 n4 + 1) binomial(2 n4, n4) n4! \| || || | | |2 (2 n3 + 1) | ) |-----------------------------------------------|| binomial(2 n3, n3) n3!|| || | |n2 - 1 | | / \(2 n4 + 9) binomial(2 n4 + 2, n4 + 1) (n4 + 1)!/| || || | |----- | |----- | || || | | \ | \n4 = 0 / || || |2 (2 n2 + 1) | ) |----------------------------------------------------------------------------------------------|| binomial(2 n2, n2) n2!|| | | / \ (2 n3 + 9) binomial(2 n3 + 2, n3 + 1) (n3 + 1)! /| || | |----- | || | \n3 = 0 / || |---------------------------------------------------------------------------------------------------------------------------------------------|| \ (2 n2 + 9) binomial(2 n2 + 2, n2 + 1) (n2 + 1)! /| | / \\ || || || || || || || || || || binomial(2 n1, n1) n1!/((2 n1 + 9) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!)||} || || // "A028353" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 2) 2 2 2| n 2 n 2 | \ 2 (2 n1 + 1) binomial(2 n1, n1) (n1!) | {4 (n!) , 4 (n!) | ) ---------------------------------------------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A028575" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A029571" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 4) = 0 |----- { \n1 = 0 0 , irem(n1, 4) = 1 0 , irem(n1, 4) = 2 / n1 \ |---- - 3/4| \ 4 / n1 n1 2 // n1 \ \3 n1 / 1/2 (-1/4) binomial(---- - 3/2, ---- - 3/4) ||---- - 3/4|!| binomial(n1 - 3, ---- - 3/2) n1 (n1 - 2) (n1 - 1) , irem(n1, 4) = 3 2 4 \\ 4 / / 2 / { 0 irem(n1, 4) = 0\ | { | | { 0 irem(n1, 4) = 1| | { | | { / n1 \ | | { |---- - 1/2| | | { \ 4 / n1 n1 n1 | | { (-64) GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 3/4) irem(n1, 4) = 2| \ |n - 1 { 4 4 4 | | |----- { | | | \ { 0 irem(n1, 4) = 3| (n1 + 1)!|, n! | ) ----------------------------------------------------------------------------------------------|, | | / (n1 + 1)! | | |----- | / \n1 = 0 / / { 0 irem(n1, 4) = 0\ | { | | { / n1 \ | | { |---- - 1/4| | | { \ 4 / // n1 \ \3 3 n1 n1 n1 | | { n1 (-1) ||---- - 1/4|!| binomial(---- - 3/4, ---- - 1/4) binomial(n1 - 1, ---- - 1/4) irem(n1, 4) = 1| | { \\ 4 / / 4 4 4 | | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ---------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 4 / n1 n1 n1 | | { (-64) GAMMA(---- + 1) GAMMA(---- + 3/4) GAMMA(---- + 1/2) irem(n1, 4) = 0| | { 4 4 4 | | { | | { 0 irem(n1, 4) = 1| | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ----------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A029573" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 6) = 0 |----- { \n1 = 0 0 , irem(n1, 6) = 1 0 , irem(n1, 6) = 2 0 , irem(n1, 6) = 3 0 , irem(n1, 6) = 4 / n1 \ |---- - 5/6| \ 6 / n1 n1 2 n1 n1 2 // n1 \ \5 n1 1/12 (-1/6) binomial(---- - 5/2, ---- - 5/6) binomial(---- - 5/3, ---- - 5/6) ||---- - 5/6|!| binomial(n1 - 5, ---- - 5/2) 2 6 3 6 \\ 6 / / 2 \ | | /(n1 + 1)!|, (n1 - 4) (n1 - 3) (n1 - 2) (n1 - 1) n1 , irem(n1, 6) = 5 | | / / { 0 irem(n1, 6) = 0\ | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | | { / n1 \ | | { |---- - 2/3| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 5/6) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) GAMMA(---- + 1/2) irem(n1, 6) = 4| |n - 1 { 6 6 6 6 6 | |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 6) = 0\ | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { / n1 \ | | { |---- - 1/2| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 5/6) GAMMA(---- + 2/3) GAMMA(---- + 1/3) irem(n1, 6) = 3| | { 6 6 6 6 6 | | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / /n - 1 |----- { | \ { n! | ) { | / { 0 , irem(n1, 6) = 0 |----- { \n1 = 0 0 , irem(n1, 6) = 1 / n1 \ |---- - 1/3| \ 6 / // n1 \ \5 n1 n1 n1 n1 2 n1 1/2 (n1 - 1) n1 (-2/3) ||---- - 1/3|!| binomial(---- - 2/3, ---- - 1/3) binomial(---- - 1, ---- - 1/3) binomial(n1 - 2, ---- - 1) , \\ 6 / / 3 6 2 6 2 irem(n1, 6) = 2 0 , irem(n1, 6) = 3 0 , irem(n1, 6) = 4 \ | | /(n1 + 1)!|, 0 , irem(n1, 6) = 5 | | / / { 0 irem(n1, 6) = 0\ | { | | { / n1 \ | | { |---- - 1/6| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 5/6) GAMMA(---- + 1/2) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 6) = 1| | { 6 6 6 6 6 | | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) GAMMA(---- + 1/2) GAMMA(---- + 5/6) irem(n1, 6) = 0| | { 6 6 6 6 6 | | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A029574" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 7 to 1 (Liouvillian solutions)" /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 7) = 0 |----- { \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 / n1 \ |---- - 6/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) GAMMA(---- + 5/7) GAMMA(---- + 3/7) GAMMA(---- + 1) , irem(n1, 7) = 6 7 7 7 7 7 7 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 / n1 \ |---- - 5/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) , irem(n1, 7) = 5 7 7 7 7 7 7 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 / n1 \ |---- - 4/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 4/7) GAMMA(---- + 6/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 2/7) , irem(n1, 7) = 4 7 7 7 7 7 7 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 / n1 \ |---- - 3/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 5/7) GAMMA(---- + 1) GAMMA(---- + 4/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 3/7) , irem(n1, 7) = 3 7 7 7 7 7 7 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 / n1 \ |---- - 2/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 5/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 4/7) , irem(n1, 7) = 2 7 7 7 7 7 7 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 / n1 \ |---- - 1/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) GAMMA(---- + 5/7) , irem(n1, 7) = 1 7 7 7 7 7 7 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { | |----- { / \n1 = 0 / n1 \ |----| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 4/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) , irem(n1, 7) = 0 7 7 7 7 7 7 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ | | /(n1 + 1)!|, n!} 0 , irem(n1, 7) = 6 | | / "A029759" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) (n1 - 1)| {2 , 2 | ) --------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A029760" n (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 3) {4 , --------------------------------------------} (n + 1) (n + 2) "A029767" n n! 2 n! {----, -----} n n "A029848" {1, binomial(2 n, n), binomial(3 n, n)} "A029887" n (2 n + 1) (2 n + 3) (2 n + 5) binomial(2 n, n) {4 (n + 2), ----------------------------------------------} n + 1 "A030238" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A030297" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A030494" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { 1/4 (n/2)! (n + 2) (n + 4) n::even { 4 2 (n + 1) (n + 3) binomial(n, n/2) (n/2)! n::even {1, { , { } { (n/2 + 1/2)! (n + 3) n::odd { (-n + 1) { 2 n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A030495" {n, n! (n + 1)} "A030662" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 1 "A030980" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A030981" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A030982" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A031970" memory used=24450.5MB, alloc=1431.5MB, time=158.34 2 n (2 n + 1) binomial(2 n, n) (2 n + 5 n + 4) {4 , -------------------------------------------} (n + 1) (n + 2) "A032037" LREtools/SearchTable: "SearchTable successful" n! (n + 1) (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) n! (n + 1) (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {----------------------------------------------------, ----------------------------------------------------} n n "A032115" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ (-1) 2 | {2 n!, 2 n! | ) -----------------|, 2 n - 1} | / (n1 + 1)! | |----- | \n1 = 0 / "A032119" LREtools/SearchTable: "SearchTable successful" n! (n + 1) (LegendreP(n + 1, 2) - 2 LegendreP(n, 2)) n! (n + 1) (LegendreQ(n + 1, 2) - 2 LegendreQ(n, 2)) {----------------------------------------------------, ----------------------------------------------------} n n "A032123" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { binomial(n, n/2) n::even {binomial(2 n, n), { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A032260" 2 {(n + 1) , (2 n + 1) binomial(2 n, n)} "A032270" n n 2 2 n! {2 (n - 3 n + 1), -----, 2 n - 1} n "A032349" LREtools/SearchTable: "SearchTable successful" (2 hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-5 n - 5) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1) {---------------------------------------------------------------------------------------------------------------------------} n (n + 1) (10 n + 7) "A032351" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 6 _Z + 8 _Z - 4, index = 1) , RootOf(_Z - 6 _Z + 8 _Z - 4, index = 2) , RootOf(_Z - 6 _Z + 8 _Z - 4, index = 3) } "A032357" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A032443" n {4 , binomial(2 n, n)} "A033030" LREtools/SearchTable: "SearchTable successful" n {(-3) n! LaguerreL(n, -n - 1/3, -1/3)} "A033296" LREtools/SearchTable: "SearchTable successful" 2 {((4 n + 4 n + 2) hypergeom([2 n + 3, -n - 1], [n + 2], -1) - (3 n + 5) (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1)/(n (n + 1) (2 n + 3) (10 n + 7))} "A033297" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 3) (n1 + 2) (n1 + 1) /| |----- | \n1 = 0 / "A033312" {1, n!} "A033321" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) | n n | \ 2 ((4 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-4 n1 - 3) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) --------------------------------------------------------------------------------------------------------|} | / n1 + 2 | |----- | \n1 = 0 / "A033504" n {4 (n + 1), (2 n + 1) binomial(2 n, n)} "A033540" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A033543" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / (-n1 - 1) n1 \| n n | \ |2 3 4 (2 n1 hypergeom([-1/2, -n1 - 1], [1], -2) + (-2 n1 - 1) hypergeom([-1/2, -n1], [1], -2))|| {(3/2) , (3/2) | ) |---------------------------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A033815" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselI(n + 1/2, 1/2), (-1) n! BesselK(n + 1/2, -1/2)} "A034015" LREtools/SearchTable: "SearchTable successful" ((12 n + 8) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1) {---------------------------------------------------------------------------------------------------------------------------------} (n + 1) (2 n + 3) (10 n + 7) "A034405" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1) 2 | 2 2 | \ 2 (n1!) (n1 + 1) (2 n1 + 1) binomial(2 n1, n1)| {(n!) , (n!) | ) ----------------------------------------------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A034430" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1) | | \ 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {n! | ) ----------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A034872" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {{ , { , { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 0 irem(n, 4) = 0 { (n/2) { { 2 (n + 3) GAMMA(n/4 + 5/4) { (n - 1) { ------------------------------- irem(n, 4) = 0 { 4 2 { (n + 1) GAMMA(n/4 + 7/4) { -------------------------------------- irem(n, 4) = 1 { { (n + 1) binomial(n/2 - 1/2, n/4 - 1/4) { 0 irem(n, 4) = 1 { { { n , { (n/2 - 1) , { 2 2 { 2 2 GAMMA(n/4 + 3/4) { -------------------------------- irem(n, 4) = 2 { ----------------------------- irem(n, 4) = 2 { n binomial(n/2 - 1, - 1/2 + n/4) { GAMMA(n/4 + 5/4) { { { (n + 1) { (n/2 - 3/2) { 2 2 { 4 2 GAMMA(n/4 + 1/2) { -------------------------------------- irem(n, 4) = 3 { ------------------------------- irem(n, 4) = 3 { (n + 1) binomial(n/2 + 1/2, n/4 + 1/4) { GAMMA(n/4 + 1) { (n/2) { 2 2 GAMMA(n/4 + 3/4) { ------------------------- irem(n, 4) = 0 { GAMMA(n/4 + 5/4) { 4 binomial(n/2, n/4) irem(n, 4) = 0 { { { (n/2 - 1/2) { binomial(n/2 + 3/2, n/4 + 3/4) (n + 3) { 4 2 GAMMA(n/4 + 1/2) { -------------------------------------- irem(n, 4) = 1 { ------------------------------- irem(n, 4) = 1, { n + 1 } { GAMMA(n/4 + 1) { { { 0 irem(n, 4) = 2 { (n/2 + 1) { { 2 (n + 3) GAMMA(n/4 + 5/4) { 2 binomial(n/2 + 1/2, n/4 + 1/4) irem(n, 4) = 3 { ----------------------------------- irem(n, 4) = 2 { (n + 1) GAMMA(n/4 + 7/4) { { 0 irem(n, 4) = 3 "A034942" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (32 n + 72 n + 77 n + 17) hypergeom([-1/2, -n - 1], [1], -4) + (-32 n - 88 n - 107 n - 43) hypergeom([-1/2, -n], [1], -4) {-----------------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A035011" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {1, ---------------------------------------, ---------------------------------------} n n "A035028" LREtools/SearchTable: "SearchTable successful" (4 n + 5) LegendreP(n + 1, 3) + LegendreP(n, 3) (4 n + 5) LegendreQ(n + 1, 3) + LegendreQ(n, 3) {-----------------------------------------------, -----------------------------------------------} n + 2 n + 2 "A035029" LREtools/SearchTable: "SearchTable successful" (4 n + 5) LegendreP(n + 1, 3) + LegendreP(n, 3) (4 n + 5) LegendreQ(n + 1, 3) + LegendreQ(n, 3) {-----------------------------------------------, -----------------------------------------------} n + 2 n + 2 "A035045" memory used=24777.0MB, alloc=1431.5MB, time=160.76 LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((4 n + 9) (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 11 n - 3) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A035101" n n {(n + 1) 2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A035119" n n {(n + 1) 2 n!, (2 n + 1) (1/2) n! binomial(2 n, n) (n + 3)} "A035318" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n (-1/2) n! binomial(2 n, n) (1/2) n! binomial(2 n, n) {---------------------------, --------------------------, n + 1 n + 1 / / /n1 - 1 \ \\ | | |----- | || | | n1 | \ / 2 (2 n2 + 1) binomial(2 n2, n2) n2! \| || | |2 (-1) (2 n1 + 1) | ) |- ---------------------------------------------|| binomial(2 n1, n1) n1!|| |n - 1 | | / \ (n2 + 2) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!/| || |----- | |----- | || n | \ | \n2 = 0 / || (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------------------------------------------------------------|| | / \ binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / -------------------------------------------------------------------------------------------------------------------------------------------, n + 1 / / /n1 - 1 \ \\ | | |----- / n2 \| || | | n1 | \ | 2 (-1) (2 n2 + 1) binomial(2 n2, n2) n2! || || | |2 (-1) (2 n1 + 1) | ) |- ---------------------------------------------|| binomial(2 n1, n1) n1!|| |n - 1 | | / \ (n2 + 2) binomial(2 n2 + 2, n2 + 1) (n2 + 1)!/| || |----- | |----- | || n | \ | \n2 = 0 / || (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------------------------------------------------------------|| | / \ binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / -------------------------------------------------------------------------------------------------------------------------------------------} n + 1 "A035320" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 2 (4 n + 1) (1/4) (n!) binomial(3 n, n) binomial(4 n, n) n 2 {--------------------------------------------------------, (4 n + 1) (1/4) (n!) binomial(3 n, n) binomial(4 n, n) n + 1 /n - 1 \ |----- / 2 \| | \ | 4 (n1!) (4 n1 + 1) (4 n1 + 3) binomial(3 n1, n1) binomial(4 n1, n1) || | ) |-----------------------------------------------------------------------------||/(n + 1)} | / | 2|| |----- \(2 n1 + 3) binomial(3 n1 + 3, n1 + 1) binomial(4 n1 + 4, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A035330" 2 n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 27 n + 122) {4 (n + 8), ------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A035610" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 3 2 binomial(2 n1, n1)| {16 , 16 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A035929" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n n n | \ | \ (1/2 + 1/2 I) binomial(2 n2, n2) n2|| {(1/2 - 1/2 I) , (1/2 + 1/2 I) , (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) --------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A036242" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((2 n + 1) BesselI(n, 1) + (-2 n - 2) BesselI(n - 1, 1)), (-1) ((2 n + 1) BesselK(n, -1) + (-2 n - 2) BesselK(n - 1, -1))} "A036243" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((2 n + 1) BesselI(n, 1) + (-2 n - 2) BesselI(n - 1, 1)), (-1) ((2 n + 1) BesselK(n, -1) + (-2 n - 2) BesselK(n - 1, -1))} "A036244" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) ((2 n + 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A036245" LREtools/SearchTable: "SearchTable not successful" {} "A036246" LREtools/SearchTable: "SearchTable not successful" {} "A036256" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | |{ 1/2 --------------------------- n1::even| /{ n1 \ n - 1 |{ n1 | n - 1 |{ 4 binomial(n1, ----) (n1 + 1) | ----- |{ (n1 + 1) binomial(n1, ----) | ----- |{ 2 | \ |{ 2 | \ |{ ----------------------------- n1::even| {1, ) |{ |, ) |{ n1 + 2 |} / |{ (2 n1 - 2) | / |{ | ----- |{ 2 (n1 + 1) | ----- |{ n1 | n1 = 0 |{ ---------------------------------------- n1::odd | n1 = 0 |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd | |{ n1 | \{ 2 / |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A036692" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A036757" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n n 2 ((3 n - 2) LegendreP(n + 1, 2) + 4 LegendreP(n, 2)) 2 ((3 n - 2) LegendreQ(n + 1, 2) + 4 LegendreQ(n, 2)) {------------------------------------------------------, ------------------------------------------------------, (n + 3) n (n + 2) (n + 3) n (n + 2) { (n/2) { 48 2 (LegendreP(n/2 + 1, 2) - 2 LegendreP(n/2, 2)) { - ------------------------------------------------------- n::even { n { , { (n/2 + 1/2) { 4 2 ((9 n - 1) LegendreP(n/2 + 3/2, 2) + (-21 n - 1) LegendreP(n/2 + 1/2, 2)) { - ---------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { (n/2) { 48 2 (LegendreQ(n/2 + 1, 2) - 2 LegendreQ(n/2, 2)) { - ------------------------------------------------------- n::even { n { , { (n/2 + 1/2) { 4 2 ((9 n - 1) LegendreQ(n/2 + 3/2, 2) + (-21 n - 1) LegendreQ(n/2 + 1/2, 2)) { - ---------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { (n/2) { 2 2 ((9 n - 1) LegendreP(n/2 + 3/2, 2) + (-21 n - 1) LegendreP(n/2 + 1/2, 2)) { - ---------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { , { (n/2 - 1/2) { 24 2 (LegendreP(n/2 + 1, 2) - 2 LegendreP(n/2, 2)) { - ------------------------------------------------------------- n::odd { n { (n/2) { 2 2 ((9 n - 1) LegendreQ(n/2 + 3/2, 2) + (-21 n - 1) LegendreQ(n/2 + 1/2, 2)) { - ---------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { } { (n/2 - 1/2) { 24 2 (LegendreQ(n/2 + 1, 2) - 2 LegendreQ(n/2, 2)) { - ------------------------------------------------------------- n::odd { n "A036765" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A036766" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A036767" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A036768" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A036774" LREtools/SearchTable: "SearchTable successful" n n (-I) n! (LegendreP(n, I) + LegendreP(n + 1, I) I) (-I) n! (LegendreQ(n, I) + LegendreQ(n + 1, I) I) {- --------------------------------------------------, - --------------------------------------------------} n n "A036781" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, n, ) (n1 + 1) n1!} / ----- n1 = 0 "A036782" memory used=25097.2MB, alloc=1431.5MB, time=163.12 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, n, ) (n1 + 1) n1!} / ----- n1 = 0 "A036829" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(3 n1, n1) | {(27/4) , (27/4) | ) -------------------------|} | / (n1 + 1)| |----- (n1 + 1/2) (27/4) | \n1 = 0 / "A036908" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)) {3 , ----------------------------------------------------------------------------------------} n "A036916" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A036917" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) hypergeom([1/2, 1/2, -n, -n], [1, -n + 1/2, -n + 1/2], 1)} "A036918" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A036919" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A037184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 3) (n + 2) (n + 1) n! (n + 5 n + 5), (n + 3) (n + 2) (n + 1) n! (n + 5 n + 5) | ) --------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 5 n1 + 10) (n1 + 5 n1 + 5)| \n1 = 0 / "A037223" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A037256" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {(n + 1) n!, n! (n + 3), n! (n + 3) | ) ---------------------------|} | / (n1 + 1)! (n1 + 4) (n1 + 3)| |----- | \n1 = 0 / "A037951" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 2 binomial(n, n/2) (n - 2) n { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n - 1) { 2 2 (n - 2) { ---------------------------------- n::odd { ------------------------------------------ n::odd { n + 3 { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A037952" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { 2 binomial(n, n/2) n { ------------------------ n::even { -------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {{ , { } { (2 n + 2) { 4 binomial(n - 1, n/2 - 1/2) n { 2 n { ------------------------------ n::odd { ------------------------------------------ n::odd { n + 1 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A037953" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 3) (n - 1) { 2 binomial(n, n/2) n (n - 4) (n - 2) { ---------------------------------------- n::even { ------------------------------------ n::even { (n + 1) (n + 3) (n + 5) binomial(n, n/2) { (n + 2) (n + 6) (n + 4) {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n - 3) (n - 1) { 2 2 (n - 4) (n - 2) { ------------------------------------------ n::odd { -------------------------------------------------- n::odd { (n + 5) (n + 3) { (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A037955" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 2 binomial(n, n/2) n { -------------------------------- n::even { -------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n - 1) { 2 2 { ---------------------------------- n::odd { ---------------------------------- n::odd { n + 3 { (n + 2) binomial(n - 1, n/2 - 1/2) "A037956" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 3) (n - 1) { 2 binomial(n, n/2) (n - 2) n { ---------------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) (n + 5) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n - 3) (n - 1) { 2 2 (n - 2) { ------------------------------------------ n::odd { ------------------------------------------ n::odd { (n + 5) (n + 3) { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A037957" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 5) (n - 3) (n - 1) { 2 binomial(n, n/2) n (n - 4) (n - 2) { ------------------------------------------------ n::even { ------------------------------------ n::even { (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) { (n + 2) (n + 6) (n + 4) {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (n - 5) (n - 3) (n - 1) { 2 2 (n - 4) (n - 2) { -------------------------------------------------- n::odd { -------------------------------------------------- n::odd { (n + 3) (n + 5) (n + 7) { (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A037958" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 7) (n - 5) (n - 3) (n - 1) { -------------------------------------------------------- n::even { (n + 1) (n + 3) (n + 5) (n + 7) (n + 9) binomial(n, n/2) {{ , { (2 n - 2) { 2 2 (n - 6) (n - 4) (n - 2) { ---------------------------------------------------------- n::odd { (n + 2) (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) { 2 binomial(n, n/2) n (n - 6) (n - 4) (n - 2) { -------------------------------------------- n::even { (n + 2) (n + 4) (n + 6) (n + 8) { } { binomial(n + 1, n/2 + 1/2) (n - 7) (n - 5) (n - 3) (n - 1) { ---------------------------------------------------------- n::odd { (n + 3) (n + 5) (n + 7) (n + 9) "A037967" 2 {binomial(2 n, n) , binomial(2 n, n)} "A038033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A038035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / /3 n1 \ \ |n - 1 |---- + 1| | |----- \ 2 / 1/2 1/2 1/2 | n n | \ 2 (n1 + 1) (2 LegendreP(n1 + 1, 2 ) - LegendreP(n1, 2 )) n1!| {(n + 1) (1/2) n!, (n + 1) (1/2) n! | ) -----------------------------------------------------------------------------|, | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / / /3 n1 \ \ |n - 1 |---- + 1| | |----- \ 2 / 1/2 1/2 1/2 | n | \ 2 (n1 + 1) (2 LegendreQ(n1 + 1, 2 ) - LegendreQ(n1, 2 )) n1!| (n + 1) (1/2) n! | ) -----------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A038112" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)} "A038151" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 3 _Z - 1, index = 1) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 2) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 3) } "A038154" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A038155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A038157" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A038158" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A038159" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A038205" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / 1/2\n1 1/2 \ |n - 1 | 2 | 2 | |----- |- ----| HermiteH(n1 + 1, ----)| | \ \ 2 / 2 | {n! | ) ---------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A038392" memory used=25421.9MB, alloc=1431.5MB, time=165.51 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" (4 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------, n { (n + 2) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-n + 2) hypergeom([-1/2, - n/2], [1], -4) { ------------------------------------------------------------------------------------------ n::even { n { , { (n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2 - 1/2], [1], -4) { -------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2 + 1) (n/2) { 5 (n + 2) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 5 (n - 2) hypergeom([3/2, - n/2], [1], 4/5) { ----------------------------------------------------------------------------------------------------------- n::even { n { , { (n/2 + 3/2) (n/2 + 1/2) { 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) { --------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2 - 1/2], [1], -4) { 1/2 -------------------------------------------------------------------------------------------------- n::even { n + 1 { , { (n + 2) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-n + 2) hypergeom([-1/2, - n/2], [1], -4) { 1/2 ------------------------------------------------------------------------------------------ n::odd { n { (n/2 + 3/2) (n/2 + 1/2) { 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) { 1/2 --------------------------------------------------------------------------------------------------------------------------- n::even { n + 1 { } { (n/2 + 1) (n/2) { 5 (n + 2) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 5 (n - 2) hypergeom([3/2, - n/2], [1], 4/5) { 1/2 ----------------------------------------------------------------------------------------------------------- n::odd { n "A038507" {1, n!} "A038602" n (2 n + 1) binomial(2 n, n) (3 n + 5) {4 , ------------------------------------} (n + 1) (n + 2) "A038679" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} (n + 1) (n + 2) "A038697" n (2 n + 3) (2 n + 1) binomial(2 n, n) {4 n, ------------------------------------} (n + 1) (n + 2) "A038806" n (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 3), ------------------------------------} n + 1 "A038836" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 5) {4 (n + 3), ------------------------------------------------------} (n + 1) (n + 2) "A039646" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" (2 n + 5) (2 n + 3) (2 n + 1) n! binomial(2 n, n) {-------------------------------------------------, n + 3 /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) (3 n1 + 10) | (2 n + 5) (2 n + 3) (2 n + 1) n! binomial(2 n, n) | ) -----------------------------------------------------------------| | / (n1 + 4) (n1 + 3) (2 n1 + 7) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / ----------------------------------------------------------------------------------------------------------------------------, (2 n + 5) (2 n + 3) n + 3 /n - 1 |----- | \ (2 n + 1) n! binomial(2 n, n) | ) (2 n1 + 1) n1! binomial(2 n1, n1) (3 n1 + 10) | / |----- \n1 = 0 /n1 - 1 \ |----- 3 2 | | \ (2 n2 + 1) n2! binomial(2 n2, n2) (55 n2 + 598 n2 + 2153 n2 + 2570) | | ) -----------------------------------------------------------------------------------------|/((n1 + 4) (n1 + 3) (2 n1 + 7) (n1 + 1)! | / (n2 + 5) (n2 + 3) (2 n2 + 7) (n2 + 1)! binomial(2 n2 + 2, n2 + 1) (3 n2 + 13) (3 n2 + 10)| |----- | \n2 = 0 / \ | | binomial(2 n1 + 2, n1 + 1))|/(n + 3)} | | / "A039647" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A039658" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 ((3 n + 4) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-3 n - 2) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) { - ---------------------------------------------------------------------------------------------------------- n::even { {1, { n + 1 , { { { { -10 hypergeom([-1/2, - n/2 - 1], [1], -4) + 10 hypergeom([-1/2, - n/2], [1], -4) n::odd { (n/2 + 3/2) (n/2 + 1/2) 2 (5 (3 n + 4) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (3 n + 2) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) - ----------------------------------------------------------------------------------------------------------------------------------- , n::even n + 1 { (n/2 + 1) (n/2) , { -10 5 hypergeom([3/2, - n/2 - 1], [1], 4/5) + 10 5 hypergeom([3/2, - n/2], [1], 4/5) , n::odd { { (n/2 + 1) (n/2) -10 5 hypergeom([3/2, - n/2 - 1], [1], 4/5) + 10 5 hypergeom([3/2, - n/2], [1], 4/5) , n::even (n/2 + 3/2) (n/2 + 1/2) 2 (5 (3 n + 4) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (3 n + 2) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) , - ----------------------------------------------------------------------------------------------------------------------------------- , n::odd n + 1 { -10 hypergeom([-1/2, - n/2 - 1], [1], -4) + 10 hypergeom([-1/2, - n/2], [1], -4) n::even { { 2 ((3 n + 4) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-3 n - 2) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) } { - ---------------------------------------------------------------------------------------------------------- n::odd { n + 1 "A039660" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 20 ((n - 3) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2], [1], -4)) { - ----------------------------------------------------------------------------------------------- n::even { n + 4 { {1, { , { { 2 { { 4 ((3 n + 5) (n - 2) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-3 n - 7 n - 10) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) { { - -------------------------------------------------------------------------------------------------------------------------- n::odd { (n + 5) (n + 1) (n/2 + 1) (n/2) 20 (5 (n - 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2], [1], 4/5)) - ---------------------------------------------------------------------------------------------------------------- , n::even n + 4 - 4 (n/2 + 3/2) (n/2 + 1/2) 2 (5 (3 n + 5) (n - 2) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (3 n + 7 n + 10) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) , /((n + 5) (n + 1)) , n::odd { 2 { 2 ((3 n + 5) (n - 2) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-3 n - 7 n - 10) hypergeom([-1/2, - n/2 - 1/2], [1], -4)) { - -------------------------------------------------------------------------------------------------------------------------- n::even { { (n + 5) (n + 1) , { { { { 10 ((n - 3) hypergeom([-1/2, - n/2 - 1], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2], [1], -4)) { { - ----------------------------------------------------------------------------------------------- n::odd { n + 4 - 2 (n/2 + 3/2) (n/2 + 1/2) 2 (5 (3 n + 5) (n - 2) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (3 n + 7 n + 10) hypergeom([3/2, - n/2 - 1/2], [1], 4/5)) /((n + 5) (n + 1)) , n::even (n/2 + 1) (n/2) 10 (5 (n - 3) hypergeom([3/2, - n/2 - 1], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2], [1], 4/5)) } - ---------------------------------------------------------------------------------------------------------------- , n::odd n + 4 "A039919" LREtools/SearchTable: "SearchTable successful" (3 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-3 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n + 2 "A041001" n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 6) (n + 3), --------------------------------------------------------} (n + 1) (n + 2) "A041005" n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 7) {4 (n + 4) (n + 3), ----------------------------------------------------------------} (n + 1) (n + 2) "A042940" n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 8) (n + 4) (n + 3), ------------------------------------------------------------------} (n + 1) (n + 2) "A042941" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 2) (n + 3), ----------------------------------------------} n + 1 "A042971" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n n (2 n + 1) binomial(2 n, n) { { 4 {2 , --------------------------, { 2 binomial(n - 1, n/2 - 1/2) n , { ------------------------ n::even} n + 1 { ------------------------------ n::odd { (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A042985" n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 4) (n + 3) (n + 2), --------------------------------------------------------} n + 1 "A043301" LREtools/SearchTable: "SearchTable successful" n n {(-2) BesselI(n + 1/2, 2), (-2) BesselK(n + 1/2, -2)} "A045406" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ / (n1 + 1) n1!\| (-1) n! | ) |- ------------|| | / \ n1 (n1 + 1)!/| n |----- | (-1) n! \n1 = 0 / {--------, ----------------------------------} n n "A045445" LREtools/SearchTable: "SearchTable successful" 2 (4 n - 2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) - (2 n - 1) (2 n + 1) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A045492" n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 4) (n + 3) (n + 2), ------------------------------------------------------------------} n + 1 "A045505" n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 5) (n + 4) (n + 3) (n + 2), ------------------------------------------------------------------} n + 1 "A045530" memory used=25748.0MB, alloc=1431.5MB, time=167.88 n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 5) (n + 4) (n + 3) (n + 2), -----------------------------------------------------------------------------} n + 1 "A045621" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A045622" n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 6) (n + 5) (n + 4) (n + 3) (n + 2), -----------------------------------------------------------------------------} n + 1 "A045635" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (4 n - 11 n + 15 n - 6) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n + 9 n - 9 n - 6) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A045720" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 5) {4 , ------------------------------------------------------} (n + 3) (n + 2) (n + 1) "A045723" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {2 , binomial(2 n, n), { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A045724" n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (n + 2) (n + 3), --------------------------------------------------------} n + 1 "A045739" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A045741" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 (3 n + 2) binomial(---, n/2) { 2 { 1/2 ------------------------------- n::even { n + 1 {{ , { (2 n - 2) 3 n { 3 2 (3 n - 1) (3 n + 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - --------------------------------------------------------------- n::odd { n (n + 1) { 3 n 3 n { 6 (3 n + 1) binomial(---, n/2) binomial(3 n, ---) { 2 2 { - ------------------------------------------------- n::even { (n + 1) binomial(n, n/2) { } { 3 n 3 n { binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { ----------------------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) "A045742" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 (3 n + 2) (5 n + 6) binomial(---, n/2) { 2 { 1/2 ----------------------------------------- n::even { (n + 1) (n + 2) {{ , { (2 n - 2) 3 n { 3 2 (3 n - 1) (3 n + 1) (3 n + 4) binomial(--- - 3/2, n/2 - 1/2) { 2 { - ------------------------------------------------------------------------- n::odd { n (n + 1) (n + 2) { 3 n 3 n { 6 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) (3 n + 4) { 2 2 { - ----------------------------------------------------------- n::even { binomial(n, n/2) (n + 1) (n + 2) { } { 3 n 3 n { (5 n + 6) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { --------------------------------------------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) "A045743" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A045744" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A045829" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 (2 n - 13 n + 39 n - 62 n + 25) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n + 12 n - 32 n + 42 n + 25) hypergeom([-1/2, -n], [1], -4) {------------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A045868" LREtools/SearchTable: "SearchTable successful" 2 2 (8 n - n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 3 n + 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------------------} n (n + 2) "A045890" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (8 n + 3 n + 8 n - 2) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7 n - 8 n - 2) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n "A045894" n (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 5) {4 (n + 11), ----------------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A045902" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 (32 n + 40 n + 105 n - 35 n + 18) hypergeom([-1/2, -n - 1], [1], -4) + (-32 n - 56 n - 119 n - 11 n + 18) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------------------------------------------------------------} n (n + 4) (n + 3) (n + 2) "A045952" n n {16 , 4 binomial(2 n, n)} "A045992" {n, binomial(2 n, n)} "A045994" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 3 _Z - 1, index = 1) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 2) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 3) } "A046032" 2 2 {1, (n + 1) (n!) } "A046033" 3 3 {1, (n + 1) (n!) } "A046126" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=26092.0MB, alloc=1431.5MB, time=170.34 { (n/2) 2 { (-4) ((n/2)!) (n + 2) { --------------------------- n::even { (n/2) 2 2 { n { (-1/4) binomial(n, n/2) ((n/2)!) (n + 1) n::even {{ , { } { (n/2 + 1/2) 2 { (n/2 - 1/2) 2 2 { (-4) ((n/2 + 1/2)!) { (n + 2) n (-1/4) ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) n::odd { ------------------------------- n::odd { n + 1 "A046212" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n n { 0 n::even { 4 {1, (-1) , { , { ---------------- n::even} { n binomial(n - 1, n/2 - 1/2) n::odd { binomial(n, n/2) { { 0 n::odd "A046222" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { binomial(n, n/2) (5 n - 8) n { { 1/2 -------------------------- n::even {1, (-1) , { (2 n - 2) , { n - 1 } { 2 2 (5 n - 8) { { ------------------------------------ n::odd { 0 n::odd { n (n - 1) binomial(n - 1, n/2 - 1/2) "A046223" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n n { { 4 4 (5 n - 13) {1, (-1) , { binomial(n - 1, n/2 - 1/2) (5 n - 13) , { -------------------------- n::even} { 1/2 ------------------------------------- n::odd { n (n - 2) binomial(n, n/2) { n - 2 { { 0 n::odd "A046662" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! (n BesselJ(n, -2) + BesselJ(n - 1, -2)), (-1) n! (n BesselY(n, -2) + BesselY(n - 1, -2))} "A046714" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 5 (2 n1 + 1) binomial(2 n1, n1)| {5 , 5 | ) ----------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A046736" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A046748" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 5 binomial(2 n1, n1)| {5 , 5 | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A046814" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 5 binomial(2 n1, n1)| {5 , 5 | ) -----------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A046885" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 5 (2 n1 + 1) binomial(2 n1, n1)| {5 , 5 | ) ----------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A046979" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n n n n 1/2 n 1/2 n | 2 | |2 | {1, (-1) , (-I) , I , (-1/2 I 2 ) n!, (1/2 I 2 ) n!, |- ----| n!, |----| n!} \ 2 / \ 2 / "A046981" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n n n n 1/2 n 1/2 n | 2 | |2 | {1, (-1) , (-I) , I , (-1/2 I 2 ) n!, (1/2 I 2 ) n!, |- ----| n!, |----| n!} \ 2 / \ 2 / "A046996" LREtools/SearchTable: "SearchTable successful" 4 3 2 {((171 n + 1540 n + 5017 n + 7016 n + 3584) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) 2 / 2 2 2 + 8 (7 n + 23) (3 n + 8) (n + 1) hypergeom([-n, -n, -n], [1, 1], -1)) / ((n + 5) (n + 4) (n + 3) (n + 2) )} / "A047001" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / /{ 0 irem(n1, 3) = 0\\ | |{ || | |{ / n1 \ || | |{ |---- - 1/3| || | |{ \ 3 / n1 n1 || | |{ 8 (2 n1 + 3) (27/4) GAMMA(---- + 2/3) GAMMA(---- + 1/3) || | |{ 3 3 || |n - 1 |{ ------------------------------------------------------------------- irem(n1, 3) = 1|| |----- |{ n1 n1 || n n | \ (-n1 - 1) |{ (n1 - 1) GAMMA(---- + 3/2) GAMMA(---- + 1) || {2 , 2 | ) 2 |{ 3 3 ||, | / |{ || |----- |{ / n1 \ || |n1 = 0 |{ |---- - 2/3| || | |{ \ 3 / n1 n1 || | |{ 9 (27/4) GAMMA(---- + 1/3) GAMMA(----) || | |{ 3 3 || | |{ -------------------------------------------------- irem(n1, 3) = 2|| | |{ n1 n1 || | |{ GAMMA(---- + 7/6) GAMMA(---- + 2/3) || \ \{ 3 3 // / /{ n1 \\ | |{ 24 binomial(n1, ----) || | |{ 3 || |n - 1 |{ --------------------- irem(n1, 3) = 0|| |----- |{ n1 - 1 || n | \ (-n1 - 1) |{ || 2 | ) 2 |{ n1 ||, | / |{ 27 binomial(n1 - 1, ---- - 1/3) || |----- |{ 3 || |n1 = 0 |{ ------------------------------- irem(n1, 3) = 1|| | |{ 2 n1 + 1 || | |{ || \ \{ 0 irem(n1, 3) = 2// / /{ / n1 \ \\ | |{ |----| || | |{ \ 3 / n1 n1 || | |{ 9 (27/4) GAMMA(----) GAMMA(---- + 1/3) || | |{ 3 3 || | |{ -------------------------------------------- irem(n1, 3) = 0|| | |{ n1 n1 || |n - 1 |{ GAMMA(---- + 2/3) GAMMA(---- + 7/6) || |----- |{ 3 3 || n | \ (-n1 - 1) |{ || 2 | ) 2 |{ 0 irem(n1, 3) = 1||} | / |{ || |----- |{ / n1 \ || |n1 = 0 |{ |---- + 1/3| || | |{ \ 3 / n1 n1 || | |{ 8 (2 n1 + 3) (27/4) GAMMA(---- + 1/3) GAMMA(---- + 2/3) || | |{ 3 3 || | |{ ------------------------------------------------------------------- irem(n1, 3) = 2|| | |{ n1 n1 || | |{ (n1 - 1) GAMMA(---- + 1) GAMMA(---- + 3/2) || \ \{ 3 3 // "A047002" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=26416.6MB, alloc=1463.5MB, time=172.71 LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (4 (2 n + 1) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) - (9 n + 5) n hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(n (5 n + 3)), binomial(2 n, n)} "A047073" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n binomial(n, n/2) { 2 4 { ------------------ n::even { ------------------ n::even { n - 1 { n binomial(n, n/2) {{ , { } { (n + 1) binomial(n + 1, n/2 + 1/2) { (2 n - 2) { 1/2 ---------------------------------- n::odd { 4 2 { n { ---------------------------------- n::odd { (n - 1) binomial(n - 1, n/2 - 1/2) "A047074" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n binomial(n, n/2) { 2 4 { ------------------ n::even { ------------------------ n::even { n - 1 { (n - 2) binomial(n, n/2) {{ , { } { (n + 1) binomial(n + 1, n/2 + 1/2) { (2 n - 2) { 1/2 ---------------------------------- n::odd { 4 2 { n - 2 { ---------------------------------- n::odd { (n - 1) binomial(n - 1, n/2 - 1/2) "A047079" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / 1/2\n / 1/2 \n / 1/2\n | 5 | |5 | | 5 | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| \ 2 / \ 2 / \ 2 / / / / /{ 0 irem(n2, 3) = 0\\\\ | | | |{ |||| | | | |{ /2 n2 \ |||| | | | |{ |---- - 2/3| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 GAMMA(---- - 1/6) |||| | | | |{ 3 |||| |n - 1 | |n1 - 1 |{ ------------------------------- irem(n2, 3) = 1|||| |----- | |----- / 1/2 \(-n2 - 1) |{ n2 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ GAMMA(---- + 4/3) |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 3 ||||, | / | | / \ 2 / |{ |||| |----- | |----- |{ /2 n2 \ |||| |n1 = 0 | |n2 = 0 |{ |---- - 4/3| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 GAMMA(---- - 1/2) |||| | | | |{ 3 |||| | | | |{ ------------------------------- irem(n2, 3) = 2|||| | | | |{ n2 |||| | | | |{ GAMMA(---- + 1) |||| \ \ \ \{ 3 //// / 1/2\n | 5 | |1/2 - ----| \ 2 / / / / /{ /2 n2\ \\\\ | | | |{ |----| |||| | | | |{ \ 3 / n2 |||| | | | |{ 2 GAMMA(---- - 1/6) |||| | | | |{ 3 |||| | | | |{ ------------------------- irem(n2, 3) = 0|||| | | | |{ n2 |||| |n - 1 | |n1 - 1 |{ GAMMA(---- + 4/3) |||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ /2 n2 \ ||||, | / | | / \ 2 / |{ |---- - 2/3| |||| |----- | |----- |{ \ 3 / n2 |||| |n1 = 0 | |n2 = 0 |{ 2 GAMMA(---- - 1/2) |||| | | | |{ 3 |||| | | | |{ ------------------------------- irem(n2, 3) = 1|||| | | | |{ n2 |||| | | | |{ GAMMA(---- + 1) |||| | | | |{ 3 |||| | | | |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// / | | | / 1/2\n | | 5 | | |1/2 - ----| | \ 2 / | | | | | \ / / /{ 2 n2 n2 \\\\ | | |{ 3 binomial(----, ----) |||| | | |{ 3 3 |||| n - 1 | |n1 - 1 |{ ---------------------- irem(n2, 3) = 0|||| ----- | |----- / 1/2 \(-n2 - 1) |{ 2 n2 - 3 |||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 0 irem(n2, 3) = 1|||| / | | / \ 2 / |{ |||| ----- | |----- |{ 2 n2 n2 |||| n1 = 0 | |n2 = 0 |{ 3 binomial(---- + 2/3, ---- + 1/3) |||| | | |{ 3 3 |||| | | |{ ---------------------------------- irem(n2, 3) = 2|||| \ \ \{ 2 n2 - 1 //// } "A047085" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {(-1) , -----------------------------------------------------------------------------------------} n "A047086" LREtools/SearchTable: "SearchTable successful" n (-1) ((5 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (n - 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n "A047087" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) (hypergeom([1/2, -n - 1], [1], 4) + (4 n + 5) hypergeom([1/2, -n], [1], 4)) {(-1) , ---------------------------------------------------------------------------------} n + 2 "A047088" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((5 n + 16 n + 15) hypergeom([1/2, -n - 1], [1], 4) + (n + 8 n + 3) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A047098" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 binomial(3 n1, n1) (5 n1 + 3)| {8 , 8 | ) ------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A047099" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 8)| {8 , 8 | ) ----------------------------------------------------------------|} | / (n1 + 2) (n1 + 1) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / "A047171" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { 2 binomial(n, n/2) n { ------------------------ n::even { -------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {1, { , { } { (2 n + 2) { 4 binomial(n - 1, n/2 - 1/2) n { 2 n { ------------------------------ n::odd { ------------------------------------------ n::odd { n + 1 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A047182" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 4 binomial(n, n/2) (n + 1) n { 1/2 ------------------------ n::even { ---------------------------- n::even { (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {1, { , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------------ n::odd { n + 3 { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A047193" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) { (27/4) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (2 n + 7) { ----------------------------------------------------- irem(n, 3) = 0 { GAMMA(5/3 + n/3) GAMMA(n/3 + 13/6) (n + 3) { 27 binomial(n, n/3) (n + 1) { { --------------------------- irem(n, 3) = 0 { (n/3 - 1/3) { 2 n + 3 {1, { 3 (27/4) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) (2 n + 5) , { , { ------------------------------------------------------------- irem(n, 3) = 1 { 3 binomial(n + 2, n/3 + 2/3) irem(n, 3) = 1 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 11/6) (n + 2) { { { 9 binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 { (n/3 - 2/3) { 9 (27/4) GAMMA(n/3 + 4/3) GAMMA(n/3 + 2/3) { ----------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 3/2) GAMMA(n/3 + 1) { (n/3) { 3 (27/4) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) (2 n + 5) { ------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 11/6) (n + 2) { { (n/3 - 1/3) { 9 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) } { ----------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 3/2) GAMMA(n/3 + 1) { { (n/3 + 1/3) { (27/4) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (2 n + 7) { ----------------------------------------------------------- irem(n, 3) = 2 { GAMMA(5/3 + n/3) GAMMA(n/3 + 13/6) (n + 3) "A047665" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, LegendreP(n, 3), LegendreQ(n, 3)} "A047749" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { 3 binomial(---, n/2) { 2 { -------------------- n::even { n + 1 {{ , { 3 n { 4 binomial(--- + 3/2, n/2 + 1/2) { 2 { -------------------------------- n::odd { 3 n + 1 { (-n) 3 n 3 n { 8 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { (-2 n + 2) 3 n 3 n { 6 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { n (n + 1) binomial(n - 1, n/2 - 1/2) "A047750" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (5 n + 2) { 2 { ---------------------------- n::even { (n + 1) (n + 2) {{ , { 3 n { binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { 3/2 -------------------------------------------------- n::odd { (n + 2) (n + 1) n { (-n) 3 n 3 n { 6 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 1) (n + 2) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 4 2 (5 n + 2) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { (n + 2) (3 n + 2) binomial(n + 1, n/2 + 1/2) "A047781" LREtools/SearchTable: "SearchTable successful" (7 n + 3) LegendreP(n, 3) + (-n - 1) LegendreP(n + 1, 3) (7 n + 3) LegendreQ(n, 3) + (-n - 1) LegendreQ(n + 1, 3) {- --------------------------------------------------------, - --------------------------------------------------------} n n "A047865" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n!} "A047889" memory used=26829.5MB, alloc=1495.5MB, time=175.51 LREtools/SearchTable: "SearchTable successful" 4 3 2 {((2 n + 1) (4 n + 52 n + 263 n + 549 n + 388) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1) 3 2 2 / 3 3 - 16 (2 n + 21 n + 76 n + 89) (n + 1) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)) binomial(2 n, n) / ((n + 1) (n + 2) (n + 3) / (n + 4))} "A047890" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A047891" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 2) - 2 LegendreP(n, 2)) 2 (LegendreQ(n + 1, 2) - 2 LegendreQ(n, 2)) {--------------------------------------------, --------------------------------------------} n n "A047904" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A047905" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A047906" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A047907" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A047908" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/3) { (n/3) { 3 (n/3)! irem(n, 3) = 0 { 3 GAMMA(n/3 + 4/3) { { ----------------------- irem(n, 3) = 0 { (n/3 + 2/3) { n + 1 { 3 (n/3 + 2/3)! { {{ ------------------------- irem(n, 3) = 1, { (n/3 - 1/3) , { n + 2 { 3 GAMMA(n/3 + 1) irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 + 1/3) { 3 (n/3 + 1/3)! { 3 GAMMA(5/3 + n/3) { ------------------------- irem(n, 3) = 2 { ----------------------------- irem(n, 3) = 2 { n + 1 { n + 2 { (n/3) { 3 GAMMA(5/3 + n/3) { ----------------------- irem(n, 3) = 0 { n + 2 { { (n/3 - 1/3) } { 3 GAMMA(n/3 + 4/3) { ----------------------------- irem(n, 3) = 1 { n + 1 { { (n/3 - 2/3) { 3 GAMMA(n/3 + 1) irem(n, 3) = 2 "A047910" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A047911" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 11 n1 + 4)| (3 n + 1) (3 n + 2) binomial(2 n, n) binomial(3 n, n) {8 , 8 | ) -------------------------------------------------------------------------|, -----------------------------------------------------} | / (n1 + 2) (n1 + 1) (2 n1 + 3) (2 n1 + 1) | 2 |----- | (n + 1) \n1 = 0 / "A047974" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, 1/2 I)} "A047990" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 2 GAMMA(n/2 + 1 + 1/2 I) GAMMA(n/2 + 1/2 - 1/2 I) n::even { 2 GAMMA(n/2 + 1 - 1/2 I) GAMMA(n/2 + 3/2 + 1/2 I) { { -------------------------------------------------- n::even {{ (n + 1) , { n + 1 + I { 2 GAMMA(n/2 + 1 - 1/2 I) GAMMA(n/2 + 3/2 + 1/2 I) { { -------------------------------------------------------- n::odd { (n - 1) { n + 1 + I { 2 GAMMA(n/2 + 1 + 1/2 I) GAMMA(n/2 + 1/2 - 1/2 I) n::odd } "A048006" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 memory used=27209.4MB, alloc=1495.5MB, time=178.15 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=27552.4MB, alloc=1495.5MB, time=180.53 memory used=27823.2MB, alloc=1495.5MB, time=182.12 memory used=28095.3MB, alloc=1495.5MB, time=183.69 memory used=28367.1MB, alloc=1495.5MB, time=185.24 memory used=28639.1MB, alloc=1495.5MB, time=186.81 memory used=28910.3MB, alloc=1495.5MB, time=188.38 memory used=29181.9MB, alloc=1495.5MB, time=189.91 memory used=29453.9MB, alloc=1495.5MB, time=191.43 memory used=29726.2MB, alloc=1495.5MB, time=192.99 { {1, { - 36 (1/9 (2 n + 15) (n + 6) hypergeom([- n/3 - 1, - n/3 - 2, - n/3 - 3/2], [1, 3/2], -1) { 2 + (-22/9 n - 70/3 n - 54) hypergeom([- n/3, - n/3 - 1, - n/3 - 1/2], [1, 3/2], -1))/(n + 6) , irem(n, 3) = 0 / - 54 |1/27 (5 n + 22) (n + 8) (2 n + 19) hypergeom([- n/3 - 5/3, - n/3 - 8/3, - n/3 - 13/6], [1, 3/2], -1) \ / 86 3 2 15628\ \ + |- -- n - 502/9 n - 2846/9 n - -----| hypergeom([- n/3 - 2/3, - n/3 - 5/3, - n/3 - 7/6], [1, 3/2], -1)|/((n + 8) (2 n + 7)) , irem(n, 3) = 1 \ 27 27 / / / - 54 |1/27 (n + 7) (5 n + 14) (2 n + 17) hypergeom([- n/3 - 4/3, - n/3 - 7/3, - n/3 - 11/6], [1, 3/2], -1) \ / 86 3 2 6356\ \ + |- -- n - 394/9 n - 186 n - ----| hypergeom([- n/3 - 1/3, - n/3 - 4/3, - n/3 - 5/6], [1, 3/2], -1)|/((n + 1) (n + 7)) , irem(n, 3) = 2} \ 27 27 / / "A048060" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 4 binomial(n, n/2) n (n + 1) (n - 2) { 1/2 -------------------------------- n::even { ------------------------------------ n::even { (n + 3) (n + 5) binomial(n, n/2) { (n + 2) (n + 6) (n + 4) {1, { , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n - 1) (n + 1) { 2 (n - 2) (n + 1) { -------------------------------------------- n::odd { -------------------------------------------------- n::odd { (n + 5) (n + 3) { (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A048775" (2 n + 3) (2 n + 1) binomial(2 n, n) {------------------------------------, n + 2} (n + 1) (n + 2) "A049027" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(9/2) , (9/2) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (9/2) | \n1 = 0 / "A049034" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 2 | \ (n1 + 1) (2 n1 + 1) (n1!) binomial(2 n1, n1) | {(2 n + 1) (n!) binomial(2 n, n), (2 n + 1) (n!) binomial(2 n, n) | ) --------------------------------------------------|} | / 2 | |----- (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A049122" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ | | |----| / 1/2 1/2 \|| |n - 1 | n1 \ 2 / (n1 + 1) | 1/2 5 5 ||| |----- | (-1) 5 2 |-5 LegendreP(n1 + 1, ----) + 5 LegendreP(n1, ----)||| n n | \ | \ 5 5 /|| {(1/2) , (1/2) | ) |- --------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ | | |----| / 1/2 1/2 \|| |n - 1 | n1 \ 2 / (n1 + 1) | 1/2 5 5 ||| |----- | (-1) 5 2 |-5 LegendreQ(n1 + 1, ----) + 5 LegendreQ(n1, ----)||| n | \ | \ 5 5 /|| (1/2) | ) |- --------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A049125" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049128" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A049130" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049133" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A049140" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049171" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049235" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (27 n1 + 135 n1 + 160)| {(27/4) , (27/4) | ) ----------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (27/4) | \n1 = 0 / "A049376" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049377" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049378" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A049401" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049402" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A049412" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A049425" memory used=30118.5MB, alloc=1495.5MB, time=195.57 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A049427" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A049428" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A049607" LREtools/SearchTable: "SearchTable successful" (4 n + 3) LegendreP(n, 3) - LegendreP(n + 1, 3) (4 n + 3) LegendreQ(n, 3) - LegendreQ(n + 1, 3) {- -----------------------------------------------, - -----------------------------------------------} n n "A049608" LREtools/SearchTable: "SearchTable successful" (n + 1) ((n + 3) LegendreP(n, 3) + (n - 1) LegendreP(n + 1, 3)) (n + 1) ((n + 3) LegendreQ(n, 3) + (n - 1) LegendreQ(n + 1, 3)) {---------------------------------------------------------------, ---------------------------------------------------------------} (n + 2) n (n + 2) n "A049609" LREtools/SearchTable: "SearchTable successful" 2 3 2 2 3 2 (2 n + 3) LegendreP(n, 3) + (4 n + 8 n - 3) LegendreP(n + 1, 3) (2 n + 3) LegendreQ(n, 3) + (4 n + 8 n - 3) LegendreQ(n + 1, 3) {------------------------------------------------------------------, ------------------------------------------------------------------} (n + 3) (n + 2) n (n + 3) (n + 2) n "A050146" LREtools/SearchTable: "SearchTable successful" (17 n + 9) LegendreP(n, 3) + (-3 n - 3) LegendreP(n + 1, 3) (17 n + 9) LegendreQ(n, 3) + (-3 n - 3) LegendreQ(n + 1, 3) {- -----------------------------------------------------------, - -----------------------------------------------------------} n - 1 n - 1 "A050147" LREtools/SearchTable: "SearchTable successful" (n + 1) ((10 n + 3) LegendreP(n, 3) + (-2 n - 1) LegendreP(n + 1, 3)) (n + 1) ((10 n + 3) LegendreQ(n, 3) + (-2 n - 1) LegendreQ(n + 1, 3)) {- ---------------------------------------------------------------------, - ---------------------------------------------------------------------} (n - 1) n (n - 1) n "A050148" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (3 n + 7 n + 3) LegendreP(n, 3) + (-n - n - 1) LegendreP(n + 1, 3) (3 n + 7 n + 3) LegendreQ(n, 3) + (-n - n - 1) LegendreQ(n + 1, 3) {- --------------------------------------------------------------------, - --------------------------------------------------------------------} (n - 1) n (n - 1) n "A050149" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (2 n + 10 n + 18 n + 9) LegendreP(n, 3) + (-2 n - 2 n - 2 n - 3) LegendreP(n + 1, 3) {- ----------------------------------------------------------------------------------------, (n - 1) (n + 2) n 3 2 3 2 (2 n + 10 n + 18 n + 9) LegendreQ(n, 3) + (-2 n - 2 n - 2 n - 3) LegendreQ(n + 1, 3) - ----------------------------------------------------------------------------------------} (n - 1) (n + 2) n "A050151" LREtools/SearchTable: "SearchTable successful" {-3 LegendreP(n, 3) + LegendreP(n + 1, 3), -3 LegendreQ(n, 3) + LegendreQ(n + 1, 3)} "A050152" LREtools/SearchTable: "SearchTable successful" (2 n + 3) LegendreP(n, 3) + (-2 n - 1) LegendreP(n + 1, 3) (2 n + 3) LegendreQ(n, 3) + (-2 n - 1) LegendreQ(n + 1, 3) {- ----------------------------------------------------------, - ----------------------------------------------------------} n + 2 n + 2 "A050168" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (3 n + 1) { -------------------------- n::even { 6 binomial(n, n/2) n::even { n (n + 1) binomial(n, n/2) { {{ , { binomial(n + 1, n/2 + 1/2) (3 n + 1) } { (2 n - 2) { ------------------------------------ n::odd { 6 2 { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A050253" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A050262" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" n {(-1) } "A050511" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) hypergeom([-1/2, -n - 1], [1], -4)} "A050984" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([-2 n, -2 n, -2 n, -2 n, -2 n], [1, 1, 1, 1], 1)} "A051033" memory used=30519.2MB, alloc=1495.5MB, time=198.32 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) { 3 (27/4) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { --------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) { { (n/3 - 1/3) {{ 9 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1/3) , { ----------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) { { (n/3 + 1/3) { 2 (n + 2) (27/4) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) { ----------------------------------------------------------- irem(n, 3) = 2 { (n + 1) GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) { (n/3) { 2 (n + 2) (27/4) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { ----------------------------------------------------- irem(n, 3) = 0 { (n + 1) GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) { 9 binomial(n, n/3) irem(n, 3) = 0 { { { (n/3 - 1/3) { 2 (n + 2) binomial(n + 2, n/3 + 2/3) { 3 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1) , { ------------------------------------ irem(n, 3) = 1} { --------------------------------------------------- irem(n, 3) = 1 { n + 1 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) { { { 3 binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 { (n/3 - 2/3) { 9 (27/4) GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { ----------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) "A051195" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 binomial(n, n/2) (n + 1) (2 n + 1) binomial(2 n, n) { { -------------------------- n::even {--------------------------, { (2 n - 2) , { n + 2 } n + 2 { 2 (n + 1) { { 1/2 ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A051286" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A051291" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A051292" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A051396" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ (2 n1 + 1) (n1 + 1) | (n!) binomial(2 n, n) | ) ---------------------------------------| | / 2 | 2 |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| (n!) binomial(2 n, n) \n1 = 0 / {----------------------, -----------------------------------------------------------------------} (2 n - 1) n (2 n - 1) n "A051397" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1 + 1 | (n!) binomial(2 n, n) | ) ---------------------------------------| | / 2 | 2 |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| (n!) binomial(2 n, n) \n1 = 0 / {----------------------, -----------------------------------------------------------------------} n n "A051398" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2\| n n | \ | (-1) (n1 + 2) || {(-1) n!, (-1) n! | ) |- ----------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A051403" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {(n + 1) | ) n1!|, n + 1} | / | |----- | \n1 = 0 / "A051524" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / "A051545" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / "A051560" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 8) (n1 + 1)!| |----- | \n1 = 0 / "A051562" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- | | \ (n1 + 1) n1! | (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 9) (n1 + 1)!| |----- | \n1 = 0 / "A051564" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- | | \ (n1 + 1) n1! | (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) -------------------|} | / (n1 + 10) (n1 + 1)!| |----- | \n1 = 0 / "A051684" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 (- n/2) 2 {1, 2 HermiteH(n + 1, ----)} 2 "A051708" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (16 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-16 n - 23 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------------------------} n "A051789" 2 binomial(2 n, n) binomial(2 n, n) {-----------------, ----------------} 2 n + 1 (n + 1) "A051920" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A052124" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {n! (n + 5) (n + 2), n! (n + 5) (n + 2) | ) ---------------------------------------------|} | / (n1 + 1)! (n1 + 6) (n1 + 3) (n1 + 5) (n1 + 2)| |----- | \n1 = 0 / "A052127" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) (-2) n1! | {(n!) (n + 5) (n + 2), (n!) (n + 5) (n + 2) | ) ------------------------------------------------|} | / 2 | |----- ((n1 + 1)!) (n1 + 6) (n1 + 3) (n1 + 5) (n1 + 2)| \n1 = 0 / "A052141" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 3), 2 LegendreQ(n, 3)} "A052143" LREtools/SearchTable: "SearchTable successful" n {(-4) ((4 n + 4) LaguerreL(n + 1, -n - 1/2, 1/4) + LaguerreL(n, -n + 1/2, 1/4)) n!} "A052169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! (n + 2), n! (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2)| |----- | \n1 = 0 / "A052177" LREtools/SearchTable: "SearchTable successful" n 2 2 ((2 n - 3 n + 2) hypergeom([-1/2, -n - 1], [1], -2) - 2 (n - 1) n hypergeom([-1/2, -n], [1], -2)) {-----------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A052201" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {(n + 2) (n + 4), (n + 2) (n + 4) | ) (n1 + 1) n1!|} | / | |----- | \n1 = 0 / "A052225" {1, (n + 2) (n + 1) n! (n + 4)} "A052392" LREtools/SearchTable: "SearchTable successful" n n 2 ((n - 1) LegendreP(n + 1, 2) + (n + 2) LegendreP(n, 2)) 2 ((n - 1) LegendreQ(n + 1, 2) + (n + 2) LegendreQ(n, 2)) {----------------------------------------------------------, ----------------------------------------------------------} n n "A052399" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052473" (n - 1) n binomial(2 n, n) {1, --------------------------} (2 n - 1) (2 n - 3) "A052501" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A052503" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) n! binomial(2 n, n) HermiteH(n, 1/2 I)} "A052517" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1)| n! | ) ------------| | / n1 (n1 + 1)!| |----- | n! \n1 = 0 / {----, ------------------------} n n "A052518" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1) | n! | ) ------------------| | / (n1 - 1) (n1 + 1)!| |----- | n! \n1 = 0 / {----, ------------------------------} n n "A052519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1) | n! | ) ------------------| | / (n1 - 2) (n1 + 1)!| |----- | n! \n1 = 0 / {----, ------------------------------} n n "A052520" {n! (n - 3)} "A052521" {n! (n - 5)} "A052524" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052554" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A052555" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 + 2 ) n!, (-2 + 1) n!} "A052556" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(RootOf(_Z + _Z + 1, index = 1) + 1) n!, (RootOf(_Z + _Z + 1, index = 2) + 1) n!, (RootOf(_Z + _Z + 1, index = 3) + 1) n!} "A052558" n {(-1) n!, n! (2 n + 3)} "A052559" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(-RootOf(_Z + 2 _Z - _Z - 1, index = 1)) n!, (-RootOf(_Z + 2 _Z - _Z - 1, index = 2)) n!, (-RootOf(_Z + 2 _Z - _Z - 1, index = 3)) n!} "A052561" n {2 n!, n!} "A052566" n {(-1) n!, n!} "A052567" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| n!, |3/2 + ----| n!} \ 2 / \ 2 / "A052568" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| n!, |3/2 + ----| n!} \ 2 / \ 2 / "A052571" {n! (n - 2)} "A052573" n {3 n!, n!} "A052574" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| n!, |3/2 + ----| n!} \ 2 / \ 2 / "A052577" n {3 n!, n!} "A052579" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) n!, (- 1/2 + 1/2 I 3 ) n!, n!} "A052580" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 + 2 ) n!, (-2 + 1) n!} "A052583" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A052584" n {2 n!, n!} "A052585" n n {(-1) n!, 2 n!} "A052589" n {2 n!, n!} "A052590" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 2 ) n!, (2 + 2 ) n!} "A052591" n {(-1) n!, n! (2 n + 1)} "A052595" memory used=30900.3MB, alloc=1495.5MB, time=201.12 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | {|3/2 - -----| n!, |3/2 + -----| n!} \ 2 / \ 2 / "A052598" n n {(-1) n!, 2 n!} "A052600" n n {(-1) n!, 2 n!, n!} "A052606" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 3 ) n!, (2 + 3 ) n!} "A052608" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 - 2 ) n!, (1 + 2 ) n!} "A052609" {n! (n - 1)} "A052611" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 + 3 ) n!, (-3 + 1) n!} "A052612" n {(-1) n!, n!} "A052616" n {(-1) n!, n!} "A052617" n {2 n!, n!} "A052618" n 2 {(-1) n!, n! (2 n + 8 n + 7)} "A052622" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 - 2 ) n!, (1 + 2 ) n!} "A052626" n {2 n!, n!} "A052629" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | {|5/2 - -----| n!, |5/2 + -----| n!} \ 2 / \ 2 / "A052630" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 5 ) n!, (2 + 5 ) n!} "A052631" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 - 2 ) n!, (1 + 2 ) n!} "A052634" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 ) n!, (-2 ) n!, n!} "A052635" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | {|3/2 - -----| n!, |3/2 + -----| n!} \ 2 / \ 2 / "A052650" n {2 n!, n! (n + 3)} "A052651" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n n n n {(-RootOf(%1, index = 1)) n!, (-RootOf(%1, index = 2)) n!, (-RootOf(%1, index = 3)) n!, (-RootOf(%1, index = 4)) n!, (-RootOf(%1, index = 5)) n! } 5 4 2 %1 := _Z + _Z + _Z - _Z - 1 "A052653" n n {(-1) n!, 2 n!} "A052654" n {4 n!, n!} "A052657" n {(-1) n!, n! (2 n - 1)} "A052658" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(-RootOf(_Z + 2 _Z - _Z - 1, index = 1)) n!, (-RootOf(_Z + 2 _Z - _Z - 1, index = 2)) n!, (-RootOf(_Z + 2 _Z - _Z - 1, index = 3)) n!} "A052659" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 2 ) n!, (2 + 2 ) n!} "A052665" n {2 n!, n!} "A052666" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | {|1/2 - -----| n!, |1/2 + -----| n!} \ 2 / \ 2 / "A052677" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 3 ) n!, (2 + 3 ) n!} "A052680" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 2 ) n!, (2 + 2 ) n!} "A052682" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | {|1/2 - -----| n!, |1/2 + -----| n!} \ 2 / \ 2 / "A052687" n {(-1) n!, n! (2 n + 7)} "A052689" n {(-1) n!, n! (2 n + 5)} "A052695" n {4 n!, n!} "A052698" n {3 n!, n!} "A052702" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052705" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (2 I) (LegendreP(n1 + 1, I) I + 3 LegendreP(n1, I)) I| {(-1) (n + 3), (-1) (n + 3) | ) -------------------------------------------------------|, | / (n1 + 3) (n1 + 4) | |----- | \n1 = 0 / /n - 1 \ |----- n1 | n | \ (2 I) (LegendreQ(n1 + 1, I) I + 3 LegendreQ(n1, I)) I| (-1) (n + 3) | ) -------------------------------------------------------|} | / (n1 + 3) (n1 + 4) | |----- | \n1 = 0 / "A052706" LREtools/SearchTable: "SearchTable successful" 2 {(10 (2 n + 1) (19 n - 9) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (-553 n - 11 n + 114) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((3 n - 2) (3 n - 1) (3 n + 1))} "A052709" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ (2 I) (LegendreP(n1, I) + LegendreP(n1 + 1, I) I) I| n | \ (2 I) (LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I) I| {(-1) , (-1) | ) -----------------------------------------------------|, (-1) | ) -----------------------------------------------------|} | / n1 | | / n1 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A052716" LREtools/SearchTable: "SearchTable successful" n! ((17 n + 9) LegendreP(n, 3) + (-3 n - 3) LegendreP(n + 1, 3)) n! ((17 n + 9) LegendreQ(n, 3) + (-3 n - 3) LegendreQ(n + 1, 3)) {- ----------------------------------------------------------------, - ----------------------------------------------------------------} (n - 1) n (n - 1) n "A052719" n! binomial(2 n, n) (n - 2) {---------------------------} (2 n - 1) (2 n - 3) "A052727" LREtools/SearchTable: "SearchTable successful" n n (-2 I) n! ((3 n + 1) LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (-2 I) n! ((3 n + 1) LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- ----------------------------------------------------------------------, - ----------------------------------------------------------------------} (n - 1) n (n - 1) n "A052735" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) n! {--------------------------------------------------------------------------------------------} n (n - 1) "A052753" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \ \ \ | | |----- | | | | | | \ n3! (n3 + 1)| | | | | n2! | ) ------------| (n2 + 1)| | / /n1 - 1 \ \ | |n1 - 1 | / n3 (n3 + 1)!| | | | |----- | | | |----- |----- | | | | | \ n2! (n2 + 1)| | | | \ \n3 = 0 / | | | n1! | ) ------------| (n1 + 1)| | n1! | ) ----------------------------------| (n1 + 1)| /n - 1 \ |n - 1 | / n2 (n2 + 1)!| | |n - 1 | / n2 (n2 + 1)! | | |----- | |----- |----- | | |----- |----- | | | \ n1! (n1 + 1)| | \ \n2 = 0 / | | \ \n2 = 0 / | n! | ) ------------| n! | ) ----------------------------------| n! | ) --------------------------------------------------------| | / n1 (n1 + 1)!| | / n1 (n1 + 1)! | | / n1 (n1 + 1)! | |----- | |----- | |----- | n! \n1 = 0 / \n1 = 0 / \n1 = 0 / {----, ------------------------, ----------------------------------------------, -------------------------------------------------------------------- n n n n } "A052754" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | n! | ) ------------------| | / (n1 - 2) (n1 + 1)!| |----- | n! \n1 = 0 / {-----, ------------------------------} n - 2 n - 2 "A052758" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| /n - 1 \ |n - 1 | / (n2 - 1) (n2 + 1)!|| |----- | |----- |----- || | \ (n1 + 1) n1! | | \ \n2 = 0 /| n! | ) ------------------| n! | ) ----------------------------------------| | / (n1 - 1) (n1 + 1)!| | / (n1 - 1) (n1 + 1)! | |----- | |----- | n! \n1 = 0 / \n1 = 0 / {-----, ------------------------------, ----------------------------------------------------} n - 1 n - 1 n - 1 "A052765" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| /n - 1 \ |n - 1 | / (n2 - 2) (n2 + 1)!|| |----- | |----- |----- || | \ (n1 + 1) n1! | | \ \n2 = 0 /| n! | ) ------------------| n! | ) ----------------------------------------| | / (n1 - 2) (n1 + 1)!| | / (n1 - 2) (n1 + 1)! | |----- | |----- | n! \n1 = 0 / \n1 = 0 / {-----, ------------------------------, ----------------------------------------------------} n - 2 n - 2 n - 2 "A052766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | n! | ) ------------------| | / (n1 - 3) (n1 + 1)!| |----- | n! \n1 = 0 / {-----, ------------------------------} n - 3 n - 3 "A052799" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | n! | ) ------------------| | / (n1 - 4) (n1 + 1)!| |----- | n! \n1 = 0 / {-----, ------------------------------} n - 4 n - 4 "A052844" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052845" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052850" n 2 n! n! (n - 1) {-----, ----------} n n "A052852" LREtools/SearchTable: "SearchTable successful" n! (n + 1) (LaguerreL(n + 1, -1) - 2 LaguerreL(n, -1)) {------------------------------------------------------} n "A052866" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------, n!} n "A052867" n n! 2 n! {----, -----} n n "A052881" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1)| {n! | ) ------------|, n!} | / n1 (n1 + 1)!| |----- | \n1 = 0 / "A052883" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ / /n1 - 1 \ \ | |----- | | | |----- n2 | | | (-n1 - 1) | \ n2! | | | (-n1 - 1) | \ 2 n2! | | | 2 (n1 + 1) | ) ---------| n1!| | 2 (n1 + 1) | ) ---------| n1!| |n - 1 | / (n2 + 1)!| | |n - 1 | / (n2 + 1)!| | |----- |----- | | |----- |----- | | n | \ \n2 = 0 / | n | \ \n2 = 0 / | 2 n! | ) ------------------------------------------| 2 n! | ) ------------------------------------------| | / (n1 + 1)! | | / (n1 + 1)! | n |----- | |----- | n! 2 n! \n1 = 0 / \n1 = 0 / {----, -----, ---------------------------------------------------------, ---------------------------------------------------------} n n n n "A052887" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A052897" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -2) + (-n - 3) LaguerreL(n, -2)) n! {-------------------------------------------------------------} n "A052898" {1, n!} "A053151" memory used=31267.6MB, alloc=1495.5MB, time=203.90 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 6 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / 1/2\n / 1/2 \n n | 5 | |5 | 3 2 n 3 2 n {(-1) , |1/2 - ----| , |---- + 1/2| , RootOf(_Z - _Z - 2 _Z - 1, index = 1) , RootOf(_Z - _Z - 2 _Z - 1, index = 2) , \ 2 / \ 2 / 3 2 n RootOf(_Z - _Z - 2 _Z - 1, index = 3) } "A053175" LREtools/SearchTable: "SearchTable successful" n 2 {2 ((n + 3) (2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-2 n - 2 n + 2) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1)) binomial(2 n, n)/(n + 1)} "A053481" n {n! n, 2 n! (2 n + 1)} "A053482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ | |----- | | | (-n1 - 1) | \ 1 | | | 2 (n1 + 1) | ) ------------------| n1!| |n - 1 | / (n2 + 2) (n2 + 1)!| | |----- |----- | | n n | \ \n2 = 0 / | {2 n!, 2 n! | ) ---------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A053486" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 3 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A053487" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 4 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A053496" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A053497" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A053498" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A053532" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A053533" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A053553" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A053792" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" n (-4) ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (-2 n + 1) hypergeom([3/4, -n], [1], 2)) {--------------------------------------------------------------------------------------------, n n 4 ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (2 n + 1) hypergeom([3/4, -n], [1], 2)) ----------------------------------------------------------------------------------------} n "A053817" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A053871" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, -n - 1/2, -1/2)} "A053983" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n + 1/2, -1), (-1) BesselY(n + 1/2, -1)} "A053984" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n + 1/2, -1), (-1) BesselY(n + 1/2, -1)} "A053987" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n + 2) BesselJ(n + 1/2, -1/2) + BesselJ(n - 1/2, -1/2)), (-1) ((4 n + 2) BesselY(n + 1/2, -1/2) + BesselY(n - 1/2, -1/2))} "A053988" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n + 2) BesselJ(n + 1/2, -1/2) + BesselJ(n - 1/2, -1/2)), (-1) ((4 n + 2) BesselY(n + 1/2, -1/2) + BesselY(n - 1/2, -1/2))} "A054091" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | n! | ) ---------| | / (n1 + 1)!| |----- | n! \n1 = 0 / {----, ---------------------} n n "A054092" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \\\ | | |----- ||| | | n1 | \ n2 + 1 ||| | | (-1) n1! | ) ---------||| /n - 1 \ |n - 1 | | / (n2 + 1)!||| |----- / n1 \| |----- | |----- ||| n n | \ | (-1) n1!|| n | \ | \n2 = 0 /|| {(-1) , (-1) | ) |- ----------||, (-1) | ) |- -----------------------------||} | / \ n1 /| | / \ n1 /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A054093" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) n1!)|, (-1) | ) |-(-1) n1! | ) ---------|||} | / | | / | | / (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054094" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) (n1 + 1) n1!)|, (-1) | ) |-(-1) (n1 + 1) n1! | ) ------------------|||} | / | | / | | / (n2 + 2) (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054095" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) n1! (n1 + 1) (n1 + 2))|, (-1) | ) |-(-1) n1! (n1 + 1) (n1 + 2) | ) ---------------------------|||} | / | | / | | / (n2 + 1)! (n2 + 2) (n2 + 3)||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054096" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | n! (n - 1) | ) ---------------------| | / (n1 + 1)! n1 (n1 - 1)| |----- | n! (n - 1) \n1 = 0 / {----------, -----------------------------------------} n n "A054097" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 | \ (n1 + n1 + 4) (n1 + 1) | n! (n + 1) | ) -----------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) + 1) (n1 + 1)| n! (n + 1) \n1 = 0 / {-----------, --------------------------------------------------------} n n "A054099" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | n! | ) ---------| | / (n1 + 1)!| |----- | n! \n1 = 0 / {----, ---------------------} n n "A054100" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \\\ | | |----- ||| | | n1 | \ n2 + 1 ||| | | (-1) n1! | ) ---------||| /n - 1 \ |n - 1 | | / (n2 + 1)!||| |----- / n1 \| |----- | |----- ||| n n | \ | (-1) n1!|| n | \ | \n2 = 0 /|| {(-1) , (-1) | ) |- ----------||, (-1) | ) |- -----------------------------||} | / \ n1 /| | / \ n1 /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A054101" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) n1!)|, (-1) | ) |-(-1) n1! | ) ---------|||} | / | | / | | / (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054102" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) (n1 + 1) n1!)|, (-1) | ) |-(-1) (n1 + 1) n1! | ) ------------------|||} | / | | / | | / (n2 + 2) (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054103" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ 1 ||| {(-1) , (-1) | ) (-(-1) n1! (n1 + 1) (n1 + 2))|, (-1) | ) |-(-1) n1! (n1 + 1) (n1 + 2) | ) ---------------------------|||} | / | | / | | / (n2 + 1)! (n2 + 2) (n2 + 3)||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A054104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | n! (n - 1) | ) ---------------------| | / (n1 + 1)! n1 (n1 - 1)| |----- | n! (n - 1) \n1 = 0 / {----------, -----------------------------------------} n n "A054108" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A054109" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A054113" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / n1 + 1 ----- n1 = 0 "A054116" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1! (n1 + 1) (n1 + 2)} / ----- n1 = 0 "A054117" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 3) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A054118" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A054122" LREtools/SearchTable: "SearchTable successful" n n 2 (n + 1) (LegendreP(n + 1, 2) - 2 LegendreP(n, 2)) 2 (n + 1) (LegendreQ(n + 1, 2) - 2 LegendreQ(n, 2)) {----------------------------------------------------, ----------------------------------------------------} n n "A054341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) | | { 2 | | { 2 | | { ---------------------------------------- n1::odd | | { -------------------- n1::even| |n - 1 { n1 | |n - 1 { n1 + 2 | |----- { n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | |----- { | n n | \ { 2 | n | \ { 0 n1::odd | {(5/2) , (5/2) | ) ----------------------------------------------------------|, (5/2) | ) --------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A054391" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 4 _Z + 3 _Z - 1, index = 1) , RootOf(_Z - 4 _Z + 3 _Z - 1, index = 2) , RootOf(_Z - 4 _Z + 3 _Z - 1, index = 3) } "A054392" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 6 _Z + 11 _Z - 7, index = 1) , RootOf(_Z - 6 _Z + 11 _Z - 7, index = 2) , RootOf(_Z - 6 _Z + 11 _Z - 7, index = 3) } "A054393" memory used=31624.8MB, alloc=1495.5MB, time=206.68 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 2 %1 := _Z - 8 _Z + 22 _Z - 24 _Z + 8 _Z - 1 "A054394" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 2 %1 := _Z - 10 _Z + 37 _Z - 62 _Z + 46 _Z - 13 "A054441" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) || {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) (2 + 5 ) binomial(2 n2, n2)||} | / | / || |----- |----- || \n1 = 0 \n2 = 0 // "A054442" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) (n + 4) { 4 binomial(n, n/2) (2 n + 7) (n + 1) { -------------------------------- n::even { ------------------------------------ n::even n { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {2 , { , { } { (2 n + 2) { 16 binomial(n - 1, n/2 - 1/2) n (n + 2) (n + 4) { 2 (2 n + 7) { ----------------------------------------------- n::odd { 1/4 ---------------------------------- n::odd { (n + 1) (n + 3) { (n + 2) binomial(n + 1, n/2 + 1/2) "A054443" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) (n + 4) { 4 binomial(n, n/2) (2 n + 9) (n + 1) (n + 3) { -------------------------------- n::even { -------------------------------------------- n::even n { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {2 (n + 7), { , { } { (2 n + 2) { 16 binomial(n - 1, n/2 - 1/2) n (n + 2) (n + 4) { 2 (n + 3) (2 n + 9) { ----------------------------------------------- n::odd { 1/4 ------------------------------------------ n::odd { (n + 1) (n + 3) { (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) "A054479" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) binomial(2 n, n) LaguerreL(n, -n - 1/2, -1/2)} "A054516" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 1) (n1 + 2)|| {(-1) n!, (-1) n! | ) |- ------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A054652" LREtools/SearchTable: "SearchTable successful" 2 {(n!) LaguerreL(n, -1)} "A054726" LREtools/SearchTable: "SearchTable successful" n 2 2 ((n + 1) (17 n - 8) LegendreP(n + 1, 3) + (-99 n - 3 n + 24) LegendreP(n, 3)) {---------------------------------------------------------------------------------, n (n - 1) (n - 2) n 2 2 ((n + 1) (17 n - 8) LegendreQ(n + 1, 3) + (-99 n - 3 n + 24) LegendreQ(n, 3)) ---------------------------------------------------------------------------------} n (n - 1) (n - 2) "A054727" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A054765" LREtools/SearchTable: "SearchTable successful" n n {(-I) n! LegendreP(n, I), (-I) n! LegendreQ(n, I)} "A054766" LREtools/SearchTable: "SearchTable successful" n n {(-I) n! LegendreP(n, I), (-I) n! LegendreQ(n, I)} "A054768" LREtools/SearchTable: "SearchTable successful" n n {(-I) n! LegendreP(n, I), (-I) n! LegendreQ(n, I)} "A054872" LREtools/SearchTable: "SearchTable successful" n n 2 ((2 n + 2) LegendreP(n + 1, 2) + (-7 n - 4) LegendreP(n, 2)) 2 ((2 n + 2) LegendreQ(n + 1, 2) + (-7 n - 4) LegendreQ(n, 2)) {---------------------------------------------------------------, ---------------------------------------------------------------} n (n - 1) n (n - 1) "A055022" 2 n binomial(2 n, n) (n - 3) (n - 5 n + 10) {4 , ----------------------------------------} (2 n - 1) (2 n - 3) (2 n - 5) "A055113" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (4 (2 n + 1) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + n (31 n + 19) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n) binomial(2 n, n)/(n (2 n - 1) (5 n + 3)), 1/2 ----------------} n - 1/2 "A055142" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, -n + 1/2, 1/2)} "A055214" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1)| n n | \ 2 | {2 n!, 2 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A055217" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) hypergeom([1/2, -n - 1], [1], 4)} "A055218" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((7 n + 28 n + 27) hypergeom([1/2, -n - 1], [1], 4) - 3 (2 n + 5) (n + 1) hypergeom([1/2, -n], [1], 4)) {3 , --------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A055219" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 2 2 n (-1) ((34 n + 255 n + 608 n + 459) hypergeom([1/2, -n - 1], [1], 4) - 3 (11 n + 66 n + 97) (n + 1) hypergeom([1/2, -n], [1], 4)) {3 , ------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 3) (n + 2) "A055220" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=31990.1MB, alloc=1495.5MB, time=209.36 LREtools/SearchTable: "SearchTable successful" n n 5 4 3 2 {3 , (-1) ((142 n + 2485 n + 16760 n + 54113 n + 83112 n + 48276) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 - 3 (47 n + 752 n + 4417 n + 11248 n + 10452) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 7) (n + 6) (n + 4) (n + 3) (n + 2))} "A055221" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n 7 6 5 4 3 2 {3 , (-1) ((547 n + 16410 n + 206542 n + 1405794 n + 5555791 n + 12671904 n + 15345180 n + 7558272) hypergeom([1/2, -n - 1], [1], 4) 6 5 4 3 2 - 3 (182 n + 5187 n + 60932 n + 376083 n + 1281488 n + 2277852 n + 1645056) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 9) (n + 8) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A055222" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=32248.2MB, alloc=1495.5MB, time=211.42 LREtools/SearchTable: "SearchTable successful" n n 9 8 7 6 5 4 3 2 {3 , (-1) ((2005 n + 96240 n + 2017464 n + 24142932 n + 180924357 n + 876128436 n + 2727281414 n + 5233276392 n + 5583279240 n + 2506826880) hypergeom([1/2, -n - 1], [1], 4) - 6 8 7 6 5 4 3 2 (334 n + 15531 n + 312762 n + 3551844 n + 24794928 n + 108579585 n + 290319476 n + 432006180 n + 273136320) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 11) (n + 10) (n + 9) (n + 8) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A055392" LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) hypergeom([1/2, -n], [-2 n], -4) {-------------------------------------------------} n + 1 "A055395" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {----------------} n + 1 "A055452" LREtools/SearchTable: "SearchTable successful" 3 2 2 (32 n - 48 n + 32 n - 13) hypergeom([-1/2, -n - 1], [1], -4) - (2 n + 1) (16 n - 24 n + 13) hypergeom([-1/2, -n], [1], -4) {-----------------------------------------------------------------------------------------------------------------------------} n (n + 2) (n - 1) "A055453" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 {((64 n - 248 n + 367 n - 328 n + 118) hypergeom([-1/2, -n - 1], [1], -4) + (-64 n + 216 n - 231 n + 96 n + 118) hypergeom([-1/2, -n], [1], -4) )/((n + 2) n (n - 1) (n - 2))} "A055454" LREtools/SearchTable: "SearchTable successful" 5 4 3 2 {((256 n - 1856 n + 5236 n - 8209 n + 7945 n - 2748) hypergeom([-1/2, -n - 1], [1], -4) 5 4 3 2 + (-256 n + 1728 n - 4260 n + 5235 n - 2819 n - 2748) hypergeom([-1/2, -n], [1], -4))/((n + 2) n (n - 1) (n - 2) (n - 3))} "A055596" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A055613" LREtools/SearchTable: "SearchTable successful" {((48 n + 229) LaguerreL(n + 1, -8) + (-16 n - 13) LaguerreL(n, -8)) n! (n + 1)} "A055634" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) { -2 (n/2)! n::even { 2 binomial(n, n/2) (n/2)! n::even { {{ , { (n/2 + 1/2) } { (- n/2 + 1/2) { 2 (n/2 + 1/2)! { -2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { ------------------------- n::odd { n + 1 "A055787" n n {16 , 4 binomial(2 n, n)} "A055790" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (-1) (n1 + 1) | n! (n + 3 n + 1) | ) -------------------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) + 3 n1 + 4) (n1 + 3 n1 + 1)| n! (n + 3 n + 1) \n1 = 0 / {-----------------, ----------------------------------------------------------------------------} n n "A055814" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A055815" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (2 n + 2 n + 2 n + 3) LegendreQ(n + 1, 3) + (-2 n - 10 n - 18 n - 9) LegendreQ(n, 3) {----------------------------------------------------------------------------------------, n (n + 2) (n - 1) 3 2 3 2 (-2 n - 2 n - 2 n - 3) LegendreP(n + 1, 3) + (2 n + 10 n + 18 n + 9) LegendreP(n, 3) - ----------------------------------------------------------------------------------------} (n - 1) (n + 2) n "A055816" LREtools/SearchTable: "SearchTable successful" 2 2 (n + 1) ((3 n - 3 n + 3) LegendreP(n + 1, 3) + (-n - 3 n - 9) LegendreP(n, 3)) {--------------------------------------------------------------------------------, (n + 3) n (n - 1) 2 2 (n + 1) ((3 n - 3 n + 3) LegendreQ(n + 1, 3) + (-n - 3 n - 9) LegendreQ(n, 3)) --------------------------------------------------------------------------------} (n + 3) n (n - 1) "A055817" memory used=32547.0MB, alloc=1495.5MB, time=213.64 LREtools/SearchTable: "SearchTable successful" 5 4 3 2 4 3 2 (10 n + 41 n + 26 n - 29 n + 12 n + 30) LegendreP(n + 1, 3) - (2 n + 3) (n + 4 n + 11 n + 32 n + 30) LegendreP(n, 3) {---------------------------------------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) n (n - 1) 5 4 3 2 4 3 2 (10 n + 41 n + 26 n - 29 n + 12 n + 30) LegendreQ(n + 1, 3) - (2 n + 3) (n + 4 n + 11 n + 32 n + 30) LegendreQ(n, 3) ---------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) n (n - 1) "A055824" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (4 n - 1) LegendreP(n + 1, 3) + (-22 n - 7 n + 3) LegendreP(n, 3) {--------------------------------------------------------------------------, n (n - 1) 2 (n + 1) (4 n - 1) LegendreQ(n + 1, 3) + (-22 n - 7 n + 3) LegendreQ(n, 3) --------------------------------------------------------------------------} n (n - 1) "A055825" LREtools/SearchTable: "SearchTable successful" 2 2 (3 n + 2 n + 1) LegendreP(n + 1, 3) + (-13 n - 10 n - 3) LegendreP(n, 3) {--------------------------------------------------------------------------, n (n - 1) 2 2 (3 n + 2 n + 1) LegendreQ(n + 1, 3) + (-13 n - 10 n - 3) LegendreQ(n, 3) --------------------------------------------------------------------------} n (n - 1) "A055826" LREtools/SearchTable: "SearchTable successful" 2 2 (n + 1) ((4 n + n + 4) LegendreP(n + 1, 3) + (-8 n - 19 n - 12) LegendreP(n, 3)) {----------------------------------------------------------------------------------, (n + 2) n (n - 1) 2 2 (n + 1) ((4 n + n + 4) LegendreQ(n + 1, 3) + (-8 n - 19 n - 12) LegendreQ(n, 3)) ----------------------------------------------------------------------------------} (n + 2) n (n - 1) "A055834" LREtools/SearchTable: "SearchTable successful" ((20 n + 10) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (5 n - 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------} 3 n + 1 "A055835" LREtools/SearchTable: "SearchTable successful" ((20 n + 10) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (5 n - 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------} 3 n + 1 "A055836" LREtools/SearchTable: "SearchTable successful" ((20 n + 10) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (5 n + 6) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------} n + 1 "A055837" LREtools/SearchTable: "SearchTable successful" {(10 (2 n + 1) (3 n + 4) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (3 n + 2) (5 n + 4) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((n + 2) (n + 1))} "A055838" LREtools/SearchTable: "SearchTable successful" ((12 n + 6) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (3 n + 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------------------} n + 3 "A055839" LREtools/SearchTable: "SearchTable successful" 2 {(10 (2 n + 1) (27 n + 118 n + 120) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + 3 (5 n + 14) (3 n + 2) (3 n + 4) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((n + 4) (n + 3) (n + 2))} "A055840" LREtools/SearchTable: "SearchTable successful" 3 2 {(10 (2 n + 1) (81 n + 589 n + 1340 n + 960) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) 2 + 9 (3 n + 4) (3 n + 2) (5 n + 29 n + 40) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((n + 5) (n + 4) (n + 3) (n + 2))} "A055879" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2 - 1/2], [1], -4) { - -------------------------------------------------------------------------------------------------- n::even {{ n + 1 , { { -5 hypergeom([-1/2, - n/2 - 1], [1], -4) + 5 hypergeom([-1/2, - n/2], [1], -4) n::odd { (n/2 + 3/2) (n/2 + 1/2) { 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) { - --------------------------------------------------------------------------------------------------------------------------- n::even { n + 1 , { { (n/2 + 1) (n/2) { -5 5 hypergeom([3/2, - n/2 - 1], [1], 4/5) + 5 5 hypergeom([3/2, - n/2], [1], 4/5) n::odd { (n/2 + 1) (n/2) { -5 5 hypergeom([3/2, - n/2 - 1], [1], 4/5) + 5 5 hypergeom([3/2, - n/2], [1], 4/5) n::even { { (n/2 + 3/2) (n/2 + 1/2) , { 5 (n + 3) hypergeom([3/2, - n/2 - 3/2], [1], 4/5) - 5 (n - 1) hypergeom([3/2, - n/2 - 1/2], [1], 4/5) { - --------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { -5 hypergeom([-1/2, - n/2 - 1], [1], -4) + 5 hypergeom([-1/2, - n/2], [1], -4) n::even { { (n + 3) hypergeom([-1/2, - n/2 - 3/2], [1], -4) + (-n + 1) hypergeom([-1/2, - n/2 - 1/2], [1], -4) } { - -------------------------------------------------------------------------------------------------- n::odd { n + 1 "A056010" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A056040" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even { binomial(n, n/2) { 4 binomial(n, n/2) n::even {{ , { } { (2 n - 2) { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { 4 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A056158" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 3)|| {(-1) n!, (-1) n! | ) |- ---------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A056199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A056541" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1)| n n | \ 2 | {2 n!, 2 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A056542" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A056543" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A056545" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 2)| n n | \ 2 | {4 n!, 4 n! | ) ------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A056546" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1)| n n | \ 5 | {5 n!, 5 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A056547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1)| n n | \ 6 | {6 n!, 6 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A056889" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=32934.6MB, alloc=1495.5MB, time=216.43 { (-2 n + 2) BesselY(n/2 - 1/2, -2) - 4 BesselY(n/2 + 1/2, -2) n::even {{ , { 2 BesselY(n/2 + 1, -2) n + (n + 2) (n - 2) BesselY(n/2, -2) n::odd { 2 BesselJ(n/2 + 1, -2) n + (n + 2) (n - 2) BesselJ(n/2, -2) n::even { , { -4 BesselJ(n/2 + 1/2, -2) + (-2 n + 2) BesselJ(n/2 - 1/2, -2) n::odd { 2 BesselY(n/2 + 1, -2) n + (n + 2) (n - 2) BesselY(n/2, -2) n::even { , { (-2 n + 2) BesselY(n/2 - 1/2, -2) - 4 BesselY(n/2 + 1/2, -2) n::odd { -4 BesselJ(n/2 + 1/2, -2) + (-2 n + 2) BesselJ(n/2 - 1/2, -2) n::even { } { 2 BesselJ(n/2 + 1, -2) n + (n + 2) (n - 2) BesselJ(n/2, -2) n::odd "A056890" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (-2 n + 2) BesselY(n/2 - 1/2, -2) - 4 BesselY(n/2 + 1/2, -2) n::even {{ , { 2 BesselY(n/2 + 1, -2) n + (n + 2) (n - 2) BesselY(n/2, -2) n::odd { 2 BesselJ(n/2 + 1, -2) n + (n + 2) (n - 2) BesselJ(n/2, -2) n::even { , { -4 BesselJ(n/2 + 1/2, -2) + (-2 n + 2) BesselJ(n/2 - 1/2, -2) n::odd { 2 BesselY(n/2 + 1, -2) n + (n + 2) (n - 2) BesselY(n/2, -2) n::even { , { (-2 n + 2) BesselY(n/2 - 1/2, -2) - 4 BesselY(n/2 + 1/2, -2) n::odd { -4 BesselJ(n/2 + 1/2, -2) + (-2 n + 2) BesselJ(n/2 - 1/2, -2) n::even { } { 2 BesselJ(n/2 + 1, -2) n + (n + 2) (n - 2) BesselJ(n/2, -2) n::odd "A056919" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1/4) binomial(n, n/2) (n/2)! ((-n - 3) KummerM(n/2 + 5/2, 1, -1) + KummerM(n/2 + 3/2, 1, -1) (n + 1)) n::even {{ , { (n/2 - 1/2) { -(-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n KummerM(n/2 + 1, 1, -1) n::odd { (n/2) { (-1/4) binomial(n, n/2) (n/2)! ((-n - 3) LaguerreL(n/2 + 3/2, 1) + LaguerreL(n/2 + 1/2, 1) (n + 1)) n::even { , { (n/2 - 1/2) { -(-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n LaguerreL(n/2, 1) n::odd { (n/2) { -(-1) (n/2)! KummerM(n/2 + 1, 1, -1) n::even { { (n/2 + 1/2) , { (-1) (n/2 + 1/2)! ((n + 3) KummerM(n/2 + 5/2, 1, -1) + (-n - 1) KummerM(n/2 + 3/2, 1, -1)) { - ----------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { -(-1) (n/2)! LaguerreL(n/2, 1) n::even { { (n/2 + 1/2) } { (-1) (n/2 + 1/2)! ((n + 3) LaguerreL(n/2 + 3/2, 1) + (-n - 1) LaguerreL(n/2 + 1/2, 1)) { - ------------------------------------------------------------------------------------------------- n::odd { n + 1 "A056920" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1/4) binomial(n, n/2) (n/2)! ((-n - 3) KummerM(n/2 + 5/2, 1, -1) + KummerM(n/2 + 3/2, 1, -1) (n + 1)) n::even {{ , { (n/2 - 1/2) { -(-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n KummerM(n/2 + 1, 1, -1) n::odd { (n/2) { (-1/4) binomial(n, n/2) (n/2)! ((-n - 3) LaguerreL(n/2 + 3/2, 1) + LaguerreL(n/2 + 1/2, 1) (n + 1)) n::even { , { (n/2 - 1/2) { -(-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n LaguerreL(n/2, 1) n::odd { (n/2) { -(-1) (n/2)! KummerM(n/2 + 1, 1, -1) n::even { { (n/2 + 1/2) , { (-1) (n/2 + 1/2)! ((n + 3) KummerM(n/2 + 5/2, 1, -1) + (-n - 1) KummerM(n/2 + 3/2, 1, -1)) { - ----------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { -(-1) (n/2)! LaguerreL(n/2, 1) n::even { { (n/2 + 1/2) } { (-1) (n/2 + 1/2)! ((n + 3) LaguerreL(n/2 + 3/2, 1) + (-n - 1) LaguerreL(n/2 + 1/2, 1)) { - ------------------------------------------------------------------------------------------------- n::odd { n + 1 "A056921" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::even {{ , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::odd { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::even { , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::even { , { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::even { } { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::odd "A056922" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::even {{ , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::odd { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::even { , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::even { , { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::even { } { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::odd "A056952" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2)! KummerM(n/2 + 1, 1, 1) n::even { {{ (n/2 + 1/2)! ((n + 3) KummerM(n/2 + 5/2, 1, 1) + (-n - 5) KummerM(n/2 + 3/2, 1, 1)) , { ----------------------------------------------------------------------------------- n::odd { n + 1 { (n/2)! LaguerreL(n/2, -1) n::even { { (n/2 + 1/2)! ((n + 3) LaguerreL(n/2 + 3/2, -1) + (-n - 5) LaguerreL(n/2 + 1/2, -1)) , { ----------------------------------------------------------------------------------- n::odd { n + 1 { (-n) { 2 ((n + 3) KummerM(n/2 + 5/2, 1, 1) + (-n - 5) KummerM(n/2 + 3/2, 1, 1)) binomial(n, n/2) (n/2)! n::even { , { (-n + 1) { 2 n KummerM(n/2 + 1, 1, 1) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { (-n) { 2 ((n + 3) LaguerreL(n/2 + 3/2, -1) + (-n - 5) LaguerreL(n/2 + 1/2, -1)) binomial(n, n/2) (n/2)! n::even { } { (-n + 1) { 2 n LaguerreL(n/2, -1) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A056953" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2)! KummerM(n/2 + 1, 1, 1) n::even { {{ (n/2 + 1/2)! ((n + 3) KummerM(n/2 + 5/2, 1, 1) + (-n - 5) KummerM(n/2 + 3/2, 1, 1)) , { ----------------------------------------------------------------------------------- n::odd { n + 1 { (n/2)! LaguerreL(n/2, -1) n::even { { (n/2 + 1/2)! ((n + 3) LaguerreL(n/2 + 3/2, -1) + (-n - 5) LaguerreL(n/2 + 1/2, -1)) , { ----------------------------------------------------------------------------------- n::odd { n + 1 { (-n) { 2 ((n + 3) KummerM(n/2 + 5/2, 1, 1) + (-n - 5) KummerM(n/2 + 3/2, 1, 1)) binomial(n, n/2) (n/2)! n::even { , { (-n + 1) { 2 n KummerM(n/2 + 1, 1, 1) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { (-n) { 2 ((n + 3) LaguerreL(n/2 + 3/2, -1) + (-n - 5) LaguerreL(n/2 + 1/2, -1)) binomial(n, n/2) (n/2)! n::even { } { (-n + 1) { 2 n LaguerreL(n/2, -1) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A056986" (2 n + 1) binomial(2 n, n) {(n + 1) n!, --------------------------} (n + 1) (n + 2) "A057552" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) ----------------------------------------} / (n1 + 3) (n1 + 1) ----- n1 = 0 "A057553" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) (5 n1 + 6) {1, ) ----------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A057571" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n + 3) { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (2 n + 3) (n + 1) n n { (n + 1) binomial(n, n/2) { ------------------------------------ n::even {(-2) , 2 (2 n + 3), { , { n + 2 } { (2 n - 2) { { 2 (n + 1) (2 n + 3) { binomial(n + 1, n/2 + 1/2) (4 n + 6) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A057585" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n (-1) ((5 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) {(-1) , 3 , --------------------------------------------------------------------------------------------} n + 3 "A057648" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A057693" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A057718" LREtools/SearchTable: "SearchTable successful" memory used=33283.8MB, alloc=1495.5MB, time=219.03 2 2 (2 n + 1) binomial(2 n, n) hypergeom([1/2, 1/2, -n - 1, -n - 1], [1, -n - 1/2, -n - 1/2], 1) {----------------------------------------------------------------------------------------------} 2 (n + 1) "A057977" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { 8 binomial(n, n/2) { (n + 1) binomial(n, n/2) { ------------------ n::even {{ , { n + 2 } { (2 n - 2) { { 4 2 { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A058006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) (n1 + 1) (-1) n1!} / ----- n1 = 0 "A058279" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((n + n + 1) BesselI(n, 2) + (-n - 1) BesselI(n - 1, 2)), (-1) ((n + n + 1) BesselK(n, -2) + (-n - 1) BesselK(n - 1, -2))} "A058307" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((n + n + 1) BesselI(n, 2) + (-n - 1) BesselI(n - 1, 2)), (-1) ((n + n + 1) BesselK(n, -2) + (-n - 1) BesselK(n - 1, -2))} "A058308" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((n + 2 n + 2) (n + 1) BesselI(n, 2) + (-n - 3 n - 3) BesselI(n - 1, 2)), n 2 2 (-1) ((n + 2 n + 2) (n + 1) BesselK(n, -2) + (-n - 3 n - 3) BesselK(n - 1, -2))} "A058309" LREtools/SearchTable: "SearchTable successful" n 4 3 2 2 {(-1) ((n + 6 n + 14 n + 15 n + 7) BesselI(n, 2) - (n + 2) (n + 4 n + 5) BesselI(n - 1, 2)), n 4 3 2 2 (-1) ((n + 6 n + 14 n + 15 n + 7) BesselK(n, -2) - (n + 2) (n + 4 n + 5) BesselK(n - 1, -2))} "A058607" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) ---------------------------------------------|} | / (n1 + 2) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A058622" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {2 , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A058797" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -2), (-1) BesselY(n, -2)} "A058798" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselJ(n, -2) + BesselJ(n - 1, -2)), (-1) (n BesselY(n, -2) + BesselY(n - 1, -2))} "A058799" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((n + 1) (n + 2 n - 2) BesselJ(n, -2) + (n + 3 n + 1) BesselJ(n - 1, -2)), n 2 2 (-1) ((n + 1) (n + 2 n - 2) BesselY(n, -2) + (n + 3 n + 1) BesselY(n - 1, -2))} "A058806" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) --------------------------------------------------------|} | / (n1 + 1) (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A058859" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A058860" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A058861" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A058987" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) (2 n + 1) binomial(2 n, n) {-------------------------------------------------------------------------, --------------------------} n + 2 (n + 1) (n + 2) "A059019" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z + 2 _Z + 1, index = 1) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 2) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 3) } "A059027" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 6 _Z + 1, index = 1) , RootOf(_Z - 5 _Z + 6 _Z + 1, index = 2) , RootOf(_Z - 5 _Z + 6 _Z + 1, index = 3) } "A059204" n {2 , n!} "A059227" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A059229" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n - 3} "A059231" LREtools/SearchTable: "SearchTable successful" memory used=33639.2MB, alloc=1527.5MB, time=221.68 (8 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------} n "A059272" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A059275" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) } "A059278" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A059279" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059280" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059281" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059284" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- 1/2 n1 1/2 | |----- 1/2 n1 1/2 | n n | \ (-I 3 ) LegendreP(n1, 3 I)| n | \ (-I 3 ) LegendreQ(n1, 3 I)| {(-1) (n + 1), (-1) (n + 1) | ) ---------------------------------|, (-1) (n + 1) | ) ---------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-1) (n1 + 2) (n1 + 1) | |----- (-1) (n1 + 2) (n1 + 1) | \n1 = 0 / \n1 = 0 / "A059345" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059348" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 4 { 4 binomial(n, n/2) (5 n + 8) { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n + 2) { 32 binomial(n - 1, n/2 - 1/2) n { 2 (5 n + 8) { ------------------------------- n::odd { -------------------------------------------------- n::odd { (n + 1) (n + 3) { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) "A059371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) n1! | {(n + 3) (n + 2) (n + 1) (1/2) n!, (n + 3) (n + 2) (n + 1) (1/2) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / "A059375" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! n BesselI(n, 2), (-1) n! n BesselK(n, -2)} "A059398" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059422" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059435" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) 2 (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {--------------------------------------------, --------------------------------------------} n n "A059480" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {(-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I) 2 I} "A059593" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A059710" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A059712" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) ((n1 - 3) hypergeom([1/2, -n1 - 1], [1], 4) + (9 n1 + 33) hypergeom([1/2, -n1], [1], 4))| {(7/2) , (7/2) | ) -----------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (7/2) | \n1 = 0 / "A059714" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (n1 + 9) | {(9/2) , (9/2) | ) ----------------------------------------|} | / (n1 + 1)| |----- (n1 + 3) (n1 + 2) (n1 + 1) (9/2) | \n1 = 0 / "A059728" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2, 2 LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n / 1/2\n / 1/2 \n n | 5 | | 5 | | 5 | |5 | n {(-1) , |1/2 - ----| , |3/2 - ----| , |3/2 + ----| , |---- + 1/2| , (-1) hypergeom([1/2, -n - 1], [1], 4)} \ 2 / \ 2 / \ 2 / \ 2 / "A059738" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4))| {(7/2) , (7/2) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (7/2) | \n1 = 0 / "A059760" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- 2 || | 2 | \ (n2 + 1) n2!|| | (n1 + 1) (n1!) | ) -------------|| /n - 1 \ |n - 1 | / 2 || |----- 2| |----- |----- ((n2 + 1)!) || | \ (n1 + 1) (n1!) | | \ \n2 = 0 /| {n! | ) ---------------|, n! | ) --------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A059837" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 2 { 4 ((n/2)!) { 2 ((n/2)!) binomial(n, n/2) n::even { ------------ n::even { {{ n + 1 , { 2 2 } { { ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) { (2 n - 2) 2 { ------------------------------------------- n::odd { 2 2 ((n/2 - 1/2)!) n::odd { n + 1 "A059838" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 { 2 ((n/2)!) { 2 ((n/2)!) binomial(n, n/2) n::even { ------------ n::even { {n!, { n , { (-n - 1) 2 2 } { { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) { (n - 1) 2 { ----------------------------------------------------- n::odd { 2 ((n/2 - 1/2)!) n::odd { n "A059944" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(3 ) n!, (-3 ) n!} "A060237" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- 2 2 || | 2 2 | \ (n2 + 1) (n2!) || | (n1 + 1) (n1!) | ) ---------------------|| /n - 1 \ |n - 1 | / 2|| |----- 2 2 | |----- |----- (n2 + 3) ((n2 + 1)!) || 2 2 2 2 | \ (n1 + 1) (n1!) | 2 2 | \ \n2 = 0 /| {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|, (n + 1) (n!) | ) -----------------------------------------------|} | / 2| | / 2 | |----- (n1 + 2) ((n1 + 1)!) | |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / \n1 = 0 / "A060696" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 2 (n/2)! { (- n/2) { --------------- n::even { 2 binomial(n, n/2) (n/2)! n::even { n {{ , { } { (- n/2 + 1/2) { (n/2 + 1/2) { 2 binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 2 2 (n/2 + 1/2)! { --------------------------- n::odd { n + 1 "A060725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" / { 0 irem(n1, 5) = 0\ | { | | { 0 irem(n1, 5) = 1| | { | | { 0 irem(n1, 5) = 2| | { | | { 0 irem(n1, 5) = 3| | { | | { / n1 \ | | { |---- - 4/5| | |n - 1 { \ 5 / n1 n1 n1 n1 | |----- { (-625) GAMMA(---- + 1) GAMMA(---- + 4/5) GAMMA(---- + 3/5) GAMMA(---- + 2/5) irem(n1, 5) = 4| | \ { 5 5 5 5 | {n! | ) -----------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 5) = 0\ | { | | { 0 irem(n1, 5) = 1| | { | | { 0 irem(n1, 5) = 2| | { | | { / n1 \ | | { |---- - 3/5| | | { \ 5 / n1 n1 n1 n1 | | { (-625) GAMMA(---- + 2/5) GAMMA(---- + 1) GAMMA(---- + 4/5) GAMMA(---- + 3/5) irem(n1, 5) = 3| |n - 1 { 5 5 5 5 | |----- { | | \ { 0 irem(n1, 5) = 4| n! | ) -----------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 5) = 0\ | { | | { 0 irem(n1, 5) = 1| | { | | { / n1 \ | | { |---- - 2/5| | | { \ 5 / n1 n1 n1 n1 | | { (-625) GAMMA(---- + 3/5) GAMMA(---- + 2/5) GAMMA(---- + 1) GAMMA(---- + 4/5) irem(n1, 5) = 2| | { 5 5 5 5 | | { | |n - 1 { 0 irem(n1, 5) = 3| |----- { | | \ { 0 irem(n1, 5) = 4| n! | ) -----------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 5) = 0\ | { | | { / n1 \ | | { |---- - 1/5| | | { \ 5 / n1 n1 n1 n1 | | { (-625) GAMMA(---- + 2/5) GAMMA(---- + 4/5) GAMMA(---- + 3/5) GAMMA(---- + 1) irem(n1, 5) = 1| | { 5 5 5 5 | | { | | { 0 irem(n1, 5) = 2| | { | |n - 1 { 0 irem(n1, 5) = 3| |----- { | | \ { 0 irem(n1, 5) = 4| n! | ) -----------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 5 / n1 n1 n1 n1 | | { (-625) GAMMA(---- + 3/5) GAMMA(---- + 1) GAMMA(---- + 4/5) GAMMA(---- + 2/5) irem(n1, 5) = 0| | { 5 5 5 5 | | { | | { 0 irem(n1, 5) = 1| | { | | { 0 irem(n1, 5) = 2| | { | |n - 1 { 0 irem(n1, 5) = 3| |----- { | | \ { 0 irem(n1, 5) = 4| n! | ) -----------------------------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A060726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" memory used=34031.5MB, alloc=1527.5MB, time=224.56 /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 6) = 0 |----- { \n1 = 0 0 , irem(n1, 6) = 1 0 , irem(n1, 6) = 2 0 , irem(n1, 6) = 3 0 , irem(n1, 6) = 4 / n1 \ |---- - 5/6| \ 6 / n1 n1 2 n1 n1 2 // n1 \ \5 n1 1/12 (-1/6) binomial(---- - 5/2, ---- - 5/6) binomial(---- - 5/3, ---- - 5/6) ||---- - 5/6|!| binomial(n1 - 5, ---- - 5/2) n1 2 6 3 6 \\ 6 / / 2 \ | | /(n1 + 1)!|, (n1 - 1) (n1 - 2) (n1 - 3) (n1 - 4) , irem(n1, 6) = 5 | | / / { 0 irem(n1, 6) = 0\ | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | | { / n1 \ | | { |---- - 2/3| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 5/6) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) GAMMA(---- + 1/2) irem(n1, 6) = 4| |n - 1 { 6 6 6 6 6 | |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 6) = 0\ | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { / n1 \ | | { |---- - 1/2| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 5/6) GAMMA(---- + 2/3) GAMMA(---- + 1/3) irem(n1, 6) = 3| | { 6 6 6 6 6 | | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / /n - 1 |----- { | \ { n! | ) { | / { 0 , irem(n1, 6) = 0 |----- { \n1 = 0 0 , irem(n1, 6) = 1 / n1 \ |---- - 1/3| \ 6 / // n1 \ \5 n1 n1 n1 n1 2 n1 1/2 (n1 - 1) n1 (-2/3) ||---- - 1/3|!| binomial(---- - 2/3, ---- - 1/3) binomial(---- - 1, ---- - 1/3) binomial(n1 - 2, ---- - 1) , \\ 6 / / 3 6 2 6 2 irem(n1, 6) = 2 0 , irem(n1, 6) = 3 0 , irem(n1, 6) = 4 \ | | /(n1 + 1)!|, 0 , irem(n1, 6) = 5 | | / / { 0 irem(n1, 6) = 0\ | { | | { / n1 \ | | { |---- - 1/6| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 5/6) GAMMA(---- + 1/2) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 6) = 1| | { 6 6 6 6 6 | | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 6 / n1 n1 n1 n1 n1 | | { (-7776) GAMMA(---- + 1/3) GAMMA(---- + 1) GAMMA(---- + 2/3) GAMMA(---- + 1/2) GAMMA(---- + 5/6) irem(n1, 6) = 0| | { 6 6 6 6 6 | | { | | { 0 irem(n1, 6) = 1| | { | | { 0 irem(n1, 6) = 2| | { | | { 0 irem(n1, 6) = 3| | { | |n - 1 { 0 irem(n1, 6) = 4| |----- { | | \ { 0 irem(n1, 6) = 5| n! | ) ------------------------------------------------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A060727" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 7 to 1 (Liouvillian solutions)" /n - 1 |----- { | \ { {n! | ) { | / { 0 , irem(n1, 7) = 0 |----- { \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 / n1 \ |---- - 6/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) GAMMA(---- + 5/7) GAMMA(---- + 3/7) GAMMA(---- + 1) , irem(n1, 7) = 6 7 7 7 7 7 7 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 / n1 \ |---- - 5/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) , irem(n1, 7) = 5 7 7 7 7 7 7 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 / n1 \ |---- - 4/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 4/7) GAMMA(---- + 6/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 2/7) , irem(n1, 7) = 4 7 7 7 7 7 7 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 / n1 \ |---- - 3/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 5/7) GAMMA(---- + 1) GAMMA(---- + 4/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 3/7) , irem(n1, 7) = 3 7 7 7 7 7 7 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 0 , irem(n1, 7) = 1 / n1 \ |---- - 2/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 5/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 4/7) , irem(n1, 7) = 2 7 7 7 7 7 7 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { 0 , irem(n1, 7) = 0 | |----- { / \n1 = 0 / n1 \ |---- - 1/7| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 6/7) GAMMA(---- + 2/7) GAMMA(---- + 4/7) GAMMA(---- + 5/7) , irem(n1, 7) = 1 7 7 7 7 7 7 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ /n - 1 | |----- { | | \ { /(n1 + 1)!|, n! | ) { 0 , irem(n1, 7) = 6 | | / { | |----- { / \n1 = 0 / n1 \ |----| \ 7 / n1 n1 n1 n1 n1 n1 (-117649) GAMMA(---- + 4/7) GAMMA(---- + 1) GAMMA(---- + 3/7) GAMMA(---- + 5/7) GAMMA(---- + 2/7) GAMMA(---- + 6/7) , irem(n1, 7) = 0 7 7 7 7 7 7 0 , irem(n1, 7) = 1 0 , irem(n1, 7) = 2 0 , irem(n1, 7) = 3 0 , irem(n1, 7) = 4 0 , irem(n1, 7) = 5 \ | | /(n1 + 1)!|, n!} 0 , irem(n1, 7) = 6 | | / "A060774" {binomial(2 n, n), binomial(3 n, n)} "A060899" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 { ------------------------ n::even { n { (n + 1) binomial(n, n/2) { 4 2 binomial(n, n/2) n::even {{ , { } { (3 n - 3) { (n + 1) { 4 2 { 2 binomial(n + 1, n/2 + 1/2) n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A060900" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 4 (3 n + 5) GAMMA(n/2 + 1) GAMMA(n/2 + 5/3) { 4 4 GAMMA(n/2 + 1/2) GAMMA(n/2 + 7/6) { -------------------------------------------- n::even { -------------------------------------- n::even { (3 n + 4) GAMMA(n/2 + 3/2) GAMMA(n/2 + 11/6) { GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) {{ , { } { (2 n - 2) { (2 n + 2) { 4 2 GAMMA(n/2 + 1/2) GAMMA(n/2 + 7/6) { 2 (3 n + 5) GAMMA(n/2 + 1) GAMMA(n/2 + 5/3) { ---------------------------------------------- n::odd { ---------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) { (3 n + 4) GAMMA(n/2 + 3/2) GAMMA(n/2 + 11/6) "A060941" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable not successful" /3125\n |----| GAMMA(n + 3/5) GAMMA(n + 1/5) GAMMA(n + 4/5) GAMMA(n + 2/5) \108 / {-------------------------------------------------------------------} GAMMA(n + 1) GAMMA(n + 4/3) GAMMA(n + 2/3) GAMMA(n + 1/2) "A060944" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- 2 2 || | 2 2 | \ (n2 + 1) (n2!) || | (n1 + 1) (n1!) | ) ----------------------|| |n - 1 | / 2 2|| |----- |----- (n2 + 3) ((n2 + 1)!) || 2 2 2 2 | \ \n2 = 0 /| 2 2 {(n + 1) (n!) , (n + 1) (n!) | ) ------------------------------------------------|, (n + 2) (n + 1) (n!) } | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A061062" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 2 {1, ) (n1 + 1) (n1!) } / ----- n1 = 0 "A061213" 2 n 2 {1, (n + 2) (n + 1) (1/2) (n!) } "A061572" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 2 2 | \ (n1 + 1) n1! | {(n + 1) (n!) , (n + 1) (n!) | ) ----------------------|} | / 2 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A061573" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 2 2 | \ (n1 + 1) n1! | {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A061575" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A061639" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A061640" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) (n1!) | {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A061714" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \ \\ |----- | |----- | |----- n2 (-n2 - 1)| || n n | \ n1 n1 | n | \ | n1 n1 | \ (-1) 2 | || {(-1) , (-1) | ) (-(-1) 2 n1!)|, (-1) | ) |-(-1) 2 | ) -----------------| n1!||} | / | | / | | / (n2 + 1)! | || |----- | |----- | |----- | || \n1 = 0 / \n1 = 0 \ \n2 = 0 / // "A061834" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {(n - 1) n, (n - 1) n | ) n1!|} | / | |----- | \n1 = 0 / "A062106" LREtools/SearchTable: "SearchTable successful" n 2 3 2 (-1) ((n + 1) (5 n - 13 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-5 n + 24 n - 22 n + 9) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------------------------------} n (n - 1) (n - 2) "A062146" LREtools/SearchTable: "SearchTable successful" {((n - 4) LaguerreL(n + 1, 1) + (-n + 1) LaguerreL(n, 1)) n! (n + 1)} "A062147" LREtools/SearchTable: "SearchTable successful" {((n + 2) LaguerreL(n + 1, -1) + (-n - 3) LaguerreL(n, -1)) n! (n + 1)} "A062191" LREtools/SearchTable: "SearchTable successful" 2 2 {((n - 30 n + 34) LaguerreL(n + 1, 1) + (-n + 22 n - 1) LaguerreL(n, 1)) n! (n + 1)} "A062192" LREtools/SearchTable: "SearchTable successful" {((n + 10) (n + 2) LaguerreL(n + 1, -1) - (n + 13) (n + 3) LaguerreL(n, -1)) n! (n + 1)} "A062197" LREtools/SearchTable: "SearchTable successful" {(n + 1) n! (2 LaguerreL(n + 1, 1) - LaguerreL(n, 1))} "A062236" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) binomial(3 n1, n1) | {(27/4) , (27/4) | ) ------------------------------------------------------|} | / (n1 + 1)| |----- (n1 + 2) (n1 + 1) (2 n1 + 3) (2 n1 + 1) (27/4) | \n1 = 0 / "A062265" LREtools/SearchTable: "SearchTable successful" {((6 n - 10) LaguerreL(n + 1, 1) + (-5 n + 1) LaguerreL(n, 1)) n! (n + 1)} "A062266" LREtools/SearchTable: "SearchTable successful" memory used=34484.3MB, alloc=1559.5MB, time=227.60 {((2 n + 4) LaguerreL(n + 1, -1) + (-3 n - 9) LaguerreL(n, -1)) n! (n + 1)} "A062267" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 1)} "A062282" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A062747" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 4 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) (2 n1 + 3) /| |----- | \n1 = 0 / "A062870" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 4 n binomial(n, n/2) ((n/2)!) n::even {{ 2 , { } { 4 n ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { ------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 2 { (n + 1) "A062992" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A063016" n binomial(2 n, n) 2 binomial(2 n, n) {----------------, -------------------} n + 1 n + 1 "A063017" n n binomial(2 n, n) 2 binomial(2 n, n) 3 binomial(2 n, n) {----------------, -------------------, -------------------} n + 1 n + 1 n + 1 "A063018" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063019" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063020" LREtools/SearchTable: "SearchTable successful" {(24 (3 n + 2) (3 n + 4) (17 n + 7) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) 3 2 / 2 + (-1819 n - 4387 n - 3070 n - 624) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1) / ((2 n + 1) (4 n - 1) (4 n + 1) / (17 n + 24))} "A063022" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063023" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063028" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063030" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063033" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063083" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A063090" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! (2 n1 + 3) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A063395" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 4 n + 3) hypergeom([-1/2, -n], [1], -8) {1, -------------------------------------------------------------------------------------------------------} n "A063419" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A063549" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 binomial(n, n/2) (n + 1) { 4 { -------------------------- n::even { 1/2 -------------------------------- n::even { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {{ , { } { 2 binomial(n + 1, n/2 + 1/2) { (2 n - 2) { ---------------------------- n::odd { 2 (n + 1) { n + 3 { 1/2 ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A063886" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ------------------ n::even { n binomial(n, n/2) { binomial(n, n/2) n::even {{ , { } { (2 n + 2) { 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A064036" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=34957.7MB, alloc=1559.5MB, time=230.49 { 2 2 {{ - 64 ((n + 4) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) + (-n - 24 n - 56) hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) { / 2 binomial(n, n/2) / (n + 2) , n::even / - 384 ((n + 3) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) + (-n - 7) hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)) { 2 { binomial(n - 1, n/2 - 1/2) n/(n + 1) , n::odd, { { { n 96 4 ((n + 3) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) + (-n - 7) hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)) - -------------------------------------------------------------------------------------------------------------------------------------- , 2 (n + 1) binomial(n, n/2) n::even (2 n + 2) - 16 2 2 2 / ((n + 4) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) + (-n - 24 n - 56) hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) / ((n + 1) / 2 (n + 2) binomial(n + 1, n/2 + 1/2)) , n::odd} "A064037" LREtools/SearchTable: "SearchTable successful" ((7 n + 11) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------} 2 (n + 3) (n + 2) (n + 1) "A064062" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064063" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- -------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 3 (-1) 12 binomial(2 n1, n1)|| {(-1/3) , (-1/3) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064088" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 4 (-1) 20 binomial(2 n1, n1)|| {(-1/4) , (-1/4) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064089" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 5 (-1) 30 binomial(2 n1, n1)|| {(-1/5) , (-1/5) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064090" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 6 (-1) 42 binomial(2 n1, n1)|| {(-1/6) , (-1/6) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064091" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 7 (-1) 56 binomial(2 n1, n1)|| {(-1/7) , (-1/7) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064092" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 8 (-1) 72 binomial(2 n1, n1)|| {(-1/8) , (-1/8) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064093" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 9 (-1) 90 binomial(2 n1, n1)|| {(-1/9) , (-1/9) | ) |- --------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064306" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A064310" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (n1 + 1) | n n | \ (-1) 2 binomial(2 n1, n1)| {(1/2) , (1/2) | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A064311" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |3 (-1) 6 binomial(2 n1, n1)|| {(1/3) , (1/3) | ) |-------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064325" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |4 (-1) 12 binomial(2 n1, n1)|| {(1/4) , (1/4) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064326" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |5 (-1) 20 binomial(2 n1, n1)|| {(1/5) , (1/5) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064327" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |6 (-1) 30 binomial(2 n1, n1)|| {(1/6) , (1/6) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064328" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |7 (-1) 42 binomial(2 n1, n1)|| {(1/7) , (1/7) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064329" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |8 (-1) 56 binomial(2 n1, n1)|| {(1/8) , (1/8) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064330" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |9 (-1) 72 binomial(2 n1, n1)|| {(1/9) , (1/9) | ) |--------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064331" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |10 (-1) 90 binomial(2 n1, n1)|| {(1/10) , (1/10) | ) |---------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064332" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |11 (-1) 110 binomial(2 n1, n1)|| {(1/11) , (1/11) | ) |----------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064333" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |12 (-1) 132 binomial(2 n1, n1)|| {(1/12) , (1/12) | ) |----------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A064340" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 4 binomial(2 n1, n1) | {(-2) (9 n - 4), (-2) (9 n - 4) | ) ----------------------------------|} | / (n1 + 1) | |----- (-2) (9 n1 + 5) (9 n1 - 4)| \n1 = 0 / "A064341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 6 binomial(2 n1, n1)|| {(-3/2) (25 n - 12), (-3/2) (25 n - 12) | ) |-2/3 -----------------------------||} | / \ (25 n1 + 13) (50 n1 - 24) /| |----- | \n1 = 0 / "A064342" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 12 binomial(2 n1, n1)|| {(-4/3) (49 n - 24), (-4/3) (49 n - 24) | ) |-3/4 ------------------------------||} | / \ (49 n1 + 25) (147 n1 - 72) /| |----- | \n1 = 0 / "A064343" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 20 binomial(2 n1, n1)|| {(-5/4) (81 n - 40), (-5/4) (81 n - 40) | ) |-4/5 ------------------------------||} | / \ (81 n1 + 41) (324 n1 - 160) /| |----- | \n1 = 0 / "A064344" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 30 binomial(2 n1, n1)|| {(-6/5) (121 n - 60), (-6/5) (121 n - 60) | ) |-5/6 ------------------------------||} | / \ (121 n1 + 61) (605 n1 - 300) /| |----- | \n1 = 0 / "A064345" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 42 binomial(2 n1, n1)|| {(-7/6) (169 n - 84), (-7/6) (169 n - 84) | ) |-6/7 ------------------------------||} | / \ (169 n1 + 85) (1014 n1 - 504) /| |----- | \n1 = 0 / "A064346" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 56 binomial(2 n1, n1)|| {(-8/7) (225 n - 112), (-8/7) (225 n - 112) | ) |-7/8 ------------------------------||} | / \ (225 n1 + 113) (1575 n1 - 784)/| |----- | \n1 = 0 / "A064347" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 72 binomial(2 n1, n1) || {(-9/8) (289 n - 144), (-9/8) (289 n - 144) | ) |-8/9 -------------------------------||} | / \ (289 n1 + 145) (2312 n1 - 1152)/| |----- | \n1 = 0 / "A064570" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! BesselJ(n + 1/2, -1), (-2) n! BesselY(n + 1/2, -1)} "A064613" LREtools/SearchTable: "SearchTable successful" n {2 (hypergeom([-1/2, -n - 1], [1], -2) - hypergeom([-1/2, -n], [1], -2))} "A064641" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n n | \ (-I 3 ) (3 LegendreP(n1 + 1, 3 I) I - LegendreP(n1, 3 I))| {(-1) , (-1) | ) ----------------------------------------------------------------------|, | / (n1 + 1) | |----- (n1 + 2) (-1) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n | \ (-I 3 ) (3 LegendreQ(n1 + 1, 3 I) I - LegendreQ(n1, 3 I))| (-1) | ) ----------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1) | \n1 = 0 / "A064815" memory used=35413.5MB, alloc=1559.5MB, time=233.36 LREtools/SearchTable: "SearchTable successful" / 1/2 n | {- (-2 2 + 2) | \ 1/2 1/2 \ 5 2 1/2 1/2 5 2 1/2 | (6 n + 6) hypergeom([-n - 1/2, 3/4 + ------], [3/2], 4 + 2 2 ) + (1 + 2 ) (13 n + 18) hypergeom([-n + 1/2, 3/4 + ------], [3/2], 4 + 2 2 )| 8 8 / 1/2 n! (2 - 1)/(n (n - 1))} "A064878" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 90 binomial(2 n1, n1) || {(-10/9) (361 n - 180), (-10/9) (361 n - 180) | ) |-9/10 -------------------------------||} | / \ (361 n1 + 181) (3249 n1 - 1620)/| |----- | \n1 = 0 / "A065058" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" (-n) 2 2 3 ((4 n - 3 n - 5) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 3 n - 5) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------------} n (n - 1) "A065065" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A065087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n! n, (n + 1) n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A065088" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n (n - 1) n!, n (n - 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A065096" LREtools/SearchTable: "SearchTable successful" (4 n - 1) LegendreP(n + 1, 3) + 3 LegendreP(n, 3) (4 n - 1) LegendreQ(n + 1, 3) + 3 LegendreQ(n, 3) {-------------------------------------------------, -------------------------------------------------} (n + 3) (n + 2) (n + 3) (n + 2) "A065355" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1! (n1 - 1)} / ----- n1 = 0 "A065409" LREtools/SearchTable: "SearchTable successful" n 2 {2 ((n + 3) (2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-2 n - 6 n - 6) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1)) binomial(2 n, n) (n + 1)/n} "A065601" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 2) | {(-1/2) (9 n + 2), (-1/2) (9 n + 2) | ) --------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) (9 n1 + 11) (18 n1 + 4)| \n1 = 0 / "A065707" LREtools/SearchTable: "SearchTable successful" {(2 n + 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2), (2 n + 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2)} "A065919" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/4), (-1) BesselK(n + 1/2, -1/4)} "A065920" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n - 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((2 n - 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A065921" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((3 n - 1) BesselI(n + 1/2, 1/3) - BesselI(n - 1/2, 1/3)), (-1) ((3 n - 1) BesselK(n + 1/2, -1/3) - BesselK(n - 1/2, -1/3))} "A065922" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n - 1) BesselI(n + 1/2, 1/4) - BesselI(n - 1/2, 1/4)), (-1) ((4 n - 1) BesselK(n + 1/2, -1/4) - BesselK(n - 1/2, -1/4))} "A065923" LREtools/SearchTable: "SearchTable successful" {BesselI(n + 1/2, 1/3), BesselK(n + 1/2, -1/3)} "A065942" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 3 n 3 n { 2 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------- n::even { binomial(n, n/2) {{ , { (-2 n + 2) 3 n 3 n { 3 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 3 n { 3 binomial(---, n/2) n::even { 2 { { 3 n } { 2 binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------- n::odd { 3 n + 1 "A065944" LREtools/SearchTable: "SearchTable successful" {(n + 1) n BesselI(n + 1/2, 1), (n + 1) n BesselK(n + 1/2, -1)} "A065945" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((2 n - 4 n + 3) BesselI(n + 1/2, 1/2) + 3 BesselI(n - 1/2, 1/2)), (-1) ((2 n - 4 n + 3) BesselK(n + 1/2, -1/2) + 3 BesselK(n - 1/2, -1/2)) } "A065946" LREtools/SearchTable: "SearchTable successful" 2 2 {(2 n - 1) BesselI(n + 1/2, 1/2) + BesselI(n - 1/2, 1/2), (2 n - 1) BesselK(n + 1/2, -1/2) + BesselK(n - 1/2, -1/2)} "A065947" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) ((9 n - 15 n + 8) BesselI(n + 1/2, 1/3) + 8 BesselI(n - 1/2, 1/3)), n 2 (-1) ((9 n - 15 n + 8) BesselK(n + 1/2, -1/3) + 8 BesselK(n - 1/2, -1/3))} "A065948" LREtools/SearchTable: "SearchTable successful" 2 2 {(9 n - 3 n - 4) BesselI(n + 1/2, 1/3) + 4 BesselI(n - 1/2, 1/3), (9 n - 3 n - 4) BesselK(n + 1/2, -1/3) + 4 BesselK(n - 1/2, -1/3)} "A065950" LREtools/SearchTable: "SearchTable successful" n 2 3 2 {(-1) ((-n - n - 22) BesselI(n - 1/2, 1) + (n - 6 n + 15 n - 22) BesselI(n + 1/2, 1)), n 3 2 2 (-1) ((n - 6 n + 15 n - 22) BesselK(n + 1/2, -1) + (-n - n - 22) BesselK(n - 1/2, -1))} "A065951" LREtools/SearchTable: "SearchTable successful" 2 {(n - 2) (n + 1) BesselK(n + 1/2, -1) - (n + 2) (n - 1) BesselK(n - 1/2, -1), 2 -(n + 2) (n - 1) BesselI(n - 1/2, 1) + (n - 2) (n + 1) BesselI(n + 1/2, 1)} "A065982" n (2 n + 1) binomial(2 n, n) (n + 2) {4 , ----------------------------------} n + 1 "A066052" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 ), (n + 1) n!} "A066084" 2 {n, (n!) , n!} "A066142" 2 {1, (n!) , n!} "A066143" {(n + 1) n, n!} "A066211" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 2 n1 + 3 | {(n!) binomial(2 n, n), (n!) binomial(2 n, n) | ) ---------------------------------------|, | / 2 | |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / /n - 1 \ |----- | 2 | \ (2 n1 + 1) n1! binomial(2 n1, n1) (n1 + 1)| (n!) binomial(2 n, n) | ) ------------------------------------------|} | / 2 | |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1) | \n1 = 0 / "A066221" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (-n) {1, 2 (2 n + 1) (hypergeom([-n - 1], [1/2], -1/2) - hypergeom([-n], [1/2], -1/2)) binomial(2 n, n) n!} "A066223" LREtools/SearchTable: "SearchTable successful" (-n) {2 binomial(2 n, n) n! hypergeom([-n], [1/2], -1/2)} "A066224" LREtools/SearchTable: "SearchTable successful" (-n) {2 (2 n + 1) (hypergeom([-n - 1], [1/2], -1/2) - hypergeom([-n], [1/2], -1/2)) binomial(2 n, n) n!} "A066237" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1!} / ----- n1 = 0 "A066357" n binomial(4 n, 2 n) 4 binomial(2 n, n) {------------------, -------------------} n + 1 n + 1 "A066380" memory used=35908.2MB, alloc=1591.5MB, time=236.74 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 binomial(3 n1, n1) (5 n1 + 11 n1 + 4)| {8 , 8 | ) ---------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A066381" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 3 2 | n n | \ 2 binomial(4 n1, n1) (44 n1 + 98 n1 + 62 n1 + 11)| {16 , 16 | ) --------------------------------------------------------------|} | / (n1 + 1) (3 n1 + 2) (3 n1 + 1) | |----- | \n1 = 0 / "A066534" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 (n1 + 1)| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A066668" LREtools/SearchTable: "SearchTable successful" {(n + 1) n! (LaguerreL(n + 1, 1) - LaguerreL(n, 1))} "A066796" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) ----------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A066797" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (4 n1 + 5) (4 n1 + 1) (4 n1 + 7) (4 n1 + 3) binomial(4 n1, 2 n1) {1, ) ----------------------------------------------------------------} / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A066798" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (6 n1 + 7) (6 n1 + 1) (2 n1 + 3) (2 n1 + 1) (6 n1 + 11) (6 n1 + 5) binomial(6 n1, 3 n1) {1, ) ---------------------------------------------------------------------------------------} / (n1 + 2) (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) ----- n1 = 0 "A066822" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((27 n + 286 n + 931 n + 918) hypergeom([1/2, -n - 1], [1], 4) + (-27 n - 276 n - 849 n - 762) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 5) (n + 6) (n + 7) "A066989" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 3 | 3 3 3 3 | \ (n1 + 1) (n1!) | {(n + 1) (n!) , (n + 1) (n!) | ) ----------------------|} | / 3 3| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A066998" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A067078" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1!} / ----- n1 = 0 "A067273" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1)| {2 n!, 2 n! | ) -------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A067297" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 {(-2) (27 n - 72 n - 11 n + 16), /n - 1 \ |----- n1 | n 3 2 | \ (2 n1 + 1) 4 binomial(2 n1, n1) (5 n1 - 12) | (-2) (27 n - 72 n - 11 n + 16) | ) -------------------------------------------------------------------------------------|} | / (n1 + 1) 3 2 3 2 | |----- (-2) (27 (n1 + 1) - 72 (n1 + 1) - 11 n1 + 5) (27 n1 - 72 n1 - 11 n1 + 16)| \n1 = 0 / "A067299" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 4 binomial(2 n1, n1) (17 n1 + 27) | {(-2) (9 n + 14), (-2) (9 n + 14) | ) ---------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-2) (9 n1 + 23) (9 n1 + 14)| \n1 = 0 / "A067300" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 | n n | \ 4 binomial(2 n1, n1) (23 n1 + 26 n1 + 12) | {(-2) (9 n + 5), (-2) (9 n + 5) | ) -----------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-2) (9 n1 + 14) (9 n1 + 5)| \n1 = 0 / "A067301" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 3 2 | n n | \ 4 binomial(2 n1, n1) (2222 n1 + 351 n1 - 5843 n1 - 3414) | {(-2) (18 n - 35), (-2) (18 n - 35) | ) -----------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (n1 + 1) (-2) (18 n1 - 17) (18 n1 - 35)| \n1 = 0 / "A067302" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 {(-2) (n + 2) (27 n - 72 n - 11 n + 16), /n - 1 \ |----- n1 | n 3 2 | \ (2 n1 + 1) 4 binomial(2 n1, n1) (5 n1 - 12) | (-2) (n + 2) (27 n - 72 n - 11 n + 16) | ) -------------------------------------------------------------------------------------|} | / (n1 + 1) 3 2 3 2 | |----- (-2) (27 (n1 + 1) - 72 (n1 + 1) - 11 n1 + 5) (27 n1 - 72 n1 - 11 n1 + 16)| \n1 = 0 / "A067305" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 2 {(-2) (n + 1) (27 n - 18 n - 20), /n - 1 \ |----- n1 | n 2 | \ (2 n1 + 1) 4 binomial(2 n1, n1) (11 n1 - 9) | (-2) (n + 1) (27 n - 18 n - 20) | ) --------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (-2) (n1 + 2) (27 (n1 + 1) - 18 n1 - 38) (n1 + 1) (27 n1 - 18 n1 - 20)| \n1 = 0 / "A067306" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 n 3 2 {(-2) (27 n + 90 n + 43 n - 32), (-2) (27 n + 90 n + 43 n - 32) /n - 1 \ |----- n1 2 | | \ (2 n1 + 1) 4 binomial(2 n1, n1) (167 n1 + 603 n1 + 460) n1 | | ) --------------------------------------------------------------------------------------------------------|} | / (n1 + 1) 3 2 3 2 | |----- (n1 + 1) (n1 + 2) (-2) (27 (n1 + 1) + 90 (n1 + 1) + 43 n1 + 11) (27 n1 + 90 n1 + 43 n1 - 32)| \n1 = 0 / "A067318" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) n1! | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A067336" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(9/2) , (9/2) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (9/2) | \n1 = 0 / "A067352" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A067353" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A067369" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ n |----- 2 | (-1) n! | \ (n1 + 1) n1! | {(n + 1) n!, --------, (n + 1) n! | ) ------------------|} n | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A067370" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ n |----- 2 | (-1) n! | \ (n1 + 1) n1! | {(n + 1) n!, --------, (n + 1) n! | ) ------------------|} n | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A067897" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ 1/2 n1 1/2 1/2 1/2 {1, ) (-1/2 I 2 ) (HermiteH(n1 + 1, 1/2 I 2 ) - 2 (n1 + 1) HermiteH(n1, 1/2 I 2 ) I)} / ----- n1 = 0 "A067955" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A068102" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1) n1!| {(n + 1) 2 n!, (n + 1) 2 n! | ) -----------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A068444" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 3 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A068551" n {4 , binomial(2 n, n)} "A068552" n {4 , n binomial(2 n, n)} "A068554" n {4 , n binomial(2 n, n)} "A068764" LREtools/SearchTable: "SearchTable successful" /3 n \ /3 n \ |--- + 1/2| |--- + 1/2| \ 2 / 1/2 1/2 1/2 \ 2 / 1/2 1/2 1/2 2 (-2 LegendreP(n + 1, 2 ) + 2 LegendreP(n, 2 )) 2 (-2 LegendreQ(n + 1, 2 ) + 2 LegendreQ(n, 2 )) {- ------------------------------------------------------------------, - ------------------------------------------------------------------} n n "A068765" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 1/2 6 6 | 1/2 n 1/2 | 1/2 6 6 | (2 6 ) 6 |6 LegendreP(n + 1, ----) - 3 LegendreP(n, ----)| (2 6 ) 6 |6 LegendreQ(n + 1, ----) - 3 LegendreQ(n, ----)| \ 2 2 / \ 2 2 / {-------------------------------------------------------------------, -------------------------------------------------------------------} n n "A068766" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n - 1) hypergeom([-1/2, -n - 1], [1], -2) - 2 n hypergeom([-1/2, -n], [1], -2)) {--------------------------------------------------------------------------------------} n "A068767" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 1/2 5 5 | 1/2 n 1/2 | 1/2 5 5 | (4 5 ) 5 |2 5 LegendreP(n + 1, ----) - 5 LegendreP(n, ----)| (4 5 ) 5 |2 5 LegendreQ(n + 1, ----) - 5 LegendreQ(n, ----)| \ 2 2 / \ 2 2 / {---------------------------------------------------------------------, ---------------------------------------------------------------------} n n "A068768" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n 1/2 | 1/2 30 30 | (2 30 ) 30 |-30 LegendreP(n + 1, -----) + 6 LegendreP(n, -----)| \ 5 5 / {- -------------------------------------------------------------------------, n / 1/2 1/2 \ 1/2 n 1/2 | 1/2 30 30 | (2 30 ) 30 |-30 LegendreQ(n + 1, -----) + 6 LegendreQ(n, -----)| \ 5 5 / - -------------------------------------------------------------------------} n "A068769" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n 1/2 | 1/2 42 42 | (2 42 ) 42 |-42 LegendreP(n + 1, -----) + 7 LegendreP(n, -----)| \ 6 6 / {- -------------------------------------------------------------------------, n / 1/2 1/2 \ 1/2 n 1/2 | 1/2 42 42 | (2 42 ) 42 |-42 LegendreQ(n + 1, -----) + 7 LegendreQ(n, -----)| \ 6 6 / - -------------------------------------------------------------------------} n "A068770" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n 1/2 | 1/2 2 14 2 14 | (4 14 ) 14 |14 LegendreP(n + 1, -------) - 4 LegendreP(n, -------)| \ 7 7 / {----------------------------------------------------------------------------, n / 1/2 1/2 \ 1/2 n 1/2 | 1/2 2 14 2 14 | (4 14 ) 14 |14 LegendreQ(n + 1, -------) - 4 LegendreQ(n, -------)| \ 7 7 / ----------------------------------------------------------------------------} n "A068771" LREtools/SearchTable: "SearchTable successful" n 12 ((2 n - 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n + 1) hypergeom([-1/2, -n], [1], -1)) {----------------------------------------------------------------------------------------------} n "A068772" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n 1/2 | 1/2 10 10 | (6 10 ) 10 |3 10 LegendreQ(n + 1, -----) - 10 LegendreQ(n, -----)| \ 3 3 / {---------------------------------------------------------------------------, n / 1/2 1/2 \ 1/2 n 1/2 | 1/2 10 10 | (6 10 ) 10 |-3 10 LegendreP(n + 1, -----) + 10 LegendreP(n, -----)| \ 3 3 / - ----------------------------------------------------------------------------} n "A069015" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 (n1 + 1) n1!| {(n + 1) 3 n!, (n + 1) 3 n! | ) -----------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A069721" n 2 binomial(2 n, n) (n - 1) {---------------------------} n + 1 "A069728" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A069729" memory used=36427.3MB, alloc=1623.5MB, time=240.26 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A069736" n n (1/2) n! binomial(2 n, n) (3/2) n! binomial(2 n, n) {--------------------------, --------------------------} n + 1 n + 1 "A069835" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 2), 2 LegendreQ(n, 2)} "A069865" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A069948" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A070190" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A070531" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A070733" n (n + 1) (-1) n! (n + 1) n! (2 n - 3) {----------------, --------------------} (n - 1) (n - 2) (n - 1) (n - 2) "A070734" n {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) (-1) n!} "A070779" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {(n + 1) n!, (n + 1) n! LaguerreL(n + 1, -1)} "A070945" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A070946" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A070947" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A070968" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n2 - 1 \\ | |----- 3 2 || | 2 2 | \ 2 n3 + 11 n3 + 18 n3 + 6|| | (n2 + 1) (n2!) | ) --------------------------|| n - 1 n - 1 |n1 - 1 | / 2 2 || n - 1 ----- ----- |----- |----- (n3 + 2) ((n3 + 1)!) || ----- \ 2 2 \ 2 2 | \ \n3 = 0 /| \ 2 2 {1, ) (n1 + 1) (n1!) , ) (n1 + 1) (n1!) | ) ----------------------------------------------------|, ) (n1 + 2) (n1 + 1) (n1!) } / / | / 2 | / ----- ----- |----- ((n2 + 1)!) | ----- n1 = 0 n1 = 0 \n2 = 0 / n1 = 0 "A071007" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A071213" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) --------------------------------------------------------|} | / (n1 + 1) (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A071214" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) --------------------------------------------------------|} | / (n1 + 2) (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A071356" LREtools/SearchTable: "SearchTable successful" n n (-2 I) (-LegendreP(n, I) + LegendreP(n + 1, I) I) (-2 I) (-LegendreQ(n, I) + LegendreQ(n + 1, I) I) {--------------------------------------------------, --------------------------------------------------} n + 2 n + 2 "A071357" LREtools/SearchTable: "SearchTable successful" n n (-2 I) (LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (-2 I) (LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- ---------------------------------------------------------, - ---------------------------------------------------------} (n + 2) (n + 3) (n + 2) (n + 3) "A071359" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A071684" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) (2 n + 1) binomial(2 n, n) { { 2 (-1) binomial(n, n/2) {--------------------------, { (n/2 - 1/2) , { ---------------------------- n::even} (n + 1) (n + 2) { (-16) { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A071688" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) (2 n + 1) binomial(2 n, n) { { 2 (-1) binomial(n, n/2) {--------------------------, { (n/2 - 1/2) , { ---------------------------- n::even} (n + 1) (n + 2) { (-16) { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A071798" 2 2 n 2 {(n + 1) (n!) , (n + 1) (2 n + 1) (1/2) (n!) binomial(2 n, n)} "A071801" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { -------------------------- n::even { 2 2 { 2 { (n + 1) binomial(n, n/2) { 4 binomial(n, n/2) n::even {binomial(2 n, n), { , { } { (4 n - 4) { 2 { 4 2 { binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------ n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) "A071803" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" (3 n + 4) (3 n + 1) (3 n + 5) (3 n + 2) binomial(2 n, n) binomial(3 n, n) {-------------------------------------------------------------------------, 2 2 (n + 1) (n + 2) { 3 n 2 3 n 2 2 { binomial(3 n, ---) binomial(---, n/2) (3 n + 7) (3 n + 5) (3 n + 1) { 2 2 { ---------------------------------------------------------------------- n::even { 4 2 4 { binomial(n, n/2) (n + 3) (n + 1) { , { 2 2 2 3 n 2 3 n 2 { 3 (3 n + 4) (3 n - 2) (3 n + 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { ------------------------------------------------------------------------------------------------ n::odd { 4 4 4 { (n + 2) n binomial(n - 1, n/2 - 1/2) { 2 3 n 2 2 2 { 3 binomial(n, n/2) binomial(---, n/2) (3 n + 2) (3 n + 4) { 2 { ------------------------------------------------------------- n::even { 4 { (n + 2) { } { 2 3 n 2 { binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) (3 n + 7) (3 n + 5) { 2 { ------------------------------------------------------------------------------- n::odd { 2 { (n + 3) "A071879" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A071896" LREtools/SearchTable: "SearchTable not successful" {} "A072100" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | |{ --------------------------- n1::even| n - 1 |{ n1 | n - 1 /{ n1 \ ----- |{ (n1 + 1) binomial(n1, ----) | ----- |{ 2 binomial(n1, ----) n1::even| \ |{ 2 | \ |{ 2 | {1, ) |{ |, ) |{ |} / |{ (2 n1 - 2) | / |{ n1 | ----- |{ 2 2 | ----- |{ binomial(n1 + 1, ---- + 1/2) n1::odd | n1 = 0 |{ ------------------------------- n1::odd | n1 = 0 \{ 2 / |{ n1 | |{ n1 binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A072131" memory used=36920.8MB, alloc=1623.5MB, time=243.65 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A072132" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A072133" memory used=37139.3MB, alloc=1623.5MB, time=245.87 memory used=37276.6MB, alloc=1623.5MB, time=247.65 memory used=37399.0MB, alloc=1623.5MB, time=249.40 memory used=37544.4MB, alloc=1623.5MB, time=251.22 memory used=37670.6MB, alloc=1623.5MB, time=252.90 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A072167" memory used=37897.8MB, alloc=1623.5MB, time=255.53 memory used=38045.3MB, alloc=1623.5MB, time=257.39 memory used=38173.4MB, alloc=1624.7MB, time=259.11 memory used=38323.2MB, alloc=1623.5MB, time=260.97 memory used=38456.1MB, alloc=1623.5MB, time=262.69 memory used=38583.6MB, alloc=1625.3MB, time=264.59 memory used=38738.5MB, alloc=1623.5MB, time=266.34 memory used=38910.3MB, alloc=1625.8MB, time=268.22 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A072331" LREtools/SearchTable: "SearchTable successful" n n {(-2) ((4 n + 2) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-2) ((4 n + 2) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A072346" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 0 n::even {{ , { , { (-n + 1) { (- n/2 + 1/2) { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { (n/2) { (n/2)! n::even { 2 (n/2)! n::even, { } { { 0 n::odd { 0 n::odd "A072371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) binomial(2 n1, n1) n1! || {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A072372" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) (2 n1 + 1) binomial(2 n1, n1) n1! || {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A072374" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) (HermiteH(n + 1, 1/2 I) + 2 I (n + 1) HermiteH(n, 1/2 I)) I} "A072547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (3 n1 + 1)| {(-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A072678" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 2 {(2 n + 1) (n!) binomial(2 n, n) LaguerreL(2 n, -1)} "A073155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 2)} "A073156" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 2 {(-1) (n + 12 n + 59 n + 84)} "A073157" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A073178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A073190" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 0 n::even { n binomial(2 n, n) n { { 2 4 {------------------, { 2 binomial(n - 1, n/2 - 1/2) , { -------------------------- n::even} (n + 1) (2 n - 1) { ---------------------------- n::odd { n (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A073192" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 0 n::even { 2 binomial(n, n/2) (2 n + 1) binomial(2 n, n) { { ------------------ n::even {--------------------------, { (2 n - 2) , { n + 2 } (n + 1) (n + 2) { 2 { { ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A073525" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 7 _Z + 14 _Z - 9, index = 1) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 2) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 3) } "A073591" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 - 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A073596" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ / n1 - 1 /n2 - 1 \\ | ----- | | ----- |----- || | \ | | \ | \ 1 || | ) (n2 + 1) n2!| | ) (n2 + 1) n2! | ) ------------------|| /n - 1 \ |n - 1 / | |n - 1 / | / (n3 + 2) (n3 + 1)!|| |----- | |----- ----- | |----- ----- |----- || | \ 1 | | \ n2 = 0 | | \ n2 = 0 \n3 = 0 /| {n! | ) ---------|, n! | ) -------------------|, n! | ) -----------------------------------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / "A073701" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A073767" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A073829" {n + 5, n!} "A073913" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 11) {4 , --------------------------------------------------------------------------------------} (n + 8) (n + 7) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A074189" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -2), (-1) BesselY(n, -2)} "A074635" LREtools/SearchTable: "SearchTable successful" hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + (-12 n - 5) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1) {---------------------------------------------------------------------------------------------------------------------} 3 n "A074702" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 2 2 | \ (-1) | {(n + 1) (n + 2) (n!) , (n + 1) (n + 2) (n!) | ) --------------------------------|} | / 2 2 2| |----- (n1 + 2) (n1 + 3) ((n1 + 1)!) | \n1 = 0 / "A074703" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 2 2 | \ 1 | {(n!) (n + 1) , (n!) (n + 1) | ) ----------------------|} | / 2 2| |----- ((n1 + 1)!) (n1 + 2) | \n1 = 0 / "A074790" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(2 n + 1) (n!) binomial(2 n, n), (2 n + 1) (n!) binomial(2 n, n) | ) --------------------------------------------------|} | / 2 | |----- (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A075045" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (7 n1 + 15) | {(27/4) , (27/4) | ) -----------------------------------------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (27/4) | \n1 = 0 / "A075125" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A075374" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -2), (-1) BesselY(n, -2)} "A075436" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) | {(16/3) , (16/3) | ) --------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (16/3) | \n1 = 0 / "A076025" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(16/3) , (16/3) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (16/3) | \n1 = 0 / "A076026" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(25/4) , (25/4) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (25/4) | \n1 = 0 / "A076035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(16/3) , (16/3) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (16/3) | \n1 = 0 / "A076036" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(25/4) , (25/4) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (25/4) | \n1 = 0 / "A076051" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (- n/2) { (n/2) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { 1/2 2 (n/2)! n::even {{ , { } { (- n/2 - 1/2) { (n/2 - 1/2) { 2 binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd { (n/2 + 1/2) 2 (n/2 - 1/2)! n::odd "A076128" {(n + 1) (n + 2), (n + 1) n!} "A076176" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ / { 0 n1::even\ | { 4 | | { | | { --------------------------- n1::even| | { n1 | | { n1 | | { 2 binomial(n1 - 1, ---- - 1/2) n1 | | { (n1 + 1) binomial(n1, ----) | /n - 1 \ |n - 1 { 2 | |n - 1 { 2 | |----- n1 | |----- { --------------------------------- n1::odd | |----- { | | \ 2 | | \ { n1 + 1 | | \ { 0 n1::odd | {n! | ) ---------|, n! | ) ---------------------------------------------------|, n! | ) ---------------------------------------------|, | / (n1 + 1)!| | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / n!} "A076177" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / { 0 irem(n1, 3) = 0\ | { | | { 0 irem(n1, 3) = 1| | { | | { n1 2 n1 n1 | | { 9 binomial(n1 - 2, ---- - 2/3) binomial(---- - 4/3, ---- - 2/3) (n1 - 1) n1 | | { 3 3 3 | /n - 1 \ |n - 1 { --------------------------------------------------------------------------- irem(n1, 3) = 2| |----- n1 | |----- { 2 | | \ 3 | | \ { (n1 + 1) | {n! | ) ---------|, n! | ) ----------------------------------------------------------------------------------------------------|, | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / { 0 irem(n1, 3) = 0\ | { | | { (n1 - 1) n1 n1 | | { 3 GAMMA(---- + 1) GAMMA(---- + 2/3) | | { 3 3 | | { ------------------------------------------- irem(n1, 3) = 1| | { n1 2 | | { GAMMA(---- + 4/3) | |n - 1 { 3 | |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) --------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { n1 n1 n1 \ | { 3 GAMMA(---- + 1) GAMMA(---- + 2/3) | | { 3 3 | | { ------------------------------------- irem(n1, 3) = 0| | { n1 2 | | { GAMMA(---- + 4/3) | | { 3 | | { | |n - 1 { 0 irem(n1, 3) = 1| /n - 1 \ |----- { | |----- n1 | | \ { 0 irem(n1, 3) = 2| | \ (-1) hypergeom([1/2, -n1 - 1], [1], 4)| n! | ) --------------------------------------------------------------|, n! | ) ----------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A076657" 3 n {binomial(2 n, n) , 16 binomial(2 n, n)} "A076729" memory used=39407.0MB, alloc=1623.5MB, time=271.77 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) --------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A076795" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (-n1) {1, ) 2 n1! binomial(2 n1, n1)} / ----- n1 = 0 "A077138" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A077419" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A077568" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 1) n1 | n n | \ 2 (-1) (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) ----------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A077611" 2 n {n! (2 n + 2 n - 1), (-1) n! (2 n + 1)} "A077613" n 2 {(-1) n!, n! (2 n + 4 n + 1)} "A077745" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) (3/2) n1! binomial(2 n1, n1)| {(-2) n!, (-2) n! | ) -----------------------------------------|} | / (n1 + 1) | |----- (-2) (n1 + 1)! | \n1 = 0 / "A078009" LREtools/SearchTable: "SearchTable successful" n n 4 (2 LegendreP(n + 1, 3/2) - 3 LegendreP(n, 3/2)) 4 (2 LegendreQ(n + 1, 3/2) - 3 LegendreQ(n, 3/2)) {--------------------------------------------------, --------------------------------------------------} n n "A078018" memory used=39830.3MB, alloc=1623.5MB, time=274.88 LREtools/SearchTable: "SearchTable successful" n n 5 (5 LegendreP(n + 1, 7/5) - 7 LegendreP(n, 7/5)) 5 (5 LegendreQ(n + 1, 7/5) - 7 LegendreQ(n, 7/5)) {--------------------------------------------------, --------------------------------------------------} n n "A078478" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 6 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" / / /{ n1 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n1::even||| |n - 1 | |{ n1 ||| |----- | |{ (n1 + 3) binomial(n1, ----) ||| n n n n | \ | n1 |{ 2 ||| {(-1) , (-I) , I , (-1) | ) |-(-1) |{ |||, | / | |{ (2 n1 - 2) ||| |----- | |{ 2 (n1 + 1) ||| |n1 = 0 | |{ ---------------------------------------- n1::odd ||| | | |{ n1 ||| | | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n1 \\\ | | |{ 4 binomial(n1, ----) (n1 + 1) ||| |n - 1 | |{ 2 ||| |----- | |{ ----------------------------- n1::even||| n | \ | n1 |{ n1 + 2 ||| (-1) | ) |-(-1) |{ |||, | / | |{ n1 ||| |----- | |{ 2 binomial(n1 + 1, ---- + 1/2) (n1 + 1) ||| |n1 = 0 | |{ 2 ||| | | |{ --------------------------------------- n1::odd ||| \ \ \{ n1 + 3 /// / / { 0 irem(n2, 4) = 0\ \ | | { | | | | { / n2 \ | | | | { |---- - 1/2| | | | | { \ 2 / n2 | | | | { 2 n2 GAMMA(---- + 3/2) | | | | { 4 | | | | { ---------------------------------- irem(n2, 4) = 1| | | | { n2 | | | | { (n2 + 2) GAMMA(---- + 2) | | | | { 4 | | | | { | | | | { 0 irem(n2, 4) = 2| | | | { | | | | { / n2 \ | | | | { |---- - 3/2| | | | | { \ 2 / n2 | | | | { 2 2 GAMMA(---- + 1) | | | | { 4 | | | | { ------------------------------- irem(n2, 4) = 3| | |n - 1 |n1 - 1 { n2 | | |----- |----- { GAMMA(---- + 3/2) | | n | \ n1 | \ { 4 | | (-I) | ) (-1) | ) -----------------------------------------------------------| I|, | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / / / { 0 irem(n2, 4) = 0\ \ | | { | | | | { / n2 \ | | | | { |---- - 1/2| | | | | { \ 2 / n2 | | | | { 2 2 GAMMA(---- + 1) | | | | { 4 | | | | { ------------------------------- irem(n2, 4) = 1| | | | { n2 | | | | { GAMMA(---- + 3/2) | | | | { 4 | | | | { | | | | { 0 irem(n2, 4) = 2| | | | { | | | | { / n2 \ | | | | { |---- + 1/2| | | | | { \ 2 / n2 | | | | { 2 n2 GAMMA(---- + 3/2) | | | | { 4 | | | | { ---------------------------------- irem(n2, 4) = 3| | |n - 1 |n1 - 1 { n2 | | |----- |----- { (n2 + 2) GAMMA(---- + 2) | | n | \ n1 | \ { 4 | | (-I) | ) (-1) | ) -----------------------------------------------------------| I|, | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { 4 2 | | | | { ----------------------------- irem(n2, 4) = 0| | | | { n2 n2 | | | | { (n2 + 2) binomial(----, ----) | | | | { 2 4 | | | | { | | | | { 0 irem(n2, 4) = 1| | | | { | | | | { n2 | | | | { 8 2 n2 | | | | { ------------------------------------------------ irem(n2, 4) = 2| | | | { n2 n2 | | | | { (n2 + 2) (n2 + 4) binomial(---- + 1, ---- + 1/2) | | |n - 1 |n1 - 1 { 2 4 | | |----- |----- { | | n | \ n1 | \ { 0 irem(n2, 4) = 3| | (-I) | ) (-1) | ) -------------------------------------------------------------------------| I|, | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / / / { n2 n2 \ \ | | { 2 binomial(----, ----) n2 | | | | { 2 4 | | | | { ------------------------- irem(n2, 4) = 0| | | | { n2 + 4 | | | | { | | | | { 0 irem(n2, 4) = 1| | | | { | | | | { n2 n2 | | | | { 4 binomial(---- - 1, ---- - 1/2) n2 | | | | { 2 4 | | | | { ----------------------------------- irem(n2, 4) = 2| | |n - 1 |n1 - 1 { n2 + 2 | | |----- |----- { | | n | \ n1 | \ { 0 irem(n2, 4) = 3| | (-I) | ) (-1) | ) ------------------------------------------------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A078480" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 {1, ) (-1) ((n1 + 1) BesselI(n1, 2) + (-n1 - 2) BesselI(n1 - 1, 2)), / ----- n1 = 0 n - 1 ----- \ n1 2 ) (-1) ((n1 + 1) BesselK(n1, -2) + (-n1 - 2) BesselK(n1 - 1, -2))} / ----- n1 = 0 "A078481" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A078482" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A078483" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A078487" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 (44 n + 73 n + 163 n + 158 n + 84) LegendreP(n + 1, 3) + (-256 n - 303 n - 833 n - 618 n - 252) LegendreP(n, 3) {---------------------------------------------------------------------------------------------------------------------, n (n - 1) (n - 2) (n - 3) (n - 4) 4 3 2 4 3 2 (44 n + 73 n + 163 n + 158 n + 84) LegendreQ(n + 1, 3) + (-256 n - 303 n - 833 n - 618 n - 252) LegendreQ(n, 3) ---------------------------------------------------------------------------------------------------------------------} n (n - 1) (n - 2) (n - 3) (n - 4) "A078509" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n + 4) BesselI(n, 2) + (n - 5) BesselI(n - 1, 2)), (-1) ((n + 4) BesselK(n, -2) + (n - 5) BesselK(n - 1, -2))} "A078531" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 3 n 3 n { { binomial(3 n, ---) binomial(---, n/2) { (2 n - 2) 3 n { 2 2 {{ 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) , { ------------------------------------- n::even} { 2 { binomial(n, n/2) (n + 1) { --------------------------------------------------- n::odd { { n (n + 1) { 0 n::odd "A078532" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { {{ n 4 n , { 9 (2 n - 1) (4 n - 5) binomial(--- - 8/3, n/3 - 2/3) { 3 { 1/243 ----------------------------------------------------- irem(n, 3) = 2 { n (n - 1) (n + 1) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) { 6912 GAMMA(n/3 + 7/12) GAMMA(n/3 + 1/12) GAMMA(n/3 + 5/6) , { -------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) { { 0 irem(n, 3) = 2 { (n/3) { 6912 GAMMA(n/3 + 1/12) GAMMA(n/3 + 7/12) GAMMA(n/3 + 5/6) { -------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) GAMMA(n/3 + 2/3) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A078533" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { { 0 irem(n, 4) = 1 { { 0 irem(n, 4) = 2 {{ , { (n/4 - 3/4) 17 13 { 800000 GAMMA(n/4 + 1/20) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + 9/20) { 20 20 { ------------------------------------------------------------------------------------- irem(n, 4) = 3 { GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { 0 irem(n, 4) = 0 { { 0 irem(n, 4) = 1 { { (n/4 - 1/2) 13 17 { 800000 GAMMA(n/4 + 9/20) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + 1/20) , { 20 20 { ------------------------------------------------------------------------------------- irem(n, 4) = 2 { GAMMA(n/4 + 1) GAMMA(n/4 + 5/4) GAMMA(n/4 + 1/2) GAMMA(n/4 + 3/4) { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 0 { { (n/4 - 1/4) 17 13 { 800000 GAMMA(n/4 + 1/20) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + 9/20) { 20 20 { ------------------------------------------------------------------------------------- irem(n, 4) = 1, { GAMMA(n/4 + 5/4) GAMMA(n/4 + 3/4) GAMMA(n/4 + 1/2) GAMMA(n/4 + 1) { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 { (n/4) 13 17 { 800000 GAMMA(n/4 + 1/20) GAMMA(n/4 + --) GAMMA(n/4 + 9/20) GAMMA(n/4 + --) { 20 20 { ------------------------------------------------------------------------------- irem(n, 4) = 0 { GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 1/2) GAMMA(n/4 + 5/4) { } { 0 irem(n, 4) = 1 { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A078534" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" memory used=40343.4MB, alloc=1623.5MB, time=278.27 { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { 0 irem(n, 5) = 2 { {{ 0 irem(n, 5) = 3, { { (n/5 - 4/5) 13 11 { 145800000 GAMMA(n/5 + 7/10) GAMMA(n/5 + --) GAMMA(n/5 + 8/15) GAMMA(n/5 + --) GAMMA(n/5 + 1/30) { 15 30 { ---------------------------------------------------------------------------------------------------------- irem(n, 5) = 4 { GAMMA(n/5 + 6/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 2/5) { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { 0 irem(n, 5) = 2 { { (n/5 - 3/5) 13 11 , { 145800000 GAMMA(n/5 + --) GAMMA(n/5 + --) GAMMA(n/5 + 1/30) GAMMA(n/5 + 7/10) GAMMA(n/5 + 8/15) { 15 30 { ---------------------------------------------------------------------------------------------------------- irem(n, 5) = 3 { GAMMA(n/5 + 2/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { (n/5 - 2/5) 11 13 { 145800000 GAMMA(n/5 + 1/30) GAMMA(n/5 + 7/10) GAMMA(n/5 + --) GAMMA(n/5 + --) GAMMA(n/5 + 8/15) { 30 15 , { ---------------------------------------------------------------------------------------------------------- irem(n, 5) = 2 { GAMMA(n/5 + 3/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { { (n/5 - 1/5) 11 13 { 145800000 GAMMA(n/5 + 1/30) GAMMA(n/5 + 8/15) GAMMA(n/5 + --) GAMMA(n/5 + --) GAMMA(n/5 + 7/10) { 30 15 { ---------------------------------------------------------------------------------------------------------- irem(n, 5) = 1 { GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 1) , { { 0 irem(n, 5) = 2 { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 { (n/5) 13 11 { 145800000 GAMMA(n/5 + 8/15) GAMMA(n/5 + --) GAMMA(n/5 + 1/30) GAMMA(n/5 + 7/10) GAMMA(n/5 + --) { 15 30 { ---------------------------------------------------------------------------------------------------- irem(n, 5) = 0 { GAMMA(n/5 + 3/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 6/5) { { 0 irem(n, 5) = 1} { { 0 irem(n, 5) = 2 { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 "A078623" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A078678" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A078679" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A078700" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A078738" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A078791" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 3| 2 | \ (2 n1 + 1) ((2 n1)!) ((n1 + 1)!) | ((2 n)!) | ) ----------------------------------| | / 2 3 2 | 2 |----- (n1 + 1) (n1!) ((2 n1 + 2)!) | ((2 n)!) \n1 = 0 / {---------, -----------------------------------------------------} 3 3 (n!) (n!) "A078995" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ (4 n1 + 1) binomial(4 n1, n1) (8 n1 + 5) | {|---| , |---| | ) --------------------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (3 n1 + 2) (3 n1 + 1) (n1 + 1) |---| | \n1 = 0 \27 / / "A079165" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/2), (-1) BesselK(n + 1/2, -1/2)} "A079280" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 6 4 { ------------------ n::even n { n binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n + 2) { 3 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A079309" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) ----------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A079489" n 4 binomial(2 n, n) (4 n + 1) binomial(4 n, 2 n) {-------------------, ----------------------------} n + 1 (n + 1) (2 n + 1) "A079514" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 n { -------------------------------- n::even (2 n + 1) binomial(2 n, n) (5 n + 9) (2 n + 1) (-1) binomial(2 n, n) (7 n + 15) { (n + 1) (n + 3) binomial(n, n/2) {------------------------------------, -------------------------------------------, { , (n + 3) (n + 2) (n + 1) (n + 3) (n + 2) (n + 1) { (3 n - 3) { 2 { - ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) { n { 2 2 binomial(n, n/2) { - --------------------- n::even { n + 2 { } { (n + 1) { 2 2 binomial(n + 1, n/2 + 1/2) { ------------------------------------- n::odd { n + 3 "A079515" n (2 n + 1) 4 binomial(2 n, n) (4 n + 1) (4 n + 3) binomial(4 n, 2 n) (16 n + 23) {-----------------------------, --------------------------------------------------} (n + 1) (n + 2) (n + 2) (2 n + 3) (n + 1) (2 n + 1) "A079516" n 2 (2 n + 1) 4 binomial(2 n, n) (19 n + 33) (4 n + 1) (4 n + 3) binomial(4 n, 2 n) (110 n + 349 n + 267) {-----------------------------------------, -------------------------------------------------------------} (n + 3) (n + 2) (n + 1) (n + 3) (n + 2) (n + 1) (2 n + 3) (2 n + 1) "A079517" n 2 (2 n + 1) 4 binomial(2 n, n) (27 n + 41) (4 n + 5) (4 n + 1) (4 n + 3) binomial(4 n, 2 n) (226 n + 955 n + 981) {-----------------------------------------, -----------------------------------------------------------------------} (n + 3) (n + 2) (n + 1) (n + 3) (n + 2) (n + 1) (2 n + 5) (2 n + 3) (2 n + 1) "A079518" n 2 (2 n + 3) (2 n + 1) 4 binomial(2 n, n) (13 n + 40) (4 n + 5) (4 n + 1) (4 n + 7) (4 n + 3) binomial(4 n, 2 n) (74 n + 407 n + 549) {---------------------------------------------------, --------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) (n + 4) (n + 3) (n + 2) (n + 1) (2 n + 5) (2 n + 3) (2 n + 1) "A079519" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 (4 n1 + 5) (4 n1 + 1) (4 n1 + 7) (4 n1 + 3) binomial(4 n1, 2 n1) (64 n1 + 277 n1 + 294)| {16 , 16 | ) -----------------------------------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / n (2 n + 3) (2 n + 1) 4 binomial(2 n, n) (11 n + 25) ---------------------------------------------------} (n + 3) (n + 2) (n + 1) "A079522" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n 2 (2 n + 1) (-1) binomial(2 n, n) (41 n + 209 n + 240) (2 n + 1) binomial(2 n, n) (3 n + 7) n {------------------------------------------------------, --------------------------------------, (n + 4) (n + 3) (n + 2) (n + 1) (n + 4) (n + 3) (n + 2) (n + 1) { n { n { 48 8 { 4 2 (17 n + 20) binomial(n, n/2) { - -------------------------------- n::even { --------------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) { , { } { (3 n + 3) { (n - 1) { 2 (17 n + 20) { 192 2 n binomial(n - 1, n/2 - 1/2) { -------------------------------------------------- n::odd { - ----------------------------------------- n::odd { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) "A079678" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2 \| /3125\n /3125\n | \ |256 (1000 n1 + 1195 n1 + 339) GAMMA(n1 + 2/5) GAMMA(n1 + 3/5) GAMMA(n1 + 4/5) GAMMA(n1 + 6/5)|| {|----| , |----| | ) |---- ------------------------------------------------------------------------------------------||} \256 / \256 / | / \3125 GAMMA(n1 + 2) GAMMA(n1 + 3/2) GAMMA(n1 + 5/4) GAMMA(n1 + 7/4) /| |----- | \n1 = 0 / "A079727" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 3 3 \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -------------------------------} / 3 ----- (n1 + 1) n1 = 0 "A079750" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A079751" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A079752" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A079753" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / "A079754" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A079756" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|, | / (n1 + 4) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / /n - 1 \ |----- | | \ 2 n1 + 1 | (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 4) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A079884" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 5 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A079949" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, 3 I)} "A080047" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {(n + 1) (n + 2) n!, (n + 1) (n + 2) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A080171" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 {(1/2 - 1/2 I 3 ) n! hypergeom([-n, 1/2 - 1/6 I 3 ], [1], 3/2 - 1/2 I 3 )} "A080227" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ (-1) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A080243" LREtools/SearchTable: "SearchTable successful" n n (-1) (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) (-1) (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {-----------------------------------------------, -----------------------------------------------} n n "A080244" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z + _Z + _Z - 1, index = 1) , RootOf(_Z + _Z + _Z - 1, index = 2) , RootOf(_Z + _Z + _Z - 1, index = 3) } "A080252" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-2) | | \ 2 | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A080568" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| , n!} \ 2 / \ 2 / "A080599" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 3 | | 3 | {|1/2 - ----| n!, |1/2 + ----| n!} \ 2 / \ 2 / "A080609" LREtools/SearchTable: "SearchTable successful" /3 n\ /3 n\ |---| |---| \ 2 / 1/2 \ 2 / 1/2 {2 LegendreP(n, 2 ), 2 LegendreQ(n, 2 )} "A080835" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A080836" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A080893" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n - 1/2, 1/2), (-1) BesselK(n - 1/2, -1/2)} "A080894" memory used=40838.6MB, alloc=1623.5MB, time=281.78 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A080896" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A080954" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 5 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A080958" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! (n1 + 3)| {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A081021" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081046" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / n1! \| {(-1) n!, (-1) n! | ) |- ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A081047" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A081048" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / n1! \| {(-1) n!, (-1) n! | ) |- ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A081051" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||} | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A081052" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||} | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A081085" LREtools/SearchTable: "SearchTable successful" 2 {((n + 3) (2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-2 n - 2 n + 2) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1)) binomial(2 n, n)/(n + 1)} "A081113" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 2) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 1) hypergeom([1/2, -n], [1], 4)), 3 (n + 2)} "A081123" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A081124" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) HermiteH(n + 1, I)} "A081125" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 (n/2)! n::even { { binomial(n, n/2) (n/2)! n::even {{ (n + 1) , { } { 2 (n/2 + 1/2)! { n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { --------------------- n::odd { n + 1 "A081126" LREtools/SearchTable: "SearchTable successful" n {(-I) (HermiteH(n + 1, 1/2 I) + HermiteH(n, 1/2 I) I)} "A081178" LREtools/SearchTable: "SearchTable successful" n n 6 (3 LegendreP(n + 1, 4/3) - 4 LegendreP(n, 4/3)) 6 (3 LegendreQ(n + 1, 4/3) - 4 LegendreQ(n, 4/3)) {--------------------------------------------------, --------------------------------------------------} n n "A081181" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 2) { 2 { ---------------------------- n::even { n + 2 {{ , { 3 n { 2 binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------- n::odd { n + 3 { (-n) 3 n 3 n { 2 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 3) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 2 (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ------------------------------------------------------------------------------------------- n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A081204" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 3 n 3 n { 2 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 1) binomial(n, n/2) {{ , { (-2 n + 2) 3 n 3 n { 3 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 3 n { 3 binomial(---, n/2) n::even { 2 { } { 3 n { 2 binomial(--- + 3/2, n/2 + 1/2) n::odd { 2 "A081205" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 4) (3 n + 2) { 2 { -------------------------------------- n::even { (n + 2) (n + 4) {{ , { 3 n { 2 binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------- n::odd { n + 3 { (-n) 3 n 3 n { 2 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 3) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 2 (3 n - 2) (3 n + 2) (3 n + 4) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A081207" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081293" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) (7 n1 + 2) {1, ) -----------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A081358" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1! | | \ (-1) n1!| {n! | ) ---------|, n! | ) ----------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A081367" LREtools/SearchTable: "SearchTable successful" n {(-2) ((n + 1) LaguerreL(n + 1, -n - 1/2, 1) + LaguerreL(n, -n + 1/2, 1)) n!} "A081405" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (- n/2) {{ , { 2 (n + 1) binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2) { { (n/2 + 1/2) 2 (n/2 - 1/2)! n::odd { 0 n::odd "A081406" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { {{ 0 irem(n, 3) = 1, { (n/3 - 1/3) , { { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 1 { (n/3 - 2/3) { { (n/3 + 1/3) 3 (n/3 - 2/3)! irem(n, 3) = 2 { 0 irem(n, 3) = 2 { (n/3) { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 0 { } { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A081407" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 {{ , { , { 0 irem(n, 4) = 2 { (n/2 - 1) { { 2 GAMMA(n/4 + 5/4) irem(n, 4) = 2 { (n/2 - 3/2) { { 2 (n/4 + 1/4) (n/4 - 3/4)! irem(n, 4) = 3 { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 0 { (n/2) { { 2 GAMMA(n/4 + 5/4) irem(n, 4) = 0 { (n/2 + 1/2) (n/4 - 1/4)! binomial(n/2 - 1/2, n/4 - 1/4) irem(n, 4) = 1 { { , { 0 irem(n, 4) = 1} { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 2 { 0 irem(n, 4) = 3 { { 0 irem(n, 4) = 3 "A081408" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { 0 irem(n, 5) = 0 { 0 irem(n, 5) = 0 { { { 0 irem(n, 5) = 1 { 0 irem(n, 5) = 1 { { {{ 0 irem(n, 5) = 2, { 0 irem(n, 5) = 2, { { { 0 irem(n, 5) = 3 { (n/5 - 3/5) { { 5 GAMMA(n/5 + 6/5) irem(n, 5) = 3 { (n/5 - 4/5) { { (n/5 + 1/5) 5 (n/5 - 4/5)! irem(n, 5) = 4 { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { 0 irem(n, 5) = 0 { { { 0 irem(n, 5) = 1 { (n/5 - 1/5) { { 5 GAMMA(n/5 + 6/5) irem(n, 5) = 1 { (n/5 - 2/5) , { , { 5 GAMMA(n/5 + 6/5) irem(n, 5) = 2 { 0 irem(n, 5) = 2 { { { 0 irem(n, 5) = 3 { 0 irem(n, 5) = 3 { { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 4 { (n/5) { 5 GAMMA(n/5 + 6/5) irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { } { 0 irem(n, 5) = 2 { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 "A081495" binomial(2 n, n) (3 n + 1) {--------------------------, 2 n - 1} n + 1 "A081668" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {1, (-1) , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A081669" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {2 , (-1) hypergeom([1/2, -n], [1], 4)} "A081670" n {1, 3 , binomial(2 n, n)} "A081671" LREtools/SearchTable: "SearchTable successful" n {2 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 3) hypergeom([-1/2, -n], [1], -2))} "A081672" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {2 , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A081673" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n {1, 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A081696" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A081698" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081798" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081919" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081920" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081921" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081922" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A081923" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 | {n! (n - 1), n! (n - 1) | ) ---------------------|} | / (n1 + 1)! n1 (n1 - 1)| |----- | \n1 = 0 / "A081924" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 3 | {n! (n - 2), n! (n - 2) | ) ---------------------------|} | / (n1 + 1)! (n1 - 1) (n1 - 2)| |----- | \n1 = 0 / "A082028" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 2 n1 + 5 | {n! (2 n - 1), n! (2 n - 1) | ) -------------------------------|} | / (n1 + 1)! (2 n1 + 1) (2 n1 - 1)| |----- | \n1 = 0 / "A082029" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 (2 n1 + 7) | {n! (2 n - 3), n! (2 n - 3) | ) -------------------------------|} | / (n1 + 1)! (2 n1 - 1) (2 n1 - 3)| |----- | \n1 = 0 / "A082030" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {n! (n + n + 1), n! (n + n + 1) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + n1 + 2) (n1 + n1 + 1)| \n1 = 0 / "A082031" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ 2 | {n! (n - n + 2), n! (n - n + 2) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) - n1 + 1) (n1 - n1 + 2)| \n1 = 0 / "A082032" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 1 \| {2 n!, 2 n! | ) |1/2 ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A082147" LREtools/SearchTable: "SearchTable successful" n n 7 (7 LegendreP(n + 1, 9/7) - 9 LegendreP(n, 9/7)) 7 (7 LegendreQ(n + 1, 9/7) - 9 LegendreQ(n, 9/7)) {--------------------------------------------------, --------------------------------------------------} n n "A082148" LREtools/SearchTable: "SearchTable successful" n n 9 (9 LegendreP(n + 1, 11/9) - 11 LegendreP(n, 11/9)) 9 (9 LegendreQ(n + 1, 11/9) - 11 LegendreQ(n, 11/9)) {-----------------------------------------------------, -----------------------------------------------------} n n "A082173" LREtools/SearchTable: "SearchTable successful" n n 10 (5 LegendreP(n + 1, 6/5) - 6 LegendreP(n, 6/5)) 10 (-5 LegendreQ(n + 1, 6/5) + 6 LegendreQ(n, 6/5)) {---------------------------------------------------, - ----------------------------------------------------} n n "A082181" LREtools/SearchTable: "SearchTable successful" n 4 ((6 n - 2) hypergeom([-1/2, -n - 1], [1], -3) + (-6 n - 1) hypergeom([-1/2, -n], [1], -3)) {---------------------------------------------------------------------------------------------} n "A082298" LREtools/SearchTable: "SearchTable successful" (8 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------} n "A082301" memory used=41345.8MB, alloc=1623.5MB, time=285.29 LREtools/SearchTable: "SearchTable successful" n n 4 (2 LegendreP(n + 1, 3/2) - 3 LegendreP(n, 3/2)) 4 (2 LegendreQ(n + 1, 3/2) - 3 LegendreQ(n, 3/2)) {--------------------------------------------------, --------------------------------------------------} n n "A082302" LREtools/SearchTable: "SearchTable successful" n n 5 (5 LegendreP(n + 1, 7/5) - 7 LegendreP(n, 7/5)) 5 (5 LegendreQ(n + 1, 7/5) - 7 LegendreQ(n, 7/5)) {--------------------------------------------------, --------------------------------------------------} n n "A082305" LREtools/SearchTable: "SearchTable successful" n n 6 (3 LegendreP(n + 1, 4/3) - 4 LegendreP(n, 4/3)) 6 (3 LegendreQ(n + 1, 4/3) - 4 LegendreQ(n, 4/3)) {--------------------------------------------------, --------------------------------------------------} n n "A082366" LREtools/SearchTable: "SearchTable successful" n n 7 (7 LegendreP(n + 1, 9/7) - 9 LegendreP(n, 9/7)) 7 (7 LegendreQ(n + 1, 9/7) - 9 LegendreQ(n, 9/7)) {--------------------------------------------------, --------------------------------------------------} n n "A082367" LREtools/SearchTable: "SearchTable successful" n 4 ((6 n - 2) hypergeom([-1/2, -n - 1], [1], -3) + (-6 n - 1) hypergeom([-1/2, -n], [1], -3)) {---------------------------------------------------------------------------------------------} n "A082395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 \ (-1) ((n1 + 3) hypergeom([1/2, -n1 - 1], [1], 4) + (3 n1 + 3) hypergeom([1/2, -n1], [1], 4)) {1, ) ----------------------------------------------------------------------------------------------} / n1 + 2 ----- n1 = 0 "A082397" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 2 2 \ (-1) ((n1 + 16 n1 + 27) hypergeom([1/2, -n1 - 1], [1], 4) + 3 (n1 + 1) hypergeom([1/2, -n1], [1], 4)) {1, ) ---------------------------------------------------------------------------------------------------------} / (n1 + 4) (n1 + 3) ----- n1 = 0 "A082425" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A082426" 2 {1, (n + 1) n!} "A082427" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A082428" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A082430" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A082448" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { (n/2)! n::even { 2 2 binomial(n, n/2) (n/2)! n::even { {{ , { 2 (n/2 + 1/2)! } { (-n + 1) { -------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A082458" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A082459" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A082488" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A082489" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A082491" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) (-1) n1!| {(n!) , (n!) | ) -------------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A082570" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A082573" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A082578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) ------------------------------------------} / 2 ----- (n1 + 2) (n1 + 1) n1 = 0 "A082579" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A082582" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A082590" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 1) binomial(2 n1, n1)| {2 , 2 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A082758" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" {hypergeom([1/2, -2 n], [1], 4)} "A082759" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A082958" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z - 1, index = 1) , RootOf(_Z - _Z - _Z - 1, index = 2) , RootOf(_Z - _Z - _Z - 1, index = 3) } "A082989" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(1/2 - 1/2 I) , (1/2 + 1/2 I) , /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) 2 || n | \ | \ (1/2 + 1/2 I) binomial(2 n2, n2) (3 n2 + 3 n2 + 2)|| (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) ------------------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A083474" 2 2 {(n + 1) n!, (n + 1) (n + 2) } "A083886" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, 3/2 I)} "A084076" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (7 n1 + 9) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (n1 + 3) /| |----- | \n1 = 0 / "A084077" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 2 | n n | \ -I (2 I) ((5 n1 + 12) (n1 + 1) LegendreQ(n1, I) - (11 n1 + 41 n1 + 36) LegendreQ(n1 + 1, I) I)| {(-1) , (-1) | ) -------------------------------------------------------------------------------------------------|, | / (n1 + 2) (n1 + 3) (n1 + 4) | |----- | \n1 = 0 / /n - 1 \ |----- n1 2 | n | \ (2 I) (-(5 n1 + 12) (n1 + 1) LegendreP(n1, I) + (11 n1 + 41 n1 + 36) LegendreP(n1 + 1, I) I) I| (-1) | ) -------------------------------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 3) (n1 + 4) | |----- | \n1 = 0 / "A084080" memory used=41829.1MB, alloc=1655.5MB, time=288.96 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 4 n { binomial(---, n/3) (4 n + 3) { 3 { 8/3 ---------------------------- irem(n, 3) = 0 { (n + 1) (n + 2) { { 4 n { 9 binomial(--- + 8/3, n/3 + 2/3) {{ 3 , { -------------------------------- irem(n, 3) = 1 { 4 n + 5 { { 4 n { 4 binomial(--- + 4/3, n/3 + 1/3) { 3 { -------------------------------- irem(n, 3) = 2 { n + 2 { /256\(n/3) 13 19 { 12 |---| GAMMA(n/3 + 5/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 3) { \27 / 12 12 { ---------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (4 n + 7) { { /256\(n/3 - 1/3) { 8 |---| GAMMA(n/3 + 5/4) GAMMA(n/3 + 3/4) GAMMA(n/3 + 1/2) { \27 / , { --------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) { { /256\(n/3 + 1/3) 17 23 { 27 |---| GAMMA(n/3 + 7/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 4) (n + 3) { \27 / 12 12 { ------------------------------------------------------------------------------------ irem(n, 3) = 2 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 7/3) (4 n + 5) (4 n + 11) { /256\(n/3) 17 23 { 27 |---| GAMMA(n/3 + 7/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 4) (n + 3) { \27 / 12 12 { ------------------------------------------------------------------------------ irem(n, 3) = 0 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 7/3) (4 n + 5) (4 n + 11) { { /256\(n/3 - 1/3) 13 19 { 12 |---| GAMMA(n/3 + 5/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 3) { \27 / 12 12 } { ---------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (4 n + 7) { { /256\(n/3 - 2/3) { 8 |---| GAMMA(n/3 + 3/4) GAMMA(n/3 + 5/4) GAMMA(n/3 + 1/2) { \27 / { --------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) GAMMA(5/3 + n/3) "A084081" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { 5 binomial(---, n/2) n (3 n + 2) { 2 { -------------------------------- n::even { (n + 3) (n + 2) (n + 1) {{ , { 3 n 2 { binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { 3/2 --------------------------------------------------- n::odd { (n + 3) (n + 2) (n + 1) n { (-n) 2 3 n 3 n { 6 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { -------------------------------------------------------- n::even { (n + 1) (n + 2) (n + 3) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 20 2 n binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ---------------------------------------------------------------------------- n::odd { (n + 2) (n + 3) binomial(n + 1, n/2 + 1/2) "A084261" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {-I (-1/2 I) (HermiteH(n + 1, I) - 2 I HermiteH(n, I))} "A084262" LREtools/SearchTable: "SearchTable successful" n {(-2) ((2 n + 2) LaguerreL(n + 1, -n - 1/2, 1/2) + LaguerreL(n, -n + 1/2, 1/2)) n!} "A084543" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 1) n1 | n n | \ 2 (-1) (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) ---------------------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A084601" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 7 ) LegendreP(n, 1/7 I 7 ), (-I 7 ) LegendreQ(n, 1/7 I 7 )} "A084603" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 11 ) LegendreP(n, 1/11 I 11 ), (-I 11 ) LegendreQ(n, 1/11 I 11 )} "A084605" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 15 ) LegendreP(n, 1/15 I 15 ), (-I 15 ) LegendreQ(n, 1/15 I 15 )} "A084607" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) LegendreP(n, I), (-2 I) LegendreQ(n, I)} "A084609" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 I 2 ) LegendreP(n, 1/2 I 2 ), (-2 I 2 ) LegendreQ(n, 1/2 I 2 )} "A084768" LREtools/SearchTable: "SearchTable successful" {LegendreP(n, 7), LegendreQ(n, 7)} "A084769" LREtools/SearchTable: "SearchTable successful" {LegendreP(n, 9), LegendreQ(n, 9)} "A084770" LREtools/SearchTable: "SearchTable successful" n n {(-4 I) LegendreP(n, 1/2 I), (-4 I) LegendreQ(n, 1/2 I)} "A084771" LREtools/SearchTable: "SearchTable successful" {(2 n + 2) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8)} "A084772" LREtools/SearchTable: "SearchTable successful" n n {4 LegendreP(n, 3/2), 4 LegendreQ(n, 3/2)} "A084773" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 3), 2 LegendreQ(n, 3)} "A084774" LREtools/SearchTable: "SearchTable successful" n n {3 LegendreP(n, 7/3), 3 LegendreQ(n, 7/3)} "A084781" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A084782" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A084868" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2 2 + 2) binomial(2 n2, n2)|| {(2 + 2 2 ) , (-2 2 + 2) , (2 + 2 2 ) | ) (-2 2 + 2) (2 + 2 2 ) | ) -----------------------------------------||} | / | / n2 + 1 || |----- |----- || \n1 = 0 \n2 = 0 // "A085110" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A085139" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A085157" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { 0 irem(n, 5) = 0 { 0 irem(n, 5) = 0 { { { 0 irem(n, 5) = 1 { 0 irem(n, 5) = 1 { { {{ 0 irem(n, 5) = 2, { 0 irem(n, 5) = 2, { { { 0 irem(n, 5) = 3 { (n/5 - 3/5) { { 5 GAMMA(n/5 + 1) irem(n, 5) = 3 { (n/5 - 4/5) { { 5 GAMMA(n/5 + 1) irem(n, 5) = 4 { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { 0 irem(n, 5) = 0 { (n/5) { { { 5 (n/5)! irem(n, 5) = 0 { 0 irem(n, 5) = 1 { (n/5 - 1/5) { { { 5 GAMMA(n/5 + 1) irem(n, 5) = 1 { 0 irem(n, 5) = 1 { (n/5 - 2/5) , { , { } { 5 GAMMA(n/5 + 1) irem(n, 5) = 2 { 0 irem(n, 5) = 2 { 0 irem(n, 5) = 2 { { { { 0 irem(n, 5) = 3 { 0 irem(n, 5) = 3 { 0 irem(n, 5) = 3 { { { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 4 "A085158" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" { 0 irem(n, 6) = 0 { 0 irem(n, 6) = 0 { { { 0 irem(n, 6) = 1 { 0 irem(n, 6) = 1 { { { 0 irem(n, 6) = 2 { 0 irem(n, 6) = 2 {{ , { , { 0 irem(n, 6) = 3 { 0 irem(n, 6) = 3 { { { 0 irem(n, 6) = 4 { (n/6 - 2/3) { { 6 GAMMA(n/6 + 1) irem(n, 6) = 4 { (n/6 - 5/6) { { 6 GAMMA(n/6 + 1) irem(n, 6) = 5 { 0 irem(n, 6) = 5 { 0 irem(n, 6) = 0 { 0 irem(n, 6) = 0 { { { 0 irem(n, 6) = 1 { 0 irem(n, 6) = 1 { { { 0 irem(n, 6) = 2 { (n/6 - 1/3) { , { 6 GAMMA(n/6 + 1) irem(n, 6) = 2, { (n/6 - 1/2) { { 1/3 n (3/2) (n/6 - 1/2)! binomial(n/3 - 1, n/6 - 1/2) irem(n, 6) = 3 { 0 irem(n, 6) = 3 { { { 0 irem(n, 6) = 4 { 0 irem(n, 6) = 4 { { { 0 irem(n, 6) = 5 { 0 irem(n, 6) = 5 { 0 irem(n, 6) = 0 { (n/6) { { 6 (n/6)! irem(n, 6) = 0 { (n/6 - 1/6) { { 6 GAMMA(n/6 + 1) irem(n, 6) = 1 { 0 irem(n, 6) = 1 { { { 0 irem(n, 6) = 2, { 0 irem(n, 6) = 2} { { { 0 irem(n, 6) = 3 { 0 irem(n, 6) = 3 { { { 0 irem(n, 6) = 4 { 0 irem(n, 6) = 4 { { { 0 irem(n, 6) = 5 { 0 irem(n, 6) = 5 "A085362" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (8 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 11 n - 1) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------} n "A085363" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (16 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-16 n - 23 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------------------------} n "A085364" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (24 n - 1) hypergeom([-1/2, -n - 1], [1], -12) + (-24 n - 35 n - 5) hypergeom([-1/2, -n], [1], -12) {------------------------------------------------------------------------------------------------------------} n "A085372" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A085386" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 n 1/2 | 2 | 2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), |- ----| HermiteH(n, ----)} \ 2 / 2 "A085387" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 { 0 n::even { (n/2) {(-1/2 I 2 ) HermiteH(n, 2 I), { , { (-1/2) (n/2)! binomial(n, n/2) n::even} { (n/2 - 1/2) { { (-2) (n/2 - 1/2)! n::odd { 0 n::odd "A085403" LREtools/SearchTable: "SearchTable successful" (3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3) (3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3) {-----------------------------------------------------------, -----------------------------------------------------------} (n - 1) n (n - 1) n "A085455" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4) {--------------------------------------------------------------------------------} n "A085456" memory used=42342.0MB, alloc=1655.5MB, time=292.56 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (7 I) 7 ((-n + 3) LegendreP(n, 3/7 I 7 ) + 7 (n + 1) LegendreP(n + 1, 3/7 I 7 ) I) {---------------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (7 I) 7 ((-n + 3) LegendreQ(n, 3/7 I 7 ) + 7 (n + 1) LegendreQ(n + 1, 3/7 I 7 ) I) ---------------------------------------------------------------------------------------------------} n "A085457" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 (11 I) 11 ((n - 5) LegendreP(n, 5/11 I 11 ) - 11 (n + 1) LegendreP(n + 1, 5/11 I 11 ) I) I {--------------------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 (11 I) 11 ((n - 5) LegendreQ(n, 5/11 I 11 ) - 11 (n + 1) LegendreQ(n + 1, 5/11 I 11 ) I) I --------------------------------------------------------------------------------------------------------} n "A085458" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 ((-n + 3) LegendreP(n, 3/7 I 7 ) + 7 (n + 1) LegendreP(n + 1, 3/7 I 7 ) I) {----------------------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 ((-n + 3) LegendreQ(n, 3/7 I 7 ) + 7 (n + 1) LegendreQ(n + 1, 3/7 I 7 ) I) ----------------------------------------------------------------------------------------------------} n "A085614" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 4 binomial(---, n/2) { 3 n 3 n { 2 {{ 2 binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) (3 n - 2) , { --------------------- n::even} { 2 2 { n + 1 { ----------------------------------------------------------------------- n::odd { { binomial(n - 1, n/2 - 1/2) (n + 1) n { 0 n::odd "A085615" LREtools/SearchTable: "SearchTable successful" n 2 2 ((n + 1) (17 n - 8) LegendreP(n + 1, 3) + (-99 n - 3 n + 24) LegendreP(n, 3)) {---------------------------------------------------------------------------------, n (n - 1) (n - 2) n 2 2 ((n + 1) (17 n - 8) LegendreQ(n + 1, 3) + (-99 n - 3 n + 24) LegendreQ(n, 3)) ---------------------------------------------------------------------------------} n (n - 1) (n - 2) "A085644" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 3)| {2 n!, 2 n! | ) ---------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A085781" n (2 n + 1) binomial(2 n, n) {2 , --------------------------} n + 1 "A085812" n n (2 n + 1) binomial(2 n, n) {2 , 4 , --------------------------} n + 1 "A085923" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A086246" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} (n - 1) n "A086325" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A086403" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n + 2) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((4 n + 2) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A086452" LREtools/SearchTable: "SearchTable successful" (4 n + 1) (4 n + 3) binomial(4 n, 2 n) hypergeom([-n - 2, -2 n - 3], [-2 n - 3/2], 1/4) {---------------------------------------------------------------------------------------} (n + 1) (2 n + 3) (2 n + 1) "A086456" LREtools/SearchTable: "SearchTable successful" (3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3) (3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3) {-----------------------------------------------------------, -----------------------------------------------------------} (n - 1) n (n - 1) n "A086581" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086611" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A086613" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A086615" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 \ (-1) ((5 n1 + 9) hypergeom([1/2, -n1 - 1], [1], 4) + (-3 n1 - 3) hypergeom([1/2, -n1], [1], 4)) {1, ) -------------------------------------------------------------------------------------------------} / (n1 + 2) (n1 + 3) ----- n1 = 0 "A086616" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ 3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3) \ 3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3) {1, ) -----------------------------------------, ) -----------------------------------------} / n1 + 2 / n1 + 2 ----- ----- n1 = 0 n1 = 0 "A086618" LREtools/SearchTable: "SearchTable successful" {hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) - 9 hypergeom([1/2, -n, -n], [1, 1], 4)} "A086622" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086625" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086677" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n {(n + 2) (n + 1) (-1) n! (4 n + 9), /n - 1 \ |----- n1 | n | \ -I (-I) n1! ((-5 n1 - 11) LegendreP(n1, 2 I) + (15 n1 + 32) LegendreP(n1 + 1, 2 I) I) (n1 + 1)| (n + 2) (n + 1) (-1) n! (4 n + 9) | ) ------------------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (-1) (n1 + 1)! (4 n1 + 13) (4 n1 + 9) | \n1 = 0 / /n - 1 \ |----- n1 | n | \ -I (-I) n1! ((-5 n1 - 11) LegendreQ(n1, 2 I) + (15 n1 + 32) LegendreQ(n1 + 1, 2 I) I) (n1 + 1)| (n + 2) (n + 1) (-1) n! (4 n + 9) | ) ------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 2) (-1) (n1 + 1)! (4 n1 + 13) (4 n1 + 9) | \n1 = 0 / "A086828" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A086852" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086853" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086855" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A086871" LREtools/SearchTable: "SearchTable successful" (8 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------} n "A086905" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n1 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n1::even||| |n - 1 | |{ n1 ||| |----- | |{ (n1 + 1) binomial(n1, ----) ||| n n | \ | n1 |{ 2 ||| {(-1) , (-1) | ) |-(-1) |{ |||, | / | |{ (2 n1 - 2) ||| |----- | |{ 2 (n1 + 1) ||| |n1 = 0 | |{ ---------------------------------------- n1::odd ||| | | |{ n1 ||| | | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n1 \\\ |n - 1 | |{ 4 binomial(n1, ----) (n1 + 1) ||| |----- | |{ 2 ||| n | \ | n1 |{ ----------------------------- n1::even||| (-1) | ) |-(-1) |{ n1 + 2 |||} | / | |{ ||| |----- | |{ n1 ||| |n1 = 0 | |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd ||| \ \ \{ 2 /// "A086990" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | | { \ 2 / n1 | | { 2 (-1) binomial(n1 - 1, ---- - 1/2) | |n - 1 { 2 | |----- { ----------------------------------------------- n1::odd | n n | \ { n1 + 1 | {(-3/2) , (-3/2) | ) -----------------------------------------------------------------|, | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / | | { 2 (-16) | | { ------------------------------ n1::even| | { n1 | | { (n1 + 1) n1 binomial(n1, ----) | |n - 1 { 2 | |----- { | n | \ { 0 n1::odd | (-3/2) | ) ------------------------------------------------|} | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / "A087137" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (-n) 2 2 {n!, { , { 2 binomial(n, n/2) ((n/2)!) n::even} { (n - 1) 2 { { 2 ((n/2 - 1/2)!) n::odd { 0 n::odd "A087208" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- / n2 \|| | n1 | \ | (-1) ||| | (n1 + 1) (-1) n1! | ) |- ------------------||| |n - 1 | / \ (n2 + 2) (n2 + 1)!/|| |----- |----- || n | \ \n2 = 0 /| {(-1) n!, n! | ) ---------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A087214" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 2 | |2 | {|- ----| n!, |----| n!} \ 2 / \ 2 / "A087299" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n {{ , { 2 (n/2)! n::even} { n (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { { 0 n::odd "A087301" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {n! (n + 1) (n + 2), n! (n + 1) (n + 2) | ) -------------------|} | / (n1 + 1)! (n1 + 2) | |----- | \n1 = 0 / "A087413" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) binomial(3 n1, n1) {1, ) --------------------------------------------------------------} / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A087457" LREtools/SearchTable: "SearchTable successful" {hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4)} "A087547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 n1! | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A087626" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A087804" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A087806" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A087809" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 binomial(3 n1, n1) (5 n1 - 9 n1 - 8)| {8 , 8 | ) --------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A087860" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, 1), n!} "A087912" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -2)} "A087923" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {n! binomial(2 n, n) (n + 3 n + 1), /n - 1 \ |----- n1 | 2 | \ (n1 + 1) 2 n1! | n! binomial(2 n, n) (n + 3 n + 1) | ) ----------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! binomial(2 n1 + 2, n1 + 1) ((n1 + 1) + 3 n1 + 4) (n1 + 3 n1 + 1)| \n1 = 0 / "A087981" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {n! (n + 3), n! (n + 3) | ) ---------------------------|} | / (n1 + 1)! (n1 + 4) (n1 + 3)| |----- | \n1 = 0 / "A088009" LREtools/ReduceToOrderTwo: "Checking Symmetric Cube... (can be time consuming...)" memory used=42858.4MB, alloc=1655.5MB, time=296.24 LREtools/ReduceToOrderTwo: "Galois group is Sp4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A088026" LREtools/SearchTable: "SearchTable successful" 2 (n!) ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) binomial(2 n, n) {---------------------------------------------------------------------------------} n "A088127" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) | {(n + 1) n!, n! (n + 3), n! (n + 3) | ) ---------------------------|} | / (n1 + 1)! (n1 + 4) (n1 + 3)| |----- | \n1 = 0 / "A088312" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! ((n + 1) LaguerreL(n + 1, 1) - n LaguerreL(n, 1)) n! {-------------------------------------------------------------, ----------------------------------------------------} n n "A088313" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! ((n + 1) LaguerreL(n + 1, 1) - n LaguerreL(n, 1)) n! {-------------------------------------------------------------, ----------------------------------------------------} n n "A088336" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |---- - 1/2| | | { (-4) |----|! n1::even| |n - 1 { \ 2 / / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 (-1) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) --------------------------------------------------------------------------------|, n! | ) ------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A088436" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | |n - 1 { \ 2 / / n1 \ | |----- { 1/2 (-2) |---- - 1/2|! (n1 + 1) (n1 + 2) n1::odd | | \ { \ 2 / | {(n + 1) n!, (n + 1) n! | ) ----------------------------------------------------------------------|, | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / n1 / n1 \ | | { 1/2 (-1/2) binomial(n1, ----) |----|! (n1 + 1) (n1 + 2) n1::even| |n - 1 { 2 \ 2 / | |----- { | | \ { 0 n1::odd | (n + 1) n! | ) -------------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A088506" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |---- - 1/2| | | { (-4) |----|! n1::even| |n - 1 { \ 2 / / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 (-1) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) --------------------------------------------------------------------------------|, n! | ) ------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A088518" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A088536" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 7)| {(1/2) , (1/2) | ) -------------------------------------------------------------|} | / (n1 + 2) (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A088594" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A088662" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { 2 { ---------------- n::even { binomial(n, n/2) (n + 4 n + 2) { binomial(n, n/2) {{ ------------------------------- n::even, { } { n + 2 { (2 n + 2) 2 { { 2 (n + 4 n + 2) { 2 n binomial(n - 1, n/2 - 1/2) n::odd { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A088718" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A088854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {2 , 2 | ) ---------------------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A088927" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A088938" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {1, (n + 1) | ) n1!|, n + 1} | / | |----- | \n1 = 0 / "A088991" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, - 1/4 - n, -1/4)} "A088992" LREtools/SearchTable: "SearchTable successful" n {(-5) n! LaguerreL(n, -n - 1/5, -1/5)} "A089022" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 binomial(2 n1, n1)| {9 , 9 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A089023" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 3 2 binomial(2 n1, n1)| {16 , 16 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A089041" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) n! (n BesselK(n, -2) - BesselK(n - 1, -2))} "A089138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 GAMMA(n1 + 4/3)|| {9 , 9 | ) |1/9 -------------------||} | / \ GAMMA(n1 + 2) /| |----- | \n1 = 0 / "A089155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 1) | n n | \ 3 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) --------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A089164" LREtools/SearchTable: "SearchTable successful" (5 n + 4) LegendreP(n + 1, 3) - LegendreP(n, 3) n (5 n + 4) LegendreQ(n + 1, 3) - LegendreQ(n, 3) n {-------------------------------------------------, -------------------------------------------------} n + 2 n + 2 "A089165" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ \ {1, ) LegendreP(n1 + 1, 3), ) LegendreQ(n1 + 1, 3)} / / ----- ----- n1 = 0 n1 = 0 "A089222" memory used=43331.4MB, alloc=1655.5MB, time=299.67 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 1)} "A089252" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (2 n1 + 1) binomial(2 n1, n1) n1!\| {2 n!, 2 n! | ) |1/2 ---------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A089324" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" /n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 5 | | 5 | | 5 | | \ {|3/2 - ----| , |3/2 + ----| , |3/2 - ----| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / /n1 - 1 /{ 0 n2::even\\\ | |----- / 1/2\(-n2 - 1) |{ ||| | 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ / n2 n2 \ ||| |2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ 2 |3 LegendreP(---- + 1/2, 3) - LegendreP(---- - 1/2, 3)| ||| | | / \ 2 / |{ \ 2 2 / ||| | |----- |{ --------------------------------------------------------- n2::odd ||| \ \n2 = 0 \{ n2 + 3 /// \ /n - 1 | / 1/2\n |----- | | 5 | | \ |, |3/2 - ----| | ) | \ 2 / | / | |----- / \n1 = 0 / /n1 - 1 /{ 0 n2::even\\\ | |----- / 1/2\(-n2 - 1) |{ ||| | 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ / n2 n2 \ ||| |2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ 2 |3 LegendreQ(---- + 1/2, 3) - LegendreQ(---- - 1/2, 3)| ||| | | / \ 2 / |{ \ 2 2 / ||| | |----- |{ --------------------------------------------------------- n2::odd ||| \ \n2 = 0 \{ n2 + 3 /// \ /n - 1 | / 1/2\n |----- | | 5 | | \ |, |3/2 - ----| | ) | \ 2 / | / | |----- / \n1 = 0 / /n1 - 1 /{ n2 n2 \\\\ | |----- / 1/2\(-n2 - 1) |{ 3 LegendreP(---- + 1/2, 3) - LegendreP(---- - 1/2, 3) |||| | 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ 2 2 |||| |2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ ----------------------------------------------------- n2::even||||, | | / \ 2 / |{ n2 + 3 |||| | |----- |{ |||| \ \n2 = 0 \{ 0 n2::odd //// /n - 1 / 1/2\n |----- | 5 | | \ |3/2 - ----| | ) \ 2 / | / |----- \n1 = 0 / /n1 - 1 /{ n2 n2 \\\\ | |----- / 1/2\(-n2 - 1) |{ 3 LegendreQ(---- + 1/2, 3) - LegendreQ(---- - 1/2, 3) |||| | 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ 2 2 |||| |2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ ----------------------------------------------------- n2::even||||} | | / \ 2 / |{ n2 + 3 |||| | |----- |{ |||| \ \n2 = 0 \{ 0 n2::odd //// "A089354" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) binomial(3 n1, n1) (7 n1 + 3) | {(-1/4) , (-1/4) | ) ---------------------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) (-1/4) | \n1 = 0 / "A089372" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ n2 {(1/4 - 1/4 I 7 ) , (1/4 + 1/4 I 7 ) , (1/4 - 1/4 I 7 ) | ) (1/4 + 1/4 I 7 ) (1/4 - 1/4 I 7 ) | ) (-1) | / | / |----- |----- \n1 = 0 \n2 = 0 1/2 (-n2 - 1) 2 (1/4 + 1/4 I 7 ) ((13 n2 + 55 n2 + 54) hypergeom([1/2, -n2 - 1], [1], 4) - 3 (5 n2 + 14) (n2 + 1) hypergeom([1/2, -n2], [1], 4))/( \\ || || (n2 + 2) (n2 + 3) (n2 + 4))||} || || // "A089380" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ {(1 - 1/2 I 2 ) , (1 + 1/2 I 2 ) , (1 - 1/2 I 2 ) | ) (1 + 1/2 I 2 ) (1 - 1/2 I 2 ) | ) | / | / |----- |----- \n1 = 0 \n2 = 0 \\ n2 1/2 (-n2 - 1) 2 2 || (-1) (1 + 1/2 I 2 ) ((2 n2 + 2 n2 - 3) hypergeom([1/2, -n2 - 1], [1], 4) + (-6 n2 - 24 n2 - 27) hypergeom([1/2, -n2], [1], 4))|| ---------------------------------------------------------------------------------------------------------------------------------------------||} (n2 + 2) (n2 + 3) (n2 + 4) || || // "A089382" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (3 n + 5 n + 1) LegendreP(n + 1, 3) + (-n - 3 n - 3) LegendreP(n, 3) (3 n + 5 n + 1) LegendreQ(n + 1, 3) + (-n - 3 n - 3) LegendreQ(n, 3) {----------------------------------------------------------------------, ----------------------------------------------------------------------} n (n + 2) n (n + 2) "A089383" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- 2 \ (12 n1 + 37 n1 + 29) LegendreP(n1 + 1, 3) - (2 n1 + 3) (n1 + 1) LegendreP(n1, 3) {1, ) ---------------------------------------------------------------------------------, / (n1 + 2) (n1 + 3) ----- n1 = 0 n - 1 ----- 2 \ (12 n1 + 37 n1 + 29) LegendreQ(n1 + 1, 3) - (2 n1 + 3) (n1 + 1) LegendreQ(n1, 3) ) ---------------------------------------------------------------------------------} / (n1 + 2) (n1 + 3) ----- n1 = 0 "A089387" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(3/4 - 1/4 I 7 ) , (3/4 + 1/4 I 7 ) , (3/4 - 1/4 I 7 ) | ) (3/4 + 1/4 I 7 ) (3/4 - 1/4 I 7 ) | / |----- \n1 = 0 /n1 - 1 \\ / |----- 1/2 (-n2 - 1) 2 2 || | | \ (3/4 + 1/4 I 7 ) ((6 n2 + 6 n2 + 3) LegendreP(n2 + 1, 3) + (-2 n2 - 6 n2 - 9) LegendreP(n2, 3))|| 1/2 n | | ) ------------------------------------------------------------------------------------------------------------||, (3/4 - 1/4 I 7 ) | | / (n2 + 3) (n2 + 2) n2 || | |----- || | \n2 = 0 // \ n - 1 ----- \ 1/2 n1 1/2 (-n1 - 1) ) (3/4 + 1/4 I 7 ) (3/4 - 1/4 I 7 ) / ----- n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 2 2 || | \ (3/4 + 1/4 I 7 ) ((6 n2 + 6 n2 + 3) LegendreQ(n2 + 1, 3) + (-2 n2 - 6 n2 - 9) LegendreQ(n2, 3))|| | ) ------------------------------------------------------------------------------------------------------------||} | / (n2 + 3) (n2 + 2) n2 || |----- || \n2 = 0 // "A089402" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 binomial(n, n/2) { 4 4 { ------------------ n::even { -------------------------- n::even binomial(2 n, n) { n - 1 { n (n + 1) binomial(n, n/2) {----------------, { , { } n + 1 { 4 binomial(n - 1, n/2 - 1/2) { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A089408" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 binomial(n, n/2) { 4 4 { ------------------ n::even { -------------------------- n::even { n - 1 { n (n + 1) binomial(n, n/2) {{ , { } { 4 binomial(n - 1, n/2 - 1/2) { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A089436" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 (5 n + 2) binomial(---, n/2) { 2 { 1/2 ------------------------------- n::even { (n + 1) (n + 2) {{ , { (2 n - 2) 3 n { 3 2 (3 n - 1) (3 n + 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - --------------------------------------------------------------- n::odd { n (n + 1) (n + 2) { 3 n 3 n { 12 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { - -------------------------------------------------- n::even { binomial(n, n/2) (n + 1) (n + 2) { } { 3 n 3 n { 2 (5 n + 2) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { ----------------------------------------------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) (3 n + 2) "A089512" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | 2 1/2 |2 | 2 1/2 {|- ----| n! (n - 2 2 + 4 n + 3), |----| n! (n + 2 2 + 4 n + 3)} \ 2 / \ 2 / "A089656" memory used=43806.6MB, alloc=1687.5MB, time=303.09 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A089664" n {n (n - 1) binomial(2 n, n), 4 (3 n + 5) n} "A089665" 2 n (n - 1) n binomial(2 n, n) (3 n - 2 n + 1) {(n + 1) n 4 (7 n + 5), -------------------------------------------} 2 n - 1 "A089708" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 2 {1, ) (-1) ((n1 + 1) (n1 + 2 n1 - 2) BesselJ(n1, -2) + (n1 + 3 n1 + 1) BesselJ(n1 - 1, -2)), / ----- n1 = 0 n - 1 ----- \ n1 2 2 ) (-1) ((n1 + 1) (n1 + 2 n1 - 2) BesselY(n1, -2) + (n1 + 3 n1 + 1) BesselY(n1 - 1, -2))} / ----- n1 = 0 "A089735" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A089737" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A089742" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A089833" binomial(2 n, n) {----------------, n! binomial(2 n, n)} n + 1 "A089849" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { -------------------------- n::even { 2 binomial(n, n/2) n::even { n (n + 1) binomial(n, n/2) { {{ , { 4 binomial(n - 1, n/2 - 1/2) } { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A089880" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { -------------------------- n::even { 2 binomial(n, n/2) n::even binomial(2 n, n) { n (n + 1) binomial(n, n/2) { {----------------, { , { 4 binomial(n - 1, n/2 - 1/2) } n + 1 { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A089914" LREtools/SearchTable: "SearchTable successful" n {3 GAMMA(n + 2/3) hypergeom([-n], [2/3], -1)} "A089915" LREtools/SearchTable: "SearchTable successful" n {4 GAMMA(n + 3/4) hypergeom([-n], [3/4], -1)} "A089916" LREtools/SearchTable: "SearchTable successful" n {5 GAMMA(n + 4/5) hypergeom([-n], [4/5], -1)} "A089917" LREtools/SearchTable: "SearchTable successful" n {6 GAMMA(n + 5/6) hypergeom([-n], [5/6], -1)} "A089941" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 2 _Z - 1, index = 1) , RootOf(_Z - _Z - 2 _Z - 1, index = 2) , RootOf(_Z - _Z - 2 _Z - 1, index = 3) } "A090010" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / | 6 5 4 3 2 6 5 4 3 2 | {(n + 1) n! (n + 33 n + 430 n + 2825 n + 9844 n + 17203 n + 11743), (n + 1) n! (n + 33 n + 430 n + 2825 n + 9844 n + 17203 n + 11743) | | | \ n - 1 ----- \ n1 / 6 5 4 3 2 ) (-1) / ((n1 + 2) (n1 + 1)! ((n1 + 1) + 33 (n1 + 1) + 430 (n1 + 1) + 2825 (n1 + 1) + 9844 (n1 + 1) + 17203 n1 + 28946) / / ----- n1 = 0 \ | 6 5 4 3 2 | (n1 + 33 n1 + 430 n1 + 2825 n1 + 9844 n1 + 17203 n1 + 11743))|} | | / "A090012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 3 2 | \ (-1) | {n! (n + 9 n + 23 n + 16), n! (n + 9 n + 23 n + 16) | ) ---------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 9 (n1 + 1) + 23 n1 + 39) (n1 + 9 n1 + 23 n1 + 16)| \n1 = 0 / "A090013" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {n! (n + 14 n + 65 n + 116 n + 65), n! (n + 14 n + 65 n + 116 n + 65) /n - 1 \ |----- n1 | | \ (-1) | | ) --------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 14 (n1 + 1) + 65 (n1 + 1) + 116 n1 + 181) (n1 + 14 n1 + 65 n1 + 116 n1 + 65)| \n1 = 0 / "A090014" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 5 4 3 2 5 4 3 2 {n! (n + 20 n + 145 n + 470 n + 669 n + 326), n! (n + 20 n + 145 n + 470 n + 669 n + 326) /n - 1 \ |----- n1 | | \ (-1) | | ) -------------------------------------------------------------------------------------------------------------------------------------|} | / 5 4 3 2 5 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 20 (n1 + 1) + 145 (n1 + 1) + 470 (n1 + 1) + 669 n1 + 995) (n1 + 20 n1 + 145 n1 + 470 n1 + 669 n1 + 326)| \n1 = 0 / "A090015" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ n1 / {n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957), n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957) | ) (-1) / ( | / / |----- \n1 = 0 6 5 4 3 2 (n1 + 1)! ((n1 + 1) + 27 (n1 + 1) + 280 (n1 + 1) + 1415 (n1 + 1) + 3634 (n1 + 1) + 4429 n1 + 6386) \ | 6 5 4 3 2 | (n1 + 27 n1 + 280 n1 + 1415 n1 + 3634 n1 + 4429 n1 + 1957))|} | | / "A090016" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 7 6 5 4 3 2 {n! (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700), n! /n - 1 |----- 7 6 5 4 3 2 | \ n1 / (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700) | ) (-1) / ((n1 + 1)! | / / |----- \n1 = 0 7 6 5 4 3 2 ((n1 + 1) + 35 (n1 + 1) + 490 (n1 + 1) + 3535 (n1 + 1) + 14084 (n1 + 1) + 30681 (n1 + 1) + 33375 n1 + 47075) \ | 7 6 5 4 3 2 | (n1 + 35 n1 + 490 n1 + 3535 n1 + 14084 n1 + 30681 n1 + 33375 n1 + 13700))|} | | / "A090317" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (n1 + 5) | {(9/2) , (9/2) | ) -------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (9/2) | \n1 = 0 / "A090344" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A090345" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A090412" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | | { \ 2 / | | { (-16) | | { ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) | n n | \ { 2 | {(3/2) , (3/2) | ) ----------------------------------------------------------|, | / (n1 + 1) | |----- (3/2) | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / n1 | | { 2 (-1) binomial(n1, ----) | | { 2 | | { ------------------------------- n1::even| |n - 1 { n1 + 2 | |----- { | n | \ { 0 n1::odd | (3/2) | ) -------------------------------------------------|} | / (n1 + 1) | |----- (3/2) | \n1 = 0 / "A090413" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | | { \ 2 / | | { (-16) | | { ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) | n n | \ { 2 | {(8/3) , (8/3) | ) ----------------------------------------------------------|, | / (n1 + 1) | |----- (8/3) | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / n1 | | { 2 (-1) binomial(n1, ----) | | { 2 | | { ------------------------------- n1::even| |n - 1 { n1 + 2 | |----- { | n | \ { 0 n1::odd | (8/3) | ) -------------------------------------------------|} | / (n1 + 1) | |----- (8/3) | \n1 = 0 / "A090442" LREtools/SearchTable: "SearchTable successful" n n 2 (3 LegendreP(n + 1, 3) - LegendreP(n, 3)) 2 (3 LegendreQ(n + 1, 3) - LegendreQ(n, 3)) {--------------------------------------------, --------------------------------------------} n + 2 n + 2 "A090470" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 2) | n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {4 n!, 4 n! | ) ----------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A090598" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-n1) \| n n | \ | 5 (2 n1 + 1) binomial(2 n1, n1) n1!|| {10 n!, 10 n! | ) |1/10 ----------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A090805" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A090826" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| , \ 2 / \ 2 / / / / / 1/2 \(-n2 - 1) \\\ |n - 1 | |n1 - 1 |5 | ||| / 1/2\n |----- | |----- |---- + 1/2| (2 n2 + 1) binomial(2 n2, n2)||| | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 2 / ||| |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) | ) ---------------------------------------------------|||} \ 2 / | / | | / (n2 + 1) (n2 + 2) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A090932" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | |2 | {|- ----| n!, |----| n!} \ 2 / \ 2 / "A091147" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 15 ) (15 LegendreP(n + 1, 1/15 I 15 ) I - 15 LegendreP(n, 1/15 I 15 )) {------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 15 ) (15 LegendreQ(n + 1, 1/15 I 15 ) I - 15 LegendreQ(n, 1/15 I 15 )) ------------------------------------------------------------------------------------} n + 2 "A091148" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 19 ) (19 LegendreP(n + 1, 1/19 I 19 ) I - 19 LegendreP(n, 1/19 I 19 )) {------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 19 ) (-19 LegendreQ(n + 1, 1/19 I 19 ) I + 19 LegendreQ(n, 1/19 I 19 )) - -------------------------------------------------------------------------------------} n + 2 "A091149" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 23 ) (23 LegendreP(n + 1, 1/23 I 23 ) I - 23 LegendreP(n, 1/23 I 23 )) {------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 23 ) (-23 LegendreQ(n + 1, 1/23 I 23 ) I + 23 LegendreQ(n, 1/23 I 23 )) - -------------------------------------------------------------------------------------} n + 2 "A091429" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 4 n1! binomial(2 n1, n1)| {18 n!, 18 n! | ) -------------------------------------|} | / (n1 + 1) | |----- 18 (n1 + 1)! | \n1 = 0 / "A091496" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /5 n \ {{ |--- - 5/2| , { \ 2 / { 5 GAMMA(n/2 + 1/10) GAMMA(n/2 + 3/10) GAMMA(n/2 + 7/10) GAMMA(n/2 + 9/10) { ------------------------------------------------------------------------------------ n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 1/4) GAMMA(n/2 + 3/4) { /5 n\ { |---| { \ 2 / { 5 GAMMA(n/2 + 1/10) GAMMA(n/2 + 3/10) GAMMA(n/2 + 7/10) GAMMA(n/2 + 9/10) } { ------------------------------------------------------------------------------ n::even { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 1/4) GAMMA(n/2 + 3/4) { { 0 n::odd "A091520" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ (-1) 2 (2 n1 + 1) binomial(2 n1, n1)| {4 , 4 | ) -------------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A091526" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (3 n1 - 1)| {(-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A091527" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 3 n 3 n { { binomial(---, n/2) binomial(3 n, ---) { (2 n - 2) 3 n { 2 2 {{ 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) , { ------------------------------------- n::even} { 2 { binomial(n, n/2) { 1/2 --------------------------------------------------- n::odd { { n { 0 n::odd "A091540" n n {3 GAMMA(n + 7/3), (n + 2) (n + 1) 3 n!} "A091561" memory used=44358.1MB, alloc=1719.5MB, time=306.87 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A091565" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A091593" LREtools/SearchTable: "SearchTable successful" n n (-3) (LegendreP(n + 1, 1/3) - 3 LegendreP(n, 1/3)) (-3) (LegendreQ(n + 1, 1/3) - 3 LegendreQ(n, 1/3)) {---------------------------------------------------, ---------------------------------------------------} n + 2 n + 2 "A091695" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A091699" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {(-1/2) , (-1/2) | ) (-2 (hypergeom([-1/2, -n1 - 1], [1], -4) - hypergeom([-1/2, -n1], [1], -4)))|} | / | |----- | \n1 = 0 / "A091814" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 3) | n n | \ 3 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {8 n!, 8 n! | ) --------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A091964" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A091993" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ 5 18 (2 n1 + 1) binomial(2 n1, n1) n1!| {18 n!, 18 n! | ) -------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A091994" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) (9/2) n1! binomial(2 n1, n1)| {20 n!, 20 n! | ) -----------------------------------------|} | / (n1 + 1) | |----- 20 (n1 + 1)! | \n1 = 0 / "A092145" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 3) | n n | \ 5 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {8 n!, 8 n! | ) --------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A092170" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 2 2 2 | {(-1) , (-1) | ) (-(-1) (n1 + 2) (n1 + 1) (n1!) )|} | / | |----- | \n1 = 0 / "A092186" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 binomial(n, n/2) ((n/2)!) (n + 1) n::even {{ 2 , { } { ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { --------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { n + 1 "A092187" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 2 { 2 2 { 4 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) n::even { 1/4 ((n/2)!) (n + 2) n::even {{ , { } { (-2 n + 2) 2 2 2 2 { 2 { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { ((n/2 + 1/2)!) (n/4 + 3/4) n::odd "A092255" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 n n { {3 , (-1) hypergeom([1/2, -n], [1], 4), { n , { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { 1/9 ------------------------------------ irem(n, 3) = 2 { 2 { GAMMA(n/3 + 1) { 0 irem(n, 3) = 0 { { 2 n { (n - 1) { binomial(---, n/3) binomial(n, n/3) irem(n, 3) = 0 { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { 3 { ------------------------------------------ irem(n, 3) = 1, { } { 2 { 0 irem(n, 3) = 1 { GAMMA(n/3 + 1) { { { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A092266" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 16 { -------------------- n::even { 2 { 2 2 { binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { 2 { (4 n + 4) { 4 binomial(n - 1, n/2 - 1/2) n::odd { 4 2 { ------------------------------------ n::odd { 2 2 { (n + 1) binomial(n + 1, n/2 + 1/2) "A092396" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) 2 {{ , { (-4) ((n/2)!) n::even} { 2 (n/2 - 1/2) 2 2 { { n (-1/4) ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) n::odd { 0 n::odd "A092472" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 - 1) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A092566" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A092634" {1, (n + 3) (n + 2) (n + 1) n!} "A092690" LREtools/SearchTable: "SearchTable successful" n {(-1) ((-n - 5) hypergeom([1/2, -n], [1], 4) + (n + 1) hypergeom([1/2, -n - 1], [1], 4))} "A092691" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1! | | \ (-1) n1!| {n! | ) ---------|, n! | ) ----------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A092692" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- / n2 \|| | n1 | \ | (-1) (n2 + 1) n2!||| | (n1 + 1) (-1) n1! | ) |- -------------------||| |n - 1 | / \ (n2 + 2) (n2 + 1)! /|| |----- |----- || n | \ \n2 = 0 /| {(-1) n!, n! | ) ----------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A092765" LREtools/SearchTable: "SearchTable successful" n {(-1) binomial(2 n, n) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)} "A092785" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (n1 + 1) | n n | \ (-1) 2 (2 n1 + 1) (3 n1 + 4) binomial(2 n1, n1)| {1, (1/2) , (1/2) | ) ---------------------------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A092822" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n n | \ 2 binomial(2 n1, n1)| {2 , 2 n, 2 n | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A093128" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A093302" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1) (2 n1 + 1)| {2 n!, 2 n! | ) ------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A093303" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A093344" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ 1 || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || | \ n1! | | \ \n2 = 0 /| {n! | ) ---------|, n! | ) ----------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A093345" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ 1 || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || | \ n1! | | \ \n2 = 0 /| {n! | ) ---------|, n! | ) ----------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A093387" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) n { (n + 1) binomial(n, n/2) { -------------------------- n::even {2 , { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A093388" LREtools/SearchTable: "SearchTable not successful" {} "A093468" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1!} / ----- n1 = 0 "A093620" LREtools/SearchTable: "SearchTable successful" (-n) {2 binomial(2 n, n) n! hypergeom([-n], [1/2], -2)} "A093856" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -1), (-1) BesselY(n, -1)} "A093858" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A093935" LREtools/SearchTable: "SearchTable successful" n 3 2 n 3 2 {(-1) ((n + 2 n - n - 1) BesselJ(n, -2) + (n + 2) n BesselJ(n - 1, -2)), (-1) ((n + 2 n - n - 1) BesselY(n, -2) + (n + 2) n BesselY(n - 1, -2)) } "A093951" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (7 n + 6) (3 n + 2) { 2 { 1/2 -------------------------------------- n::even { (n + 3) (n + 2) (n + 1) {{ , { 3 n { 3 binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { ---------------------------------------------------- n::odd { (n + 3) (n + 2) n { (-n) 3 n 3 n { 12 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { -------------------------------------------------------- n::even { (n + 2) (n + 3) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 2 2 (7 n + 6) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { (n + 2) (n + 3) binomial(n + 1, n/2 + 1/2) "A093963" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 {1, ) (-1/2 I) ((4 n1 + 14 n1 + 11) HermiteH(n1 + 1, 1/2 I) + 2 I (n1 + 1) (4 n1 + 7) HermiteH(n1, 1/2 I))} / ----- n1 = 0 "A093964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! (n - 1), n! (n - 1) | ) ---------------------|} | / (n1 + 1)! n1 (n1 - 1)| |----- | \n1 = 0 / "A093965" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n! n, (n + 1) n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A093985" LREtools/SearchTable: "SearchTable successful" n n {(-1) (2 n BesselJ(n, -1) + BesselJ(n - 1, -1)), (-1) (2 n BesselY(n, -1) + BesselY(n - 1, -1))} "A093986" LREtools/SearchTable: "SearchTable successful" n n {(-1) (2 n BesselJ(n, -1) + BesselJ(n - 1, -1)), (-1) (2 n BesselY(n, -1) + BesselY(n - 1, -1))} "A094061" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A094074" LREtools/SearchTable: "SearchTable successful" (-n) 2 2 {4 binomial(2 n, n) (n!) hypergeom([-n], [1/2], -2)} "A094113" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(3 + 2 2 ) , (-2 2 + 3) , LegendreP(n + 1, 3), LegendreQ(n + 1, 3)} "A094187" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) (17/2) n1! binomial(2 n1, n1)| {(n + 1) 162 n!, (n + 1) 162 n! | ) ------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) 162 (n1 + 1)! | \n1 = 0 / "A094294" memory used=44905.5MB, alloc=1719.5MB, time=310.65 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A094639" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -------------------------------} / 2 2 ----- (n1 + 1) (n1 + 2) n1 = 0 "A094822" LREtools/SearchTable: "SearchTable successful" n {(-3) ((n + 1) LaguerreL(n + 1, -n - 1/3, 1) + LaguerreL(n, 2/3 - n, 1)) n!} "A094856" LREtools/SearchTable: "SearchTable successful" n {(-4) ((n + 1) LaguerreL(n + 1, -n - 1/4, 1) + LaguerreL(n, 3/4 - n, 1)) n!} "A094869" LREtools/SearchTable: "SearchTable successful" n {(-5) ((n + 1) LaguerreL(n + 1, -n - 1/5, 1) + LaguerreL(n, 4/5 - n, 1)) n!} "A094876" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" { (n/3) { (n/3) { 2 (n + 7) GAMMA(n/6 + 2/3) { 2 (n + 10) GAMMA(n/6 + 7/6) { ------------------------------- irem(n, 6) = 0 { -------------------------------- irem(n, 6) = 0 { (n - 2) GAMMA(n/6 + 13/6) { (n + 1) GAMMA(n/6 + 8/3) { { { (n/3 - 1/3) { (n/3 - 1/3) { 4 2 GAMMA(n/6 + 1/2) { 2 (n + 9) GAMMA(n/6 + 1) { ------------------------------- irem(n, 6) = 1 { ----------------------------------- irem(n, 6) = 1 { GAMMA(n/6 + 2) { n GAMMA(n/6 + 5/2) { { { (n/3 + 4/3) { (n/3 - 2/3) { 2 (n + 11) GAMMA(n/6 + 4/3) { 4 2 GAMMA(n/6 + 5/6) { -------------------------------------- irem(n, 6) = 2 { ------------------------------- irem(n, 6) = 2 { (n + 2) GAMMA(n/6 + 17/6) { GAMMA(n/6 + 7/3) {{ , { , { (n/3 + 1) { (n/3 - 1) { 2 (n + 10) GAMMA(n/6 + 7/6) { 2 (n + 7) GAMMA(n/6 + 2/3) { ------------------------------------ irem(n, 6) = 3 { ----------------------------------- irem(n, 6) = 3 { (n + 1) GAMMA(n/6 + 8/3) { (n - 2) GAMMA(n/6 + 13/6) { { { (n/3 + 2/3) { (n/3 - 4/3) { 2 (n + 9) GAMMA(n/6 + 1) { 4 2 GAMMA(n/6 + 1/2) { ----------------------------------- irem(n, 6) = 4 { ------------------------------- irem(n, 6) = 4 { n GAMMA(n/6 + 5/2) { GAMMA(n/6 + 2) { { { (n/3 + 1/3) { (n/3 + 1/3) { 4 2 GAMMA(n/6 + 5/6) { 2 (n + 11) GAMMA(n/6 + 4/3) { ------------------------------- irem(n, 6) = 5 { -------------------------------------- irem(n, 6) = 5 { GAMMA(n/6 + 7/3) { (n + 2) GAMMA(n/6 + 17/6) { (n/3) { 2 (n + 11) GAMMA(n/6 + 4/3) { -------------------------------- irem(n, 6) = 0 { (n + 2) GAMMA(n/6 + 17/6) { 24 binomial(n/3, n/6) { { --------------------- irem(n, 6) = 0 { (n/3 - 1/3) { n + 6 { 2 (n + 10) GAMMA(n/6 + 7/6) { { -------------------------------------- irem(n, 6) = 1 { 6 binomial(5/3 + n/3, n/6 + 5/6) { (n + 1) GAMMA(n/6 + 8/3) { -------------------------------- irem(n, 6) = 1 { { n + 2 { (n/3 - 2/3) { { 2 (n + 9) GAMMA(n/6 + 1) { 6 binomial(n/3 + 4/3, n/6 + 2/3) { ----------------------------------- irem(n, 6) = 2 { -------------------------------- irem(n, 6) = 2 { n GAMMA(n/6 + 5/2) { n + 1 { , { , { (n/3 - 1) { 6 binomial(n/3 + 1, n/6 + 1/2) { 4 2 GAMMA(n/6 + 5/6) { ------------------------------ irem(n, 6) = 3 { ----------------------------- irem(n, 6) = 3 { n { GAMMA(n/6 + 7/3) { { { 24 binomial(n/3 + 2/3, n/6 + 1/3) { (n/3 - 4/3) { --------------------------------- irem(n, 6) = 4 { 2 (n + 7) GAMMA(n/6 + 2/3) { n + 8 { ------------------------------------- irem(n, 6) = 4 { { (n - 2) GAMMA(n/6 + 13/6) { 6 binomial(n/3 + 1/3, n/6 + 1/6) { { -------------------------------- irem(n, 6) = 5 { (n/3 - 5/3) { n - 2 { 4 2 GAMMA(n/6 + 1/2) { ------------------------------- irem(n, 6) = 5 { GAMMA(n/6 + 2) { /2 n\ { |---| { \ 3 / { 9 2 { ---------------------------- irem(n, 6) = 0 { n (n + 3) binomial(n/3, n/6) { (n/3) { { 4 2 GAMMA(n/6 + 5/6) { /2 n \ { ------------------------- irem(n, 6) = 0 { |--- - 2/3| { GAMMA(n/6 + 7/3) { \ 3 / { { 36 2 { (n/3 - 1/3) { ---------------------------------------------- irem(n, 6) = 1 { 2 (n + 7) GAMMA(n/6 + 2/3) { (n + 2) (n + 8) binomial(n/3 - 1/3, n/6 - 1/6) { ------------------------------------- irem(n, 6) = 1 { { (n - 2) GAMMA(n/6 + 13/6) { /2 n \ { { |--- - 4/3| { (n/3 - 2/3) { \ 3 / { 4 2 GAMMA(n/6 + 1/2) { 9 2 { ------------------------------- irem(n, 6) = 2 { ---------------------------------------------- irem(n, 6) = 2 { GAMMA(n/6 + 2) { (n - 2) (n + 1) binomial(n/3 - 2/3, n/6 - 1/3) { , { } { (n/3 + 1) { /2 n \ { 2 (n + 11) GAMMA(n/6 + 4/3) { |--- - 2| { ------------------------------------ irem(n, 6) = 3 { \ 3 / { (n + 2) GAMMA(n/6 + 17/6) { 36 2 { { -------------------------------------- irem(n, 6) = 3 { (n/3 + 2/3) { n (n + 6) binomial(n/3 - 1, n/6 - 1/2) { 2 (n + 10) GAMMA(n/6 + 7/6) { { -------------------------------------- irem(n, 6) = 4 { /2 n \ { (n + 1) GAMMA(n/6 + 8/3) { |--- + 4/3| { { \ 3 / { (n/3 + 1/3) { 9 2 { 2 (n + 9) GAMMA(n/6 + 1) { ---------------------------------------------- irem(n, 6) = 4 { ----------------------------------- irem(n, 6) = 5 { (n + 2) (n + 5) binomial(n/3 + 2/3, n/6 + 1/3) { n GAMMA(n/6 + 5/2) { { /2 n \ { |--- + 2/3| { \ 3 / { 9 2 { ---------------------------------------------- irem(n, 6) = 5 { (n + 1) (n + 4) binomial(n/3 + 1/3, n/6 + 1/6) "A094891" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , (-1) ((3 n + 5) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)), 3 (4 n + 9)} "A094893" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { 1/2 ------------------------ n::even n n { binomial(n, n/2) (2 n + 2) n::even { (n + 1) binomial(n, n/2) {(-2) , 2 (6 n + 13), { , { } { (n + 2) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A094905" LREtools/SearchTable: "SearchTable successful" n {(-6) ((n + 1) LaguerreL(n + 1, -n - 1/6, 1) + LaguerreL(n, 5/6 - n, 1)) n!} "A094911" LREtools/SearchTable: "SearchTable successful" n {(-7) ((n + 1) LaguerreL(n + 1, -n - 1/7, 1) + LaguerreL(n, 6/7 - n, 1)) n!} "A094935" LREtools/SearchTable: "SearchTable successful" n {(-8) ((n + 1) LaguerreL(n + 1, -n - 1/8, 1) + LaguerreL(n, 7/8 - n, 1)) n!} "A094941" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2)! binomial(n, n/2) n::even {{ , { } { (n - 1) { 0 n::odd { 2 (n/2 - 1/2)! n::odd "A095000" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 3 2 3 2 | \ 1 | {n! (n + 3 n + 5 n + 2), n! (n + 3 n + 5 n + 2) | ) -----------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 3 (n1 + 1) + 5 n1 + 7) (n1 + 3 n1 + 5 n1 + 2)| \n1 = 0 / "A095176" LREtools/SearchTable: "SearchTable successful" n {(-9) ((n + 1) LaguerreL(n + 1, -n - 1/9, 1) + LaguerreL(n, 8/9 - n, 1)) n!} "A095177" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 {n! (n + 6 n + 17 n + 20 n + 9), /n - 1 \ |----- | 4 3 2 | \ 1 | n! (n + 6 n + 17 n + 20 n + 9) | ) --------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 6 (n1 + 1) + 17 (n1 + 1) + 20 n1 + 29) (n1 + 6 n1 + 17 n1 + 20 n1 + 9)| \n1 = 0 / "A095237" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) (2 n1 + 3) | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A095722" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / | 7 6 5 4 3 2 7 6 5 4 3 2 | {n! (n + 21 n + 196 n + 1015 n + 3094 n + 5453 n + 5053 n + 1854), n! (n + 21 n + 196 n + 1015 n + 3094 n + 5453 n + 5053 n + 1854) | | | \ n - 1 ----- \ 7 6 5 4 3 2 ) 1/((n1 + 1)! ((n1 + 1) + 21 (n1 + 1) + 196 (n1 + 1) + 1015 (n1 + 1) + 3094 (n1 + 1) + 5453 (n1 + 1) + 5053 n1 + 6907) / ----- n1 = 0 \ | 7 6 5 4 3 2 | (n1 + 21 n1 + 196 n1 + 1015 n1 + 3094 n1 + 5453 n1 + 5053 n1 + 1854))|} | | / "A095740" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 8 7 6 5 4 3 2 {n! (n + 28 n + 350 n + 2492 n + 10899 n + 29596 n + 48082 n + 42048 n + 14833), n! /n - 1 |----- 8 7 6 5 4 3 2 | \ (n + 28 n + 350 n + 2492 n + 10899 n + 29596 n + 48082 n + 42048 n + 14833) | ) 1/((n1 + 1)! | / |----- \n1 = 0 8 7 6 5 4 3 2 ((n1 + 1) + 28 (n1 + 1) + 350 (n1 + 1) + 2492 (n1 + 1) + 10899 (n1 + 1) + 29596 (n1 + 1) + 48082 (n1 + 1) + 42048 n1 + 56881) \ | 8 7 6 5 4 3 2 | (n1 + 28 n1 + 350 n1 + 2492 n1 + 10899 n1 + 29596 n1 + 48082 n1 + 42048 n1 + 14833))|} | | / "A095776" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A095816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A095839" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (3/2) n1! binomial(2 n1, n1)| {(-2) n!, (-2) n! | ) ------------------------------|} | / (n1 + 1) | |----- (-2) (n1 + 1)! | \n1 = 0 / "A095922" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A095981" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A096121" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! (n + 1) (n BesselK(n, -2) - BesselK(n - 1, -2)), -(-1) n! (n + 1) (-n BesselI(n, 2) + BesselI(n - 1, 2))} "A096191" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1)} "A096192" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, hypergeom([-n - 1, -n - 1, -n - 1, -n - 1], [1, 1, 1], 1)} "A096307" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 5 4 3 2 5 4 3 2 {n! (n + 10 n + 45 n + 100 n + 109 n + 44), n! (n + 10 n + 45 n + 100 n + 109 n + 44) /n - 1 \ |----- | | \ 1 | | ) ----------------------------------------------------------------------------------------------------------------------------------|} | / 5 4 3 2 5 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 10 (n1 + 1) + 45 (n1 + 1) + 100 (n1 + 1) + 109 n1 + 153) (n1 + 10 n1 + 45 n1 + 100 n1 + 109 n1 + 44)| \n1 = 0 / "A096341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ {n! (n + 15 n + 100 n + 355 n + 694 n + 689 n + 265), n! (n + 15 n + 100 n + 355 n + 694 n + 689 n + 265) | ) 1/((n1 + 1)! | / |----- \n1 = 0 6 5 4 3 2 ((n1 + 1) + 15 (n1 + 1) + 100 (n1 + 1) + 355 (n1 + 1) + 694 (n1 + 1) + 689 n1 + 954) \ | 6 5 4 3 2 | (n1 + 15 n1 + 100 n1 + 355 n1 + 694 n1 + 689 n1 + 265))|} | | / "A096471" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=45453.8MB, alloc=1719.5MB, time=314.45 { n 2 { (-n) 2 2 2 { 2 ((n/2)!) (n/4 + 1/2) n::even { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { } { (n - 1) 2 2 { (-n - 1) 2 2 { 1/4 2 (n + 1) ((n/2 - 1/2)!) n::odd { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 2) n::odd "A096654" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n! (n + 3), (n + 1) n! (n + 3) | ) ------------------------------------|} | / (n1 + 2) (n1 + 1)! (n1 + 4) (n1 + 3)| |----- | \n1 = 0 / "A096939" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" n {(n + 1) n! (LaguerreL(n + 1, -1) - LaguerreL(n, -1)), (n + 1) (-1) n! (LaguerreL(n + 1, 1) - LaguerreL(n, 1))} "A096965" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" n {(n + 1) n! (LaguerreL(n + 1, -1) - LaguerreL(n, -1)), (n + 1) (-1) n! (LaguerreL(n + 1, 1) - LaguerreL(n, 1))} "A097180" n n n 8 GAMMA(n + 5/4) 8 GAMMA(n + 7/4) (2 n + 1) 2 binomial(2 n, n) {-----------------, -----------------, -----------------------------} GAMMA(n + 2) GAMMA(n + 2) n + 1 "A097189" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- n1 ----- n1 \ 9 GAMMA(n1 + 4/3) \ 9 GAMMA(n1 + 5/3) {1, ) -------------------, ) -------------------} / GAMMA(n1 + 2) / GAMMA(n1 + 2) ----- ----- n1 = 0 n1 = 0 "A097204" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ 1 | {2 , n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A097332" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4)) {1, ) ----------------------------------------------------------------------------} / n1 + 2 ----- n1 = 0 "A097422" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ (n2 + 1) n2! | {1, ) (n1 + 1) n1!, ) (n1 + 1) n1! | ) ------------------|} / / | / (n2 + 2) (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A097593" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, n! (n + 2), n! (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2)| |----- | \n1 = 0 / "A097625" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-2) ((n + 3 n + 4) BesselI(n, 4) + (-2 n - 6) BesselI(n - 1, 4)), (-2) ((n + 3 n + 4) BesselK(n, -4) + (-2 n - 6) BesselK(n - 1, -4))} "A097632" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(1 - 5 ) n!, (5 + 1) n!} "A097656" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ 1 | {2 , n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A097678" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- | \ 1/2 n1 1/2 1/2 {n! (n + 2), n! (n + 2) | ) (- 1/2 + 1/2 I 3 ) ((4 n1 + 10) hypergeom([-n1 - 1, 1/2 - 1/2 I 3 ], [1], 3/2 - 1/2 I 3 ) | / |----- \n1 = 0 \ | 1/2 1/2 1/2 1/2 | - (n1 + 1) (1 + 3 I) hypergeom([-n1, 1/2 - 1/2 I 3 ], [1], 3/2 - 1/2 I 3 )) n1! (3 I - 1) (n1 + 1)/((n1 + 1)! (n1 + 3) (n1 + 2))|} | | / "A097779" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A097814" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 1 \| {3 n!, 3 n! | ) |1/3 ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A097815" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 1 \| {4 n!, 4 n! | ) |1/4 ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A097816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 1 \| {5 n!, 5 n! | ) |1/5 ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A097817" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ 2 3 | {3 n!, 3 n! | ) --------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A097819" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2)| n n | \ 3 2 | {4 n!, 4 n! | ) ----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A097820" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 2)| n n | \ 2 | {4 n!, 4 n! | ) ----------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A097821" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ 2 5 | {5 n!, 5 n! | ) --------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A097861" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {1, (-1) hypergeom([1/2, -n - 1], [1], 4)} "A097893" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) hypergeom([1/2, -n1 - 1], [1], 4)} / ----- n1 = 0 "A097894" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 2 \ (-1) ((14 n1 + 71 n1 + 81) hypergeom([1/2, -n1 - 1], [1], 4) - 3 (4 n1 + 13) (n1 + 1) hypergeom([1/2, -n1], [1], 4)) {1, ) -----------------------------------------------------------------------------------------------------------------------} / (n1 + 5) (n1 + 4) ----- n1 = 0 "A097967" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ 2 n1 + 1 | {2 , n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A097999" LREtools/SearchTable: "SearchTable successful" {n! (n + 1) (3 LegendreP(n + 1, 3) - LegendreP(n, 3)), n! (n + 1) (3 LegendreQ(n + 1, 3) - LegendreQ(n, 3))} "A098051" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {n - 1} "A098057" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 7 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 2 {n + 1, n - 3 n - 12} "A098075" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098118" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1! binomial(2 n1, n1) | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) --------------------------------------------------------|} | / (n1 + 2) (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A098123" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098264" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 19 ) LegendreP(n, 1/19 I 19 ), (-I 19 ) LegendreQ(n, 1/19 I 19 )} "A098265" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 23 ) LegendreP(n, 1/23 I 23 ), (-I 23 ) LegendreQ(n, 1/23 I 23 )} "A098269" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 4), 2 LegendreQ(n, 4)} "A098270" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 5), 2 LegendreQ(n, 5)} "A098276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 n1 || {(n + 1) (-1) n! (n - 1), (n + 1) (-1) n! (n - 1) | ) |- ------------------||} | / \ (n1 + 2) (n1 + 1)!/| |----- | \n1 = 0 / "A098329" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 31 ) LegendreP(n, 1/31 I 31 ), (-I 31 ) LegendreQ(n, 1/31 I 31 )} "A098331" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 5 (n/2) 5 {5 LegendreP(n, ----), 5 LegendreQ(n, ----)} 5 5 "A098332" LREtools/SearchTable: "SearchTable successful" memory used=45979.5MB, alloc=1719.5MB, time=318.47 n n {3 LegendreP(n, 1/3), 3 LegendreQ(n, 1/3)} "A098333" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 13 (n/2) 13 {13 LegendreP(n, -----), 13 LegendreQ(n, -----)} 13 13 "A098334" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 17 (n/2) 17 {17 LegendreP(n, -----), 17 LegendreQ(n, -----)} 17 17 "A098335" LREtools/SearchTable: "SearchTable successful" /3 n\ /3 n\ |---| 1/2 |---| 1/2 \ 2 / 2 \ 2 / 2 {2 LegendreP(n, ----), 2 LegendreQ(n, ----)} 2 2 "A098336" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 1/2 n 3 1/2 n 3 {(2 3 ) LegendreP(n, ----), (2 3 ) LegendreQ(n, ----)} 3 3 "A098337" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 1/2 n 5 1/2 n 5 {(2 5 ) LegendreP(n, ----), (2 5 ) LegendreQ(n, ----)} 5 5 "A098338" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 3 13 (n/2) 3 13 {13 LegendreP(n, -------), 13 LegendreQ(n, -------)} 13 13 "A098339" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 3 17 (n/2) 3 17 {17 LegendreP(n, -------), 17 LegendreQ(n, -------)} 17 17 "A098340" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 (n/2) 21 (n/2) 21 {21 LegendreP(n, -----), 21 LegendreQ(n, -----)} 7 7 "A098341" LREtools/SearchTable: "SearchTable successful" n n {5 LegendreP(n, 3/5), 5 LegendreQ(n, 3/5)} "A098405" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 3) (2 n1 + 1) 2 binomial(2 n1, n1) {1, ) --------------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A098409" LREtools/SearchTable: "SearchTable successful" n {3 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -4/3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4/3))} "A098410" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 3) hypergeom([-1/2, -n], [1], -1))} "A098411" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 3) hypergeom([-1/2, -n], [1], -2))} "A098439" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 47 ) LegendreP(n, 1/47 I 47 ), (-I 47 ) LegendreQ(n, 1/47 I 47 )} "A098440" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 59 ) LegendreP(n, 1/59 I 59 ), (-I 59 ) LegendreQ(n, 1/59 I 59 )} "A098441" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-3 I 7 ) LegendreP(n, 1/21 I 7 ), (-3 I 7 ) LegendreQ(n, 1/21 I 7 )} "A098442" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 95 ) LegendreP(n, 1/95 I 95 ), (-I 95 ) LegendreQ(n, 1/95 I 95 )} "A098443" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) LegendreP(n, 2 I), (-2 I) LegendreQ(n, 2 I)} "A098444" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 11 ) LegendreP(n, 3/11 I 11 ), (-I 11 ) LegendreQ(n, 3/11 I 11 )} "A098453" LREtools/SearchTable: "SearchTable successful" n {(-2) hypergeom([1/2, -n], [1], 4)} "A098455" LREtools/SearchTable: "SearchTable successful" n n {(-6 I) LegendreP(n, 1/3 I), (-6 I) LegendreQ(n, 1/3 I)} "A098456" LREtools/SearchTable: "SearchTable successful" n n {(-8 I) LegendreP(n, 1/4 I), (-8 I) LegendreQ(n, 1/4 I)} "A098460" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 2 ) n! LegendreP(n, 1/2 I 2 ), (-I 2 ) n! LegendreQ(n, 1/2 I 2 )} "A098461" LREtools/SearchTable: "SearchTable successful" n {(-1) n! hypergeom([1/2, -n], [1], 4)} "A098465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4), hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)} "A098469" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) {2 (hypergeom([-1/2, -n - 1], [1], -2) - hypergeom([-1/2, -n], [1], -2)), { , { (2 n - 2) { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 2 binomial(n, n/2) n::even { } { binomial(n + 1, n/2 + 1/2) n::odd "A098470" LREtools/SearchTable: "SearchTable successful" n 4 3 2 {(-1) ((121 n + 2202 n + 13985 n + 36324 n + 32076) hypergeom([1/2, -n - 1], [1], 4) 4 3 2 + (-123 n - 2202 n - 13647 n - 34140 n - 28404) hypergeom([1/2, -n], [1], 4)) (n + 1)/((n + 6) (n + 7) (n + 8) (n + 9) (n + 10))} "A098477" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098478" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098479" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098480" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098481" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098482" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098483" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098484" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098518" memory used=46550.8MB, alloc=1751.5MB, time=322.43 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 7 ) 7 (LegendreP(n, 1/7 I 7 ) + 7 LegendreP(n + 1, 1/7 I 7 ) I), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (LegendreQ(n, 1/7 I 7 ) + 7 LegendreQ(n + 1, 1/7 I 7 ) I)} "A098519" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 11 ) 11 (LegendreP(n, 1/11 I 11 ) + 11 LegendreP(n + 1, 1/11 I 11 ) I), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 11 ) 11 (LegendreQ(n, 1/11 I 11 ) + 11 LegendreQ(n + 1, 1/11 I 11 ) I)} "A098520" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 15 ) 15 (LegendreP(n, 1/15 I 15 ) + 15 LegendreP(n + 1, 1/15 I 15 ) I), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 (LegendreQ(n, 1/15 I 15 ) + 15 LegendreQ(n + 1, 1/15 I 15 ) I)} "A098521" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 7 ) ((4 n + 1) LegendreP(n, 1/7 I 7 ) + 7 LegendreP(n + 1, 1/7 I 7 ) I) {- -------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 7 ) ((4 n + 1) LegendreQ(n, 1/7 I 7 ) + 7 LegendreQ(n + 1, 1/7 I 7 ) I) - -------------------------------------------------------------------------------------} n + 2 "A098522" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-I 11 ) ((6 n + 1) LegendreP(n, 1/11 I 11 ) + 11 LegendreP(n + 1, 1/11 I 11 ) I) {- -------------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-I 11 ) ((6 n + 1) LegendreQ(n, 1/11 I 11 ) + 11 LegendreQ(n + 1, 1/11 I 11 ) I) - -------------------------------------------------------------------------------------------} n + 2 "A098535" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A098536" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098537" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A098538" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A098557" n (-1) n! n! (2 n - 1) {---------, ------------} (n - 1) n (n - 1) n "A098614" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | |1/2 - ----| binomial(2 n, n) |---- + 1/2| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A098615" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { (2 n2 - 2) ||| | | | { 2 ||| | | | { ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) ----------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { 2 binomial(n2, ----) ||| | | | { 2 ||| | | | { -------------------- n2::even||| |n - 1 | |n1 - 1 { n2 + 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (5 ) | ) |1/5 5 (-1) | ) --------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A098616" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n (1 - 2 ) binomial(2 n, n) (1 + 2 ) binomial(2 n, n) {----------------------------, ----------------------------} n + 1 n + 1 "A098617" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /n1 - 1 / 3 n2\ /{ 0 n2::even\\\\ |----- | |----- |- 3/2 - ----| |{ |||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ (2 n2 - 2) |||| {(-2 2 ) , (2 2 ) , (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ 2 ||||, | / | | / |{ ---------------------------------------- n2::odd |||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) |||| \ \ \ \{ 2 //// /n - 1 / /n1 - 1 / 3 n2\ /{ n2 \\\\ |----- | |----- |- 3/2 - ----| |{ 2 binomial(n2, ----) |||| 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ 2 |||| (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ -------------------- n2::even||||} | / | | / |{ n2 + 2 |||| |----- | |----- |{ |||| \n1 = 0 \ \n2 = 0 \{ 0 n2::odd //// "A098618" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 17 | | 17 | |3/2 - -----| binomial(2 n, n) |3/2 + -----| binomial(2 n, n) \ 2 / \ 2 / {-------------------------------, -------------------------------} n + 1 n + 1 "A098619" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { /5 n2 \ ||| | | | { |---- - 5/2| ||| | | | { \ 2 / ||| | | | { 2 ||| | | | { ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(17 ) , (-17 ) , (17 ) | ) |1/17 17 (-1) | ) ----------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-17 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 \ \\\ | | | { |----| ||| | | | { \ 2 / n2 ||| | | | { 2 2 binomial(n2, ----) ||| | | | { 2 ||| | | | { ---------------------------- n2::even||| |n - 1 | |n1 - 1 { n2 + 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (17 ) | ) |1/17 17 (-1) | ) ----------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-17 ) ||| \n1 = 0 \ \n2 = 0 /// "A098659" LREtools/SearchTable: "SearchTable successful" n n {5 LegendreP(n, 7/5), 5 LegendreQ(n, 7/5)} "A098660" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /5 n\ { |---| { \ 2 / { 2 { ------------------------ n::even { (n/2) { (n + 1) binomial(n, n/2) { 4 2 binomial(n, n/2) n::even {{ , { } { /5 n \ { (n/2 + 1/2) { |--- - 5/2| { 2 binomial(n + 1, n/2 + 1/2) n::odd { \ 2 / { 4 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A098662" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 48 { ------------------------ n::even { (n/2) { (n + 1) binomial(n, n/2) { 6 3 binomial(n, n/2) n::even {{ , { } { (n/2 - 1/2) { (n/2 + 1/2) { 6 48 { 3 binomial(n + 1, n/2 + 1/2) n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A098663" LREtools/SearchTable: "SearchTable successful" n n {2 (LegendreP(n + 1, 2) + LegendreP(n, 2)), 2 (LegendreQ(n + 1, 2) + LegendreQ(n, 2))} "A098664" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 { ------------------------ n::even { n { (n + 1) binomial(n, n/2) { 8 2 binomial(n, n/2) n::even {{ , { } { (3 n - 3) { (n + 1) { 8 2 { 2 binomial(n + 1, n/2 + 1/2) n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A098665" LREtools/SearchTable: "SearchTable successful" {(8 n + 5) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 9) hypergeom([-1/2, -n], [1], -8)} "A098730" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 3 _Z - 1, index = 1) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 2) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 3) } "A098746" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || | \ 1/2 n1 1/2 (-n1 - 1) | \ (1/2 + 1/2 I 3 ) (3 n2 + 1) binomial(3 n2, n2) (13 n2 + 12)|| | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) ----------------------------------------------------------------------||} | / | / (n2 + 1) (2 n2 + 3) (2 n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A098772" n {(-1) binomial(2 n, n), binomial(4 n, 2 n)} "A099021" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A099022" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselI(n + 1/2, 1/2), (-1) n! BesselK(n + 1/2, -1/2)} "A099030" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 (n/2)! (n + 2) n::even {(n + 2) (n + 1) n!, { , { (n/2 + 1/2) { (n/2 + 5/2) 2 (n/2 + 1/2)! n::odd { (- n/2) { 2 (n + 1) (n + 5) binomial(n, n/2) (n/2)! n::even { } { (- n/2 + 1/2) { 2 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A099038" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A099170" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 n 3 n 3 n {RootOf(2 _Z + _Z - 1, index = 1) , RootOf(2 _Z + _Z - 1, index = 2) , RootOf(2 _Z + _Z - 1, index = 3) } "A099171" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-1) , (1/2) } "A099250" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (10 n + 9) hypergeom([1/2, -2 n - 2], [1], 4) + (-18 n - 9) hypergeom([1/2, -2 n], [1], 4) {------------------------------------------------------------------------------------------} (4 n + 3) (2 n + 3) "A099251" memory used=47128.2MB, alloc=1751.5MB, time=326.72 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (n + 1) hypergeom([1/2, -2 n - 2], [1], 4) + (-9 n - 6) hypergeom([1/2, -2 n], [1], 4) {--------------------------------------------------------------------------------------} 4 n + 3 "A099252" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (2 n + 3) hypergeom([1/2, -2 n - 2], [1], 4) + (-18 n - 9) hypergeom([1/2, -2 n], [1], 4) {-----------------------------------------------------------------------------------------} 4 n + 3 "A099323" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 1) hypergeom([1/2, -n], [1], 4) {---------------------------------------------------------------------------------} n "A099324" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / binomial(2 n1, n1)\| {(-1) , (-1) | ) |- ------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A099325" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 , { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A099326" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 (n + 1), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A099327" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (8 n + 22) { ---------------- n::even n { 2 { binomial(n, n/2) {2 (3 n + 2), { binomial(n, n/2) (4 n + 13 n + 6) n::even, { } { { (2 n + 2) 2 { 2 n binomial(n - 1, n/2 - 1/2) (4 n + 11) n::odd { 2 (4 n + 13 n + 6) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A099363" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 2 binomial(n, n/2) { -------------------------------- n::even { - ------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) { 2 { ---------------------------- n::odd { - ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A099364" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 8 binomial(n, n/2) (n + 1) { 1/2 -------------------------------- n::even { - -------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) { 2 (n + 1) { ---------------------------- n::odd { - -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A099553" LREtools/SearchTable: "SearchTable not successful" {} "A099576" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 2) { 2 { ---------------------------- n::even { n + 2 {{ , { 3 n { (n + 1) binomial(--- + 3/2, n/2 + 1/2) { 2 { -------------------------------------- n::odd { n + 2 { (-n) 3 n 3 n { 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------------- n::even { (n + 2) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 2 (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ------------------------------------------------------------------------------------------- n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A099578" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 2) { 2 { 3/2 ---------------------------- n::even {{ n + 1 , { { 3 n { binomial(--- + 3/2, n/2 + 1/2) n::odd { 2 { (-n) 3 n 3 n { 4 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 1) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 6 2 (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { --------------------------------------------------------------------------------------------- n::odd { n (n + 1) binomial(n - 1, n/2 - 1/2) "A099598" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A099601" LREtools/SearchTable: "SearchTable successful" {hypergeom([-2 n, -n, 2 n + 1], [1, 1], 1)} "A099758" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n | \ 2 (n1 + 1)| 2 n! | ) -------------------| | / (n1 + 1)! | n |----- | 2 n! \n1 = 0 / {-----, ----------------------------------} n n "A099760" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n | \ 2 (n1 + 1)| 2 n! | ) -------------------| | / (n1 + 1)! | n |----- | 2 n! \n1 = 0 / {-----, ----------------------------------} n n "A099827" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 5 | 5 5 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 5| |----- ((n1 + 1)!) | \n1 = 0 / "A099869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (4 n1 + 1) binomial(4 n1, 2 n1) (10 n1 + 9) {1, ) -------------------------------------------} / (n1 + 1) (2 n1 + 3) (2 n1 + 1) ----- n1 = 0 "A099934" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselJ(n, -2) + BesselJ(n - 1, -2)), (-1) (n BesselY(n, -2) + BesselY(n - 1, -2))} "A099953" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (-n1) {1, ) 2 (2 n1 + 1) binomial(2 n1, n1) n1!} / ----- n1 = 0 "A099975" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (4 n1 + 1) (4 n1 + 3) binomial(4 n1, 2 n1) (5 n1 + 7) {1, ) -----------------------------------------------------} / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A100066" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 ----- |{ | ----- /{ n1 \ \ |{ (2 n1 - 2) | \ |{ binomial(n1, ----) n1::even| {1, ) |{ 2 |, ) |{ 2 |} / |{ ------------------------------- n1::odd | / |{ | ----- |{ n1 | ----- \{ 0 n1::odd / n1 = 0 |{ n1 binomial(n1 - 1, ---- - 1/2) | n1 = 0 \{ 2 / "A100067" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -binomial(n1, ----) n1::even| | { 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(5/2) , (5/2) | ) ---------------------------------------------------|, (5/2) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A100068" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 3 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { 3 binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(10/3) , (10/3) | ) ---------------------------------------------------|, (10/3) | ) ------------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (10/3) | |----- (10/3) | \n1 = 0 / \n1 = 0 / "A100069" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 2 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -binomial(n1, ----) n1::even| | { 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { 2 binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(17/4) , (17/4) | ) ---------------------------------------------------|, (17/4) | ) ------------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (17/4) | |----- (17/4) | \n1 = 0 / \n1 = 0 / "A100071" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even { n binomial(n, n/2) n::even { binomial(n, n/2) {{ , { } { binomial(n + 1, n/2 + 1/2) (n/2 + 1/2) n::odd { (2 n - 2) { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A100087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ / { 0 n1::even\ | { 2 4 | | { | | { ------------------------------ n1::even| | { n1 | | { n1 | | { 2 binomial(n1 - 1, ---- - 1/2) | | { n1 (n1 + 1) binomial(n1, ----) | |n - 1 { 2 | |n - 1 { 2 | |----- { ------------------------------ n1::odd | |----- { | n n | \ { n1 + 1 | n | \ { 0 n1::odd | {(5/2) , (5/2) | ) ------------------------------------------------|, (5/2) | ) ------------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A100095" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { (2 n2 - 2) ||| | | | { 2 ||| | | | { ------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) -------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { binomial(n2, ----) n2::even||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (5 ) | ) |1/5 5 (-1) | ) ------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A100096" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { (2 n1 - 2) | | { 2 | / { n1 \ | { ------------------------------- n1::odd | | { binomial(n1, ----) n1::even| |n - 1 { n1 | |n - 1 { 2 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { | n n n | \ { 2 | n | \ { 0 n1::odd | {(-2) , (5/2) , (5/2) | ) -------------------------------------------------|, (5/2) | ) ------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A100097" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /n1 - 1 / 3 n2\ /{ 0 n2::even\\\\ |----- | |----- |- 3/2 - ----| |{ |||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ (2 n2 - 2) |||| {(-2 2 ) , (2 2 ) , (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ 2 ||||, | / | | / |{ ------------------------------- n2::odd |||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ n2 binomial(n2 - 1, ---- - 1/2) |||| \ \ \ \{ 2 //// /n - 1 / /n1 - 1 / 3 n2\ \\\ |----- | |----- |- 3/2 - ----| /{ n2 \||| 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ binomial(n2, ----) n2::even|||| (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ 2 ||||} | / | | / |{ |||| |----- | |----- \{ 0 n2::odd /||| \n1 = 0 \ \n2 = 0 /// "A100098" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(5/2) , (5/2) | ) ---------------------------------------------------|, (5/2) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A100099" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -4 binomial(n1, ----) n1::even| | { 4 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(5/2) , (5/2) | ) ---------------------------------------------------|, (5/2) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A100192" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=47693.2MB, alloc=1751.5MB, time=330.72 /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (n1 - 1)| {(9/2) , (9/2) | ) ---------------------------|} | / (n1 + 1) | |----- (n1 + 1) (9/2) | \n1 = 0 / "A100193" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (2 n1 - 1)| {(16/3) , (16/3) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (16/3) | \n1 = 0 / "A100217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A100223" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreP(n, ----) + 5 (n + 1) LegendreP(n + 1, ----)| \ 5 5 / {- ---------------------------------------------------------------------------, n (n - 1) / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreQ(n, ----) + 5 (n + 1) LegendreQ(n + 1, ----)| \ 5 5 / - ---------------------------------------------------------------------------} n (n - 1) "A100228" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n | 13 1/2 13 | (-13 ) |(11 n - 1) LegendreP(n, -----) + 13 (n + 1) LegendreP(n + 1, -----)| \ 13 13 / {- ----------------------------------------------------------------------------------, (n - 1) n / 1/2 1/2 \ 1/2 n | 13 1/2 13 | (-13 ) |(11 n - 1) LegendreQ(n, -----) + 13 (n + 1) LegendreQ(n + 1, -----)| \ 13 13 / - ----------------------------------------------------------------------------------} (n - 1) n "A100231" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n | 5 1/2 5 | (-2 5 ) |(3 n - 1) LegendreP(n, ----) + 5 (n + 1) LegendreP(n + 1, ----)| \ 5 5 / {- -------------------------------------------------------------------------------, (n - 1) n / 1/2 1/2 \ 1/2 n | 5 1/2 5 | (-2 5 ) |(3 n - 1) LegendreQ(n, ----) + 5 (n + 1) LegendreQ(n + 1, ----)| \ 5 5 / - -------------------------------------------------------------------------------} (n - 1) n "A100234" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n | 3 29 1/2 3 29 | (-29 ) |(11 n - 9) LegendreP(n, -------) + 3 29 (n + 1) LegendreP(n + 1, -------)| \ 29 29 / {- ----------------------------------------------------------------------------------------, (n - 1) n / 1/2 1/2 \ 1/2 n | 3 29 1/2 3 29 | (-29 ) |(11 n - 9) LegendreQ(n, -------) + 3 29 (n + 1) LegendreQ(n + 1, -------)| \ 29 29 / - ----------------------------------------------------------------------------------------} (n - 1) n "A100238" LREtools/SearchTable: "SearchTable successful" n n (2 I) ((3 n + 1) LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (2 I) ((3 n + 1) LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- ------------------------------------------------------------------, - ------------------------------------------------------------------} (n - 1) n (n - 1) n "A100239" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (3 I) ((7 n + 3) LegendreP(n, 3 I) + 3 (n + 1) LegendreP(n + 1, 3 I) I) {- ------------------------------------------------------------------------------------, (n - 1) n 1/2 n 1/2 1/2 1/2 (3 I) ((7 n + 3) LegendreQ(n, 3 I) + 3 (n + 1) LegendreQ(n + 1, 3 I) I) - ------------------------------------------------------------------------------------} (n - 1) n "A100240" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreP(n, ----) + 5 (n + 1) LegendreP(n + 1, ----)| \ 5 5 / {- ---------------------------------------------------------------------------, n (n - 1) / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreQ(n, ----) + 5 (n + 1) LegendreQ(n + 1, ----)| \ 5 5 / - ---------------------------------------------------------------------------} n (n - 1) "A100299" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 LegendreP(n + 1, 3) - LegendreP(n, 3) 3 LegendreQ(n + 1, 3) - LegendreQ(n, 3) {(-1) , ---------------------------------------, ---------------------------------------} n + 2 n + 2 "A100300" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 LegendreP(n + 1, 3) - LegendreP(n, 3) 3 LegendreQ(n + 1, 3) - LegendreQ(n, 3) {(-1) , ---------------------------------------, ---------------------------------------} n + 2 n + 2 "A100327" LREtools/SearchTable: "SearchTable successful" (2 n + 2) hypergeom([2 n + 3, -n - 1], [1], -1) + (-11 n - 8) hypergeom([-n, 2 n + 1], [1], -1) {-----------------------------------------------------------------------------------------------} (17 n + 11) n "A100328" LREtools/SearchTable: "SearchTable successful" (n - 1) hypergeom([2 n + 3, -n - 1], [1], -1) + (3 n + 4) hypergeom([-n, 2 n + 1], [1], -1) {-------------------------------------------------------------------------------------------} (17 n + 11) n "A100404" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A100444" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) (n!) binomial(2 n, n), (n + 1) (n!) binomial(2 n, n) | ) ------------------------------------------------|} | / 2 | |----- (n1 + 2) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A100445" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(2 n + 3) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- | 2 | \ 1 | (2 n + 3) (2 n + 1) (n!) binomial(2 n, n) | ) -------------------------------------------------------------|} | / 2 | |----- (2 n1 + 5) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A100510" LREtools/SearchTable: "SearchTable successful" (-n) {2 (2 n + 1) (hypergeom([-n - 1], [1/2], -2) - hypergeom([-n], [1/2], -2)) binomial(2 n, n) n!} "A100511" n {n binomial(2 n, n), 4 n} "A100622" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A100948" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 18 GAMMA(n/3 + 10/3) { 9 GAMMA(n/3 + 11/3) { -------------------- irem(n, 3) = 0 { ------------------- irem(n, 3) = 0 { n + 7 { n + 8 { 2/9 (n/3)! (n + 3) (n + 6) irem(n, 3) = 0 { { { {{ 2 GAMMA(n/3 + 3) irem(n, 3) = 1, { 18 GAMMA(n/3 + 10/3) , { (n/3 + 2/3)! (n + 5) irem(n, 3) = 1} { { -------------------- irem(n, 3) = 1 { { 9 GAMMA(n/3 + 11/3) { n + 7 { (n/3 + 1/3)! (2 n + 8) irem(n, 3) = 2 { ------------------- irem(n, 3) = 2 { { n + 8 { 2 GAMMA(n/3 + 3) irem(n, 3) = 2 "A101106" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A101194" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A101292" {(n + 1) n, n!} "A101308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A101487" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 binomial(2 n, n) {-------------------} n + 1 "A101488" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A101490" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {----------------} n + 1 "A101499" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A101500" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A101634" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A101682" LREtools/SearchTable: "SearchTable successful" n n {(-2) BesselI(n - 1/2, 1), (-2) BesselK(n - 1/2, -1)} "A101785" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A101786" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A101850" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 3 2 | | 3 2 | {|2 - ------| , |2 + ------| , \ 2 / \ 2 / / / / 1/2\(-n2 - 1) \\ |n - 1 |n1 - 1 | 3 2 | || / 1/2\n |----- / 1/2\n1 / 1/2\(-n1 - 1) |----- |2 + ------| (2 n2 + 1) binomial(2 n2, n2)|| | 3 2 | | \ | 3 2 | | 3 2 | | \ \ 2 / || |2 - ------| | ) |2 + ------| |2 - ------| | ) ---------------------------------------------------||} \ 2 / | / \ 2 / \ 2 / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A102038" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A102052" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ -I (2 I) (-(n1 + 1) (n1 + 8) LegendreQ(n1, I) + (3 n1 + 4) (n1 + 7) LegendreQ(n1 + 1, I) I)| {(n + 7) | ) ---------------------------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 3) (n1 + 8) (n1 + 7) | |----- | \n1 = 0 / /n - 1 \ |----- n1 | | \ (2 I) ((n1 + 1) (n1 + 8) LegendreP(n1, I) - (3 n1 + 4) (n1 + 7) LegendreP(n1 + 1, I) I) I| (n + 7) | ) -------------------------------------------------------------------------------------------|, n + 7} | / (n1 + 4) (n1 + 3) (n1 + 8) (n1 + 7) | |----- | \n1 = 0 / "A102058" LREtools/SearchTable: "SearchTable not successful" {} "A102059" LREtools/SearchTable: "SearchTable not successful" {} "A102071" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((2 n + 8 n + 7) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 6 n - 1) hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A102231" LREtools/SearchTable: "SearchTable successful" memory used=48249.8MB, alloc=1751.5MB, time=334.91 2 3 2 {(2 (37 n + 8) (2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-822 n - 995 n - 357 n - 40) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n)/(n (n - 2) (n - 1) (10 n + 7))} "A102289" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 1) LaguerreL(n1, -1) n1!|| {(-1) n!, (-1) n! | ) |- -------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A102290" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 2 \| n n | \ | (-1) (n1 + 1) (LaguerreL(n1 + 1, -1) - 2 LaguerreL(n1, -1)) n1!|| {(-1) n!, (-1) n! | ) |- ------------------------------------------------------------------||} | / \ n1 (n1 + 1)! /| |----- | \n1 = 0 / "A102307" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| binomial(2 n, n), |3/2 + ----| binomial(2 n, n)} \ 2 / \ 2 / "A102319" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([1/2, -n], [1], 4), (2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)} "A102403" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102406" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 1)} "A102407" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102692" LREtools/SearchTable: "SearchTable successful" n! ((2 n + 1) LegendreP(n + 1, 3) + (-10 n - 3) LegendreP(n, 3)) n! ((2 n + 1) LegendreQ(n + 1, 3) + (-10 n - 3) LegendreQ(n, 3)) {----------------------------------------------------------------, ----------------------------------------------------------------} 2 2 n (n - 1) n (n - 1) "A102699" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n - 1) { ------------------ n::even { 2 n binomial(n, n/2) n::even n { n binomial(n, n/2) { {2 (n + 1), { , { (2 n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) } { (2 n - 2) { 1/2 -------------------------------------------- n::odd { 4 2 { n { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A102736" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n 2 { n 2 { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) irem(n, 3) = 0 { 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { { ------------------------------------- irem(n, 3) = 0 { n 2 { n + 1 { 9 3 GAMMA(n/3 + 1) GAMMA(5/3 + n/3) { {{ ------------------------------------- irem(n, 3) = 1, { (n - 1) 2 , { n (n + 2) { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) irem(n, 3) = 1 { { { (n + 1) 2 { (n + 1) 2 { 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { 3 GAMMA(n/3 + 1) GAMMA(5/3 + n/3) { ------------------------------------------- irem(n, 3) = 2 { ----------------------------------------- irem(n, 3) = 2 { n + 1 { n (n + 2) { n 2 { 3 GAMMA(n/3 + 1) GAMMA(5/3 + n/3) { ----------------------------------- irem(n, 3) = 0 { n (n + 2) { { (n - 1) 2 } { 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { ------------------------------------------- irem(n, 3) = 1 { n + 1 { { n 2 { 1/9 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) irem(n, 3) = 2 "A102757" LREtools/SearchTable: "SearchTable successful" n {3 n! LaguerreL(n, -1/3)} "A102761" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A102773" LREtools/SearchTable: "SearchTable successful" n {4 n! LaguerreL(n, -1/4)} "A102839" LREtools/SearchTable: "SearchTable successful" n {(-1) (n + 1) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4))} "A102840" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2 + 1/2) | 1/2 5 5 | (n/2 + 1/2) | 1/2 5 5 | {5 (n + 1) |5 LegendreP(n + 1, ----) - LegendreP(n, ----)|, 5 (n + 1) |5 LegendreQ(n + 1, ----) - LegendreQ(n, ----)|} \ 5 5 / \ 5 5 / "A102879" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102880" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102881" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102882" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A102898" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ / { 0 n1::even\ | { 2 4 | | { | | { ------------------------------ n1::even| | { n1 | | { n1 | | { 2 binomial(n1 - 1, ---- - 1/2) | | { n1 (n1 + 1) binomial(n1, ----) | |n - 1 { 2 | |n - 1 { 2 | |----- { ------------------------------ n1::odd | |----- { | n n | \ { n1 + 1 | n | \ { 0 n1::odd | {(10/3) , (10/3) | ) ------------------------------------------------|, (10/3) | ) ------------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (10/3) | |----- (10/3) | \n1 = 0 / \n1 = 0 / "A103137" LREtools/SearchTable: "SearchTable successful" n n (-1) ((3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3)) (-1) ((3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3)) {-------------------------------------------------------------------, -------------------------------------------------------------------} (n - 1) n (n - 1) n "A103138" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((n + 1) (11 n + 4) LegendreP(n + 1, 3) + (-65 n - 53 n - 12) LegendreP(n, 3)) {-------------------------------------------------------------------------------------, n (n - 1) (n - 2) n 2 (-1) ((n + 1) (11 n + 4) LegendreQ(n + 1, 3) + (-65 n - 53 n - 12) LegendreQ(n, 3)) -------------------------------------------------------------------------------------} n (n - 1) (n - 2) "A103194" LREtools/SearchTable: "SearchTable successful" {((-n - 2) LaguerreL(n, -1) + (n + 1) LaguerreL(n + 1, -1)) n!} "A103210" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 5) - 5 LegendreP(n, 5) LegendreQ(n + 1, 5) - 5 LegendreQ(n, 5) {---------------------------------------, ---------------------------------------} n n "A103211" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 7) - 7 LegendreP(n, 7) LegendreQ(n + 1, 7) - 7 LegendreQ(n, 7) {---------------------------------------, ---------------------------------------} n n "A103213" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ (n1 + 1) n1! | | \ (n1 + 1) 2 n1! | {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A103370" LREtools/SearchTable: "SearchTable successful" (7 n + 11) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4) {-------------------------------------------------------------------------------------------------------} 2 (n + 3) (n + 2) "A103519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 3 | {(n + 1) n!, (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A103769" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A103779" LREtools/SearchTable: "SearchTable successful" n {(-3) GAMMA(n - 1/3) GAMMA(n + 1/3) (n + 1) n (9 (-1) hypergeom([n - 1/3, -n - 4/3], [-1/3], 8/9) - 7 (-1) hypergeom([n - 4/3, -n - 1/3], [-1/3], 8/9))/(GAMMA(n + 1) GAMMA(n + 4/3)), n (-3) GAMMA(n - 1/3) GAMMA(n + 1/3) (9 hypergeom([n - 1/3, -n - 4/3], [-1/3], 1/9) - 7 hypergeom([n - 4/3, -n - 1/3], [-1/3], 1/9)) -----------------------------------------------------------------------------------------------------------------------------------} GAMMA(n + 1) GAMMA(n + 4/3) "A103821" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ n1 \ n1 {1, ) (-2 I) LegendreP(n1 + 1, I), ) (-2 I) LegendreQ(n1 + 1, I)} / / ----- ----- n1 = 0 n1 = 0 "A103872" LREtools/SearchTable: "SearchTable successful" n (-1) ((3 n + 3) hypergeom([1/2, -n], [1], 4) + (n + 3) hypergeom([1/2, -n - 1], [1], 4)) {-----------------------------------------------------------------------------------------} n + 2 "A103882" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-8 n - 3) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) (n + 1) {---------------------------------------------------------------------------------------------------------------------------------------------} 2 n "A103885" LREtools/SearchTable: "SearchTable successful" ((8 n + 6) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-n - 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1) {---------------------------------------------------------------------------------------------------------------------------------} (n + 1) (10 n + 7) "A103943" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {4 , --------------------------, { , { } n + 1 { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A103944" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) (n + 1), (-1) (n + 1) | ) |- ----------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A103945" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 3 2 \| n n | \ | (-1) 2 (2 n1 + 1) (3 n1 + 29 n1 + 60 n1 + 36) binomial(2 n1, n1)|| {(-1) (n + 1), (-1) (n + 1) | ) |- ----------------------------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (n1 + 2) /| \n1 = 0 / "A103970" LREtools/SearchTable: "SearchTable successful" n (-2) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------} n "A103971" LREtools/SearchTable: "SearchTable successful" n n -I (-4 I) (LegendreP(n, 1/2 I) + 2 I LegendreP(n + 1, 1/2 I)) -I (-4 I) (LegendreQ(n, 1/2 I) + 2 I LegendreQ(n + 1, 1/2 I)) {--------------------------------------------------------------, --------------------------------------------------------------} n n "A103972" LREtools/SearchTable: "SearchTable successful" memory used=48804.9MB, alloc=1751.5MB, time=338.99 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 5 ) 5 (LegendreP(n, 1/5 I 5 ) + 5 LegendreP(n + 1, 1/5 I 5 ) I) {-------------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 5 ) 5 (LegendreQ(n, 1/5 I 5 ) + 5 LegendreQ(n + 1, 1/5 I 5 ) I) -------------------------------------------------------------------------------------} n "A103973" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /5 n\ { |---| { \ 2 / { (n/2) { 8 2 { 2 2 binomial(n, n/2) n::even { -------------------------- n::even { { n (n + 1) binomial(n, n/2) {{ (n/2 - 1/2) , { } { 8 2 binomial(n - 1, n/2 - 1/2) { /5 n \ { ----------------------------------------- n::odd { |--- + 5/2| { n + 1 { \ 2 / { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A103978" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (n/2) { 12 48 { 2 3 binomial(n, n/2) n::even { -------------------------- n::even { { (n + 1) n binomial(n, n/2) {{ (n/2 - 1/2) , { } { 12 3 binomial(n - 1, n/2 - 1/2) { (n/2 + 1/2) { ------------------------------------------ n::odd { 2 48 { n + 1 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A104184" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" (4 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n + 2 "A104268" n (2 n + 1) binomial(2 n, n) (3 n + 2) {4 , ------------------------------------} (n + 1) (n + 2) "A104344" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 2 2 {1, ) (n1 + 1) (n1 + 2) (n1!) } / ----- n1 = 0 "A104454" LREtools/SearchTable: "SearchTable successful" n 2 {5 (16 (n + 1) (n - 3) hypergeom([-3/2, -n - 1], [1], -4/5) + (-16 n + 8 n + 39) hypergeom([-3/2, -n], [1], -4/5))} "A104455" LREtools/SearchTable: "SearchTable successful" n {3 (hypergeom([-1/2, -n - 1], [1], -4/3) - hypergeom([-1/2, -n], [1], -4/3))} "A104470" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A104496" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (n1 - 1) binomial(2 n1, n1)\| {(-1) , (-1) | ) |- ---------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A104497" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 1) hypergeom([-1/2, -n], [1], -1)) {---------------------------------------------------------------------------------------------} n "A104498" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 1) hypergeom([-1/2, -n], [1], -1)) {---------------------------------------------------------------------------------------------} n "A104506" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2 + 1/2) | 1/2 5 5 | (n/2 + 1/2) | 1/2 5 5 | {5 |5 LegendreP(n + 1, ----) - LegendreP(n, ----)|, 5 |5 LegendreQ(n + 1, ----) - LegendreQ(n, ----)|} \ 5 5 / \ 5 5 / "A104507" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 / n1 \ ----- |---- + 1/2| / 1/2 1/2 \ \ \ 2 / | 1/2 5 5 | {1, ) 5 |5 LegendreP(n1 + 1, ----) - 3 LegendreP(n1, ----)|, / \ 5 5 / ----- n1 = 0 n - 1 / n1 \ ----- |---- + 1/2| / 1/2 1/2 \ \ \ 2 / | 1/2 5 5 | ) 5 |5 LegendreQ(n1 + 1, ----) - 3 LegendreQ(n1, ----)|} / \ 5 5 / ----- n1 = 0 "A104530" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(16/3) , (16/3) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (16/3) | \n1 = 0 / "A104531" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(25/4) , (25/4) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (25/4) | \n1 = 0 / "A104532" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(36/5) , (36/5) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (36/5) | \n1 = 0 / "A104533" LREtools/SearchTable: "SearchTable successful" n 2 ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {----------------------------------------------------------------} n "A104545" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A104547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A104550" LREtools/SearchTable: "SearchTable successful" (4 n + 3) LegendreP(n, 3) - LegendreP(n + 1, 3) (4 n + 3) LegendreQ(n, 3) - LegendreQ(n + 1, 3) {- -----------------------------------------------, - -----------------------------------------------} n n "A104551" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ 0 n1::even\ n - 1 |{ | n - 1 /{ / n1 \ \ ----- |{ / n1 \ | ----- |{ |----| | \ |{ |---- - 1/2| | \ |{ \ 2 / n1 | {1, ) |{ \ 2 / |, ) |{ (-1) binomial(n1, ----) n1::even|} / |{ (-16) | / |{ 2 | ----- |{ ------------------------------- n1::odd | ----- |{ | n1 = 0 |{ n1 | n1 = 0 \{ 0 n1::odd / |{ n1 binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A104553" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ / (n1 + 1) LegendreP(n1, 3) + (-6 n1 - 9) LegendreP(n1 + 1, 3)\ {1, n, ) |- ------------------------------------------------------------|, / \ n1 + 2 / ----- n1 = 0 n - 1 ----- \ / (n1 + 1) LegendreQ(n1, 3) + (-6 n1 - 9) LegendreQ(n1 + 1, 3)\ ) |- ------------------------------------------------------------|} / \ n1 + 2 / ----- n1 = 0 "A104565" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n | 1/2 2 2 | 1/2 n | 1/2 2 2 | (-2 2 ) |2 LegendreP(n + 1, ----) - 2 LegendreP(n, ----)| (-2 2 ) |2 LegendreQ(n + 1, ----) - 2 LegendreQ(n, ----)| \ 2 2 / \ 2 2 / {---------------------------------------------------------------, ---------------------------------------------------------------} n + 2 n + 2 "A104574" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 n \ (-1) (n1 hypergeom([1/2, -n1 - 1], [1], 4) + (3 n1 + 12) hypergeom([1/2, -n1], [1], 4)) (n1 + 1) {1, n, (-1) , ) --------------------------------------------------------------------------------------------------} / n1 + 2 ----- n1 = 0 "A104598" n (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (3 n + 7), ------------------------------------} n + 1 "A104625" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 %1 := _Z - 4 _Z - 1 "A104629" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) ---------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A104631" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A104632" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A104653" LREtools/SearchTable: "SearchTable successful" n! (n + 1) ((3 n + 5) LegendreP(n + 1, 2) + (-12 n - 10) LegendreP(n, 2)) n! (n + 1) ((3 n + 5) LegendreQ(n + 1, 2) + (-12 n - 10) LegendreQ(n, 2)) {-------------------------------------------------------------------------, ------------------------------------------------------------------------- n (n - 2) (n - 1) n (n - 2) (n - 1) } "A104722" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 4 { 4 binomial(n, n/2) (5 n + 8) { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n + 2) { 32 binomial(n - 1, n/2 - 1/2) n { 2 (5 n + 8) { ------------------------------- n::odd { -------------------------------------------------- n::odd { (n + 1) (n + 3) { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) "A104858" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ / LegendreP(n1, 3) - 3 LegendreP(n1 + 1, 3)\ \ / LegendreQ(n1, 3) - 3 LegendreQ(n1 + 1, 3)\ {1, ) |- -----------------------------------------|, ) |- -----------------------------------------|} / \ n1 + 2 / / \ n1 + 2 / ----- ----- n1 = 0 n1 = 0 "A104859" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) {1, ) ----------------------------------------} / (n1 + 1) (2 n1 + 3) (2 n1 + 1) ----- n1 = 0 "A104969" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {2 , 2 | ) 2 |{ \ 2 / n1 ||, | / |{ 2 (-1) binomial(n1 - 1, ---- - 1/2) n1 || |----- |{ 2 || |n1 = 0 |{ -------------------------------------------------- n1::odd || \ \{ n1 + 1 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / || n | \ (-n1 - 1) |{ (-16) || 2 | ) 2 |{ --------------------------- n1::even||} | / |{ n1 || |----- |{ binomial(n1, ----) (n1 + 1) || |n1 = 0 |{ 2 || | |{ || \ \{ 0 n1::odd // "A104970" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ (-1) 2 (2 n1 + 1) binomial(2 n1, n1)| {4 , 4 | ) -------------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A105151" LREtools/SolveLRE: "Reduced the order of" E^4+(-n-5)*E^3+(-n-4)*E-1 "to two: Half integer product u(n/2) * u(n/2+1/2)" E^2+(-2*n-1)*E-1 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { -%3 %4 n::even { -%3 %2 n::even { -%3 BesselI(n/2 + 1, 1) n::even {{ , { , { , { -%3 BesselI(n/2 + 1, 1) n::odd { -%1 BesselI(n/2 + 1, 1) n::odd { -%3 %4 n::odd { -%3 BesselK(n/2 + 1, -1) n::even { -%1 %4 n::even { -%1 %2 n::even { , { , { , { -%1 %4 n::odd { -%3 BesselK(n/2 + 1, -1) n::odd { -%1 BesselK(n/2 + 1, -1) n::odd { -%1 BesselI(n/2 + 1, 1) n::even { -%1 BesselK(n/2 + 1, -1) n::even { , { } { -%3 %2 n::odd { -%1 %2 n::odd %1 := (n + 1) BesselK(n/2 + 1/2, -1) - BesselK(n/2 - 1/2, -1) %2 := n BesselK(n/2, -1) - BesselK(n/2 - 1, -1) %3 := (n + 1) BesselI(n/2 + 1/2, 1) - BesselI(n/2 - 1/2, 1) %4 := n BesselI(n/2, 1) - BesselI(n/2 - 1, 1) "A105216" LREtools/SolveLRE: "Reduced the order of" E^4+(-n-5)*E^3+(-n-4)*E-1 "to two: Half integer product u(n/2) * u(n/2+1/2)" E^2+(-2*n-1)*E-1 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { -%3 %4 n::even { -%3 %2 n::even { -%3 BesselI(n/2 + 1, 1) n::even {{ , { , { , { -%3 BesselI(n/2 + 1, 1) n::odd { -%1 BesselI(n/2 + 1, 1) n::odd { -%3 %4 n::odd { -%3 BesselK(n/2 + 1, -1) n::even { -%1 %4 n::even { -%1 %2 n::even { , { , { , { -%1 %4 n::odd { -%3 BesselK(n/2 + 1, -1) n::odd { -%1 BesselK(n/2 + 1, -1) n::odd { -%1 BesselI(n/2 + 1, 1) n::even { -%1 BesselK(n/2 + 1, -1) n::even { , { } { -%3 %2 n::odd { -%1 %2 n::odd %1 := (n + 1) BesselK(n/2 + 1/2, -1) - BesselK(n/2 - 1/2, -1) %2 := n BesselK(n/2, -1) - BesselK(n/2 - 1, -1) %3 := (n + 1) BesselI(n/2 + 1/2, 1) - BesselI(n/2 - 1/2, 1) %4 := n BesselI(n/2, 1) - BesselI(n/2 - 1, 1) "A105219" LREtools/SearchTable: "SearchTable successful" {(n + 1) n! (LaguerreL(n + 1, -1) - 2 LaguerreL(n, -1))} "A105524" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=49350.9MB, alloc=1783.5MB, time=343.12 / / / /{ / n2 \ \\\\ | | | |{ |----| |||| | | | |{ \ 2 / n2 |||| | | | |{ 12 (-1) binomial(n2, ----) |||| |n - 1 | |n1 - 1 / n2 \ |{ 2 |||| / 1/2\n / 1/2\n / 1/2\n |----- | |----- |---- + 1/2| |{ -------------------------------- n2::even|||| | 2 | |2 | | 2 | | \ | 1/2 n1 | \ \ 2 / |{ n2 + 4 |||| {|- ----| , |----| , |- ----| | ) |-2 (-1) | ) 2 |{ ||||, \ 2 / \ 2 / \ 2 / | / | | / |{ / n2 \ |||| |----- | |----- |{ |---- - 1/2| |||| |n1 = 0 | |n2 = 0 |{ \ 2 / n2 |||| | | | |{ 16 (-1) binomial(n2 - 1, ---- - 1/2) n2 |||| | | | |{ 2 |||| | | | |{ --------------------------------------------------- n2::odd |||| \ \ \ \{ (n2 + 1) (n2 + 3) //// / / / /{ / n2 \ \\\\ | | | |{ |----| |||| | | | |{ \ 2 / |||| | | | |{ 4 (-16) |||| | | | |{ ------------------------------------ n2::even|||| |n - 1 | |n1 - 1 / n2 \ |{ n2 |||| / 1/2\n |----- | |----- |---- + 1/2| |{ (n2 + 3) (n2 + 1) binomial(n2, ----) |||| | 2 | | \ | 1/2 n1 | \ \ 2 / |{ 2 |||| |- ----| | ) |-2 (-1) | ) 2 |{ ||||} \ 2 / | / | | / |{ / n2 \ |||| |----- | |----- |{ |---- + 1/2| |||| |n1 = 0 | |n2 = 0 |{ \ 2 / |||| | | | |{ 3 (-16) |||| | | | |{ ---------------------------------------------- n2::odd |||| | | | |{ n2 |||| | | | |{ (n2 + 1) (n2 + 4) binomial(n2 + 1, ---- + 1/2) |||| \ \ \ \{ 2 //// "A105633" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A105641" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(- 1/4 - 1/4 I 7 ) , (- 1/4 + 1/4 I 7 ) } "A105695" LREtools/SearchTable: "SearchTable successful" n (-1) ((2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (4 n - 1) hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------------------------} (n - 1) n "A105696" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (-2 n + 9 n - 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------------} (n - 1) n "A105747" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) ((4 n1 + 2) BesselI(n1 + 1/2, 1/2) - BesselI(n1 - 1/2, 1/2)), / ----- n1 = 0 n - 1 ----- \ n1 ) (-1) ((4 n1 + 2) BesselK(n1 + 1/2, -1/2) - BesselK(n1 - 1/2, -1/2))} / ----- n1 = 0 "A105748" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ n1 \ n1 {1, ) (-1) ((2 n1 + 1) BesselI(n1 + 1/2, 1) - BesselI(n1 - 1/2, 1)), ) (-1) ((2 n1 + 1) BesselK(n1 + 1/2, -1) - BesselK(n1 - 1/2, -1))} / / ----- ----- n1 = 0 n1 = 0 "A105749" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselI(n + 1/2, 1), (-1) n! BesselK(n + 1/2, -1)} "A105750" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-I) GAMMA(n + 1 + I), I GAMMA(n + 1 - I)} "A105751" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-I) GAMMA(n + 1 + I), I GAMMA(n + 1 - I)} "A105849" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A105864" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A105865" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A105872" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (7 n2 + 19)| | {(-1/3 I 3 ) , (1/3 I 3 ) , (-1/3 I 3 ) | ) (-1) 3 | ) ----------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (n2 + 3) (n2 + 2) (n2 + 1) (1/3 I 3 ) | | \n1 = 0 \n2 = 0 / / "A105926" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (2 n1 + 1) | {n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A105927" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) n1 (-1) | {n! (n + n - 1), n! (n + n - 1) | ) -----------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + n1) (n1 + n1 - 1)| \n1 = 0 / "A105928" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 3 | \ (n1 + 1) (n1 - 1) n1 (-1) | {n! (n - 4 n + 1), n! (n - 4 n + 1) | ) -------------------------------------------------|} | / 3 3 | |----- (n1 + 1)! ((n1 + 1) - 4 n1 - 3) (n1 - 4 n1 + 1)| \n1 = 0 / "A106050" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106053" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106174" LREtools/SearchTable: "SearchTable successful" n n {(-1) (2 n BesselJ(n, -1) + BesselJ(n - 1, -1)), (-1) (2 n BesselY(n, -1) + BesselY(n - 1, -1))} "A106181" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (n/2) { 2 (-1) binomial(n, n/2) { 2 (-16) { ---------------------------- n::even { -------------------------- n::even { n + 2 { (n + 1) n binomial(n, n/2) {{ , { } { (n/2 - 1/2) { (n/2 + 1/2) { 2 (-1) binomial(n - 1, n/2 - 1/2) { 2 (-16) { -------------------------------------------- n::odd { ------------------------------------------ n::odd { n + 1 { (n + 2) (n + 1) binomial(n + 1, n/2 + 1/2) "A106183" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106184" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106185" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106186" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106188" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n \ n1 | \ | (-1) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| {1, (-1) , ) (-1) | ) |- -----------------------------------------------||} / | / \ (n2 + 1) (n2 + 2) /| ----- |----- | n1 = 0 \n2 = 0 / "A106189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)||| {(2 ) , (-2 ) , (2 ) | ) |1/2 2 (-1) | ) ----------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (n2 + 1) (n2 + 2) (-2 ) ||| \n1 = 0 \ \n2 = 0 /// "A106191" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) {1, ) ------------------} / n1 + 1 ----- n1 = 0 "A106192" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n \ n1 | \ | (-1) (2 n2 + 1) binomial(2 n2, n2)|| {1, (-1) , ) (-1) | ) |- ------------------------------------||} / | / \ (n2 + 1) (n2 + 2) /| ----- |----- | n1 = 0 \n2 = 0 / "A106193" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ (2 n2 + 1) binomial(2 n2, n2) ||| {(2 ) , (-2 ) , (2 ) | ) |1/2 2 (-1) | ) ---------------------------------|||} | / | | / 1/2 (n2 + 1)||| |----- | |----- (n2 + 1) (n2 + 2) (-2 ) ||| \n1 = 0 \ \n2 = 0 /// "A106228" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A106258" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 I 2 ) LegendreP(n, 2 I), (-2 I 2 ) LegendreQ(n, 2 I)} "A106259" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 I 3 ) LegendreP(n, 3 I), (-2 I 3 ) LegendreQ(n, 3 I)} "A106260" LREtools/SearchTable: "SearchTable successful" n n {(-4 I) LegendreP(n, 2 I), (-4 I) LegendreQ(n, 2 I)} "A106261" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 I 5 ) LegendreP(n, 5 I), (-2 I 5 ) LegendreQ(n, 5 I)} "A106269" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n \ n1 | \ | (-1) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| {1, (-1) , ) (-1) | ) |- -----------------------------------------------||} / | / \ (n2 + 1) (n2 + 2) /| ----- |----- | n1 = 0 \n2 = 0 / "A106271" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -----------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A106272" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n \ n1 | \ | (-1) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| {1, (-1) , ) (-1) | ) |- -----------------------------------------------||} / | / \ (n2 + 3) (n2 + 2) (n2 + 1) /| ----- |----- | n1 = 0 \n2 = 0 / "A106539" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (-n1 - 1) | {2 , 2 | ) 2 n1!|} | / | |----- | \n1 = 0 / "A106640" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 3) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) (2 n + 1) binomial(2 n, n) {-----------------------------------------------------------------------------------------, --------------------------} n + 2 (n + 1) (n + 2) "A106651" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=49926.0MB, alloc=1783.5MB, time=347.12 /n - 1 / /n1 - 1 |----- | |----- n n / 2 82\ n | \ | n1 2 | \ / n2 / {(-1) , (-2) |n + 11/3 n + --|, (-1) | ) |-2 (27 n1 + 207 n1 + 334) | ) |-1/2 (-1) (2 n2 + 1) (2 n2 + 3) (4 n2 + 1) (4 n2 + 3) | \ 27/ | / | | / \ \ |----- | |----- \n1 = 0 \ \n2 = 0 4 3 2 n2 n2 4 (4 n2 + 7) (4 n2 + 5) (297 n2 + 3222 n2 + 12554 n2 + 21323 n2 + 13524) hypergeom([- ----, - ---- - 1/2], [-2 n2 - 7/2], -1) 2 2 2 n2 n2 \ - 3 (3 n2 + 5) (3 n2 + 7) (n2 + 2) (n2 + 1) (339 n2 + 1945 n2 + 2828) hypergeom([- ----, - ---- + 1/2], [-2 n2 - 3/2], -1)| binomial(4 n2, n2) 2 2 / / 2 / ((n2 + 1) (n2 + 2) (n2 + 3) (3 n2 + 1) (3 n2 + 2) (3 n2 + 4) (3 n2 + 5) (3 n2 + 7) (3 n2 + 8) (7 n2 + 13) (27 (n2 + 1) + 207 n2 + 541) / \\\ ||| 2 \||| (27 n2 + 207 n2 + 334))||||} /||| ||| /// "A107103" LREtools/SearchTable: "SearchTable successful" {2 BesselI(n + 1/2, 1) + BesselI(n - 1/2, 1), 2 BesselK(n + 1/2, -1) + BesselK(n - 1/2, -1)} "A107104" LREtools/SearchTable: "SearchTable successful" n n {(-2) BesselI(n - 1/2, 2), (-2) BesselK(n - 1/2, -2)} "A107231" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { ---------------- n::even { binomial(n, n/2) { binomial(n, n/2) (n + 1) (n + 2) n::even {{ , { } { (2 n + 2) { 2 (n + 2) n binomial(n - 1, n/2 - 1/2) n::odd { 2 (n/2 + 1) { -------------------------- n::odd { binomial(n + 1, n/2 + 1/2) "A107232" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 binomial(n, n/2) (2 n + 1) { 2 4 (n + 2) { ---------------------------- n::even { -------------------------------- n::even { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {{ , { } { 16 binomial(n - 1, n/2 - 1/2) n (n + 2) { (2 n + 2) { --------------------------------------- n::odd { 2 (2 n + 1) { (n + 1) (n + 3) { 1/2 ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A107233" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n - 1) { ---------------- n::even { 2 { binomial(n, n/2) {{ n binomial(n, n/2) n::even, { } { { (2 n - 2) { 1/4 (2 n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { 4 2 n { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A107264" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 (-I 3 ) (3 LegendreP(n + 1, 3 I) I - LegendreP(n, 3 I)) (-I 3 ) (-3 LegendreQ(n + 1, 3 I) I + LegendreQ(n, 3 I)) {-------------------------------------------------------------------, - --------------------------------------------------------------------} n + 2 n + 2 "A107265" LREtools/SearchTable: "SearchTable successful" (n/2) 1/2 1/2 1/2 (n/2) 1/2 1/2 1/2 5 (5 LegendreP(n + 1, 5 ) - LegendreP(n, 5 )) 5 (5 LegendreQ(n + 1, 5 ) - LegendreQ(n, 5 )) {---------------------------------------------------------, ---------------------------------------------------------} n + 2 n + 2 "A107266" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 (2 3 ) (3 LegendreP(n + 1, 3 ) - LegendreP(n, 3 )) (2 3 ) (-3 LegendreQ(n + 1, 3 ) + LegendreQ(n, 3 )) {------------------------------------------------------------, - -------------------------------------------------------------} n + 2 n + 2 "A107283" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ n1 + 2 n1 + 3| {n! | ) --------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A107373" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (2 n + 2) n::even { binomial(n, n/2) {2 , { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A107587" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2) | 1/2 5 5 | (n/2) | 1/2 5 5 | 5 |5 LegendreP(n + 1, ----) - 5 LegendreP(n, ----)| 5 |5 LegendreQ(n + 1, ----) - 5 LegendreQ(n, ----)| \ 5 5 / \ 5 5 / {-----------------------------------------------------------, -----------------------------------------------------------, n + 2 n + 2 n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) -------------------------------------------------------------------------} n + 2 "A107597" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A107708" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A107841" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 5) - 5 LegendreP(n, 5) LegendreQ(n + 1, 5) - 5 LegendreQ(n, 5) {---------------------------------------, ---------------------------------------} n n "A107991" n (-1) n! n! (2 n + 3) {---------------, ---------------} (n + 1) (n + 2) (n + 1) (n + 2) "A108080" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (n2 - 2 n2 - 2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A108081" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2) (n2 + 4)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A108095" LREtools/SearchTable: "SearchTable successful" n n (-1) ((7 n + 7) LegendreP(n + 1, 7) + (-97 n - 49) LegendreP(n, 7)) (-1) ((7 n + 7) LegendreQ(n + 1, 7) + (-97 n - 49) LegendreQ(n, 7)) {--------------------------------------------------------------------, --------------------------------------------------------------------} (n - 1) n (n - 1) n "A108188" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n n | \ (-1/2 I 2 ) (HermiteH(n1 + 1, 1/2 I 2 ) - 2 HermiteH(n1, 1/2 I 2 ) I)| {(-1) n, (-1) n | ) --------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (-1) n1 | \n1 = 0 / "A108204" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, 1/4 - n, 1/4)} "A108205" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, - 1/4 - n, -1/4)} "A108206" LREtools/SearchTable: "SearchTable successful" n {(-3) n! LaguerreL(n, - 1/9 - n, -1/9)} "A108207" LREtools/SearchTable: "SearchTable successful" n {(-5) n! LaguerreL(n, 1/25 - n, 1/25)} "A108208" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, 1/8 - n, 1/8)} "A108209" LREtools/SearchTable: "SearchTable successful" n {(-5) n! LaguerreL(n, 2/25 - n, 2/25)} "A108210" LREtools/SearchTable: "SearchTable successful" n {(-5) n! LaguerreL(n, 3/25 - n, 3/25)} "A108246" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108296" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A108304" LREtools/SearchTable: "SearchTable successful" 6 5 4 3 2 {((4 n + 81 n + 679 n + 3042 n + 8578 n + 13806 n + 8748) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 5 4 3 2 / 2 2 2 - 9 (n + 1) (4 n + 73 n + 530 n + 1928 n + 3654 n + 2916) hypergeom([1/2, -n, -n], [1, 1], 4)) / (n (n + 2) (n + 3) (n + 5) (n + 4) )} / "A108307" LREtools/SearchTable: "SearchTable successful" 6 5 4 3 2 {((3 n + 73 n + 727 n + 1783 n - 3690 n - 18320 n - 16384) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) 5 4 3 2 / 2 2 2 - 8 (n + 1) (3 n + 67 n + 591 n + 2833 n + 6962 n + 6704) hypergeom([-n, -n, -n], [1, 1], -1)) / ((n + 2) (n + 3) (n + 4) (n + 6) / 2 (n + 5) )} "A108308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - _Z + 1, index = 1) , RootOf(2 _Z - _Z + 1, index = 2) , RootOf(2 _Z - _Z + 1, index = 3) } "A108424" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------} (10 n + 7) (n + 1) "A108427" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1) {--------------------------------------------------------------------------------------------------------------------------------} (10 n + 7) (n + 1) "A108432" memory used=50516.8MB, alloc=1783.5MB, time=351.15 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) {(3 - 17 ) , (3 + 17 ) , (3 - 17 ) | ) (3 + 17 ) (3 - 17 ) | ) (3 + 17 ) | / | / |----- |----- \n1 = 0 \n2 = 0 3 2 2 ((32 n2 + 24 n2 - 59 n2 - 42) hypergeom([2 n2 + 3, -n2 - 1], [n2 + 2], -1) + (n2 + 1) n2 hypergeom([-n2, 2 n2 + 1], [n2 + 1], -1)) \\ || || binomial(2 n2, n2) (2 n2 + 1)/((n2 + 1) (n2 + 2) (2 n2 + 3) (2 n2 + 5) (10 n2 + 7))||} || || // "A108434" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) {(1 - 2 ) , (1 + 2 ) , (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) | / | / |----- |----- \n1 = 0 \n2 = 0 2 ((168 n2 + 356 n2 + 166) hypergeom([2 n2 + 3, -n2 - 1], [n2 + 2], -1) + (n2 + 1) (4 n2 + 5) hypergeom([-n2, 2 n2 + 1], [n2 + 1], -1)) \\ || || binomial(2 n2, n2) (2 n2 + 1)/((n2 + 2) (2 n2 + 3) (2 n2 + 5) (10 n2 + 7))||} || || // "A108436" LREtools/SearchTable: "SearchTable successful" 2 {((28 n + 44 n + 18) hypergeom([2 n + 3, -n - 1], [n + 2], -1) - (n + 1) (n + 3) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1)/( (n + 1) (n + 2) (2 n + 3) (10 n + 7))} "A108442" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- n n n | \ n1 | \ {(-I) , I , (-I) | ) (-1) | ) | / | / |----- |----- \n1 = 0 \n2 = 0 (2 (8 n2 + 7) (2 n2 + 1) hypergeom([2 n2 + 3, -n2 - 1], [n2 + 2], -1) + (6 n2 + 7) (n2 + 1) hypergeom([-n2, 2 n2 + 1], [n2 + 1], -1)) \ \ | | (n2 + 1) | | binomial(2 n2, n2)/((2 n2 + 3) (10 n2 + 7) (n2 + 1) I )| I|} | | | | / / "A108444" LREtools/SearchTable: "SearchTable successful" 4 3 2 {((168 n + 748 n + 1194 n + 821 n + 204) hypergeom([2 n + 3, -n - 1], [n + 2], -1) 3 2 + (n + 1) (4 n + 16 n + 24 n + 15) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (2 n + 1)/((n + 1) (n + 2) (2 n + 3) (2 n + 5) (10 n + 7))} "A108447" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108448" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) (4 hypergeom([2 n + 3, -n - 1], [n + 2], -1) - 3 hypergeom([-n, 2 n + 1], [n + 1], -1)) {------------------------------------------------------------------------------------------------------------------} 10 n + 7 "A108449" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 9 _Z + 3 _Z - 3, index = 1) , RootOf(_Z - 9 _Z + 3 _Z - 3, index = 2) , RootOf(_Z - 9 _Z + 3 _Z - 3, index = 3) } "A108450" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ 2 {1, ) ((40 n1 + 76 n1 + 34) hypergeom([2 n1 + 3, -n1 - 1], [n1 + 2], -1) + (-n1 - 1) hypergeom([-n1, 2 n1 + 1], [n1 + 1], -1)) / ----- n1 = 0 binomial(2 n1, n1) (2 n1 + 1)/((n1 + 1) (n1 + 2) (2 n1 + 3) (10 n1 + 7))} "A108452" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 5 _Z - 2, index = 1) , RootOf(_Z - 5 _Z + 5 _Z - 2, index = 2) , RootOf(_Z - 5 _Z + 5 _Z - 2, index = 3) } "A108453" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ 2 {1, ) ((40 n1 + 76 n1 + 34) hypergeom([2 n1 + 3, -n1 - 1], [n1 + 2], -1) + (-n1 - 1) hypergeom([-n1, 2 n1 + 1], [n1 + 1], -1)) / ----- n1 = 0 binomial(2 n1, n1) (2 n1 + 1)/((n1 + 1) (n1 + 2) (2 n1 + 3) (10 n1 + 7))} "A108488" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108489" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108490" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108524" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| n n | \ |2 4 (2 LegendreP(n1 + 1, 2) - LegendreP(n1, 2))|| n | \ | 2 4 (-2 LegendreQ(n1 + 1, 2) + LegendreQ(n1, 2))|| {(1/2) , (1/2) | ) |-------------------------------------------------||, (1/2) | ) |- --------------------------------------------------||} | / \ n1 + 2 /| | / \ n1 + 2 /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A108600" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 %1 := _Z - 4 _Z - 1 "A108623" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=51107.5MB, alloc=1783.5MB, time=355.24 / n1 \ / n1 \ |----| / 1/2 1/2 \ |----| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 LegendreP(n1 + 1, ----) - 5 LegendreP(n1, ----)| ----- 5 |5 LegendreQ(n1 + 1, ----) - 5 LegendreQ(n1, ----)| \ \ 5 5 / \ \ 5 5 / {1, ) --------------------------------------------------------------, ) --------------------------------------------------------------} / n1 + 2 / n1 + 2 ----- ----- n1 = 0 n1 = 0 "A108624" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / n1 \ \ | |----| / 1/2 1/2 \| |n - 1 \ 2 / | 1/2 5 5 || |----- 5 |-5 LegendreP(n1 + 1, ----) + 5 LegendreP(n1, ----)|| n n | \ \ 5 5 /| {(-1) , (-1) | ) ---------------------------------------------------------------|, | / n1 + 2 | |----- | \n1 = 0 / / / n1 \ \ | |----| / 1/2 1/2 \| |n - 1 \ 2 / | 1/2 5 5 || |----- 5 |-5 LegendreQ(n1 + 1, ----) + 5 LegendreQ(n1, ----)|| n | \ \ 5 5 /| (-1) | ) ---------------------------------------------------------------|} | / n1 + 2 | |----- | \n1 = 0 / "A108626" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108628" LREtools/SearchTable: "SearchTable successful" ((8 n + 4) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-n - 1) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) {-----------------------------------------------------------------------------------------------------------------------------------} n + 1 "A108629" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A108630" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A108666" LREtools/SearchTable: "SearchTable successful" {(n + 1) LegendreP(n + 1, 3) + (-7 n - 3) LegendreP(n, 3), (n + 1) LegendreQ(n + 1, 3) + (-7 n - 3) LegendreQ(n, 3)} "A108704" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108781" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A108863" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 1) hypergeom([1/2, -n], [1], 4)) {3 , ------------------------------------------------------------------------------------------} n + 2 "A108869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 6 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A108895" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" /{ 0 irem(n1, 4) = 0\ /{ 0 irem(n1, 4) = 0\ |{ | |{ | n - 1 |{ 0 irem(n1, 4) = 1| n - 1 |{ 0 irem(n1, 4) = 1| ----- |{ | ----- |{ | \ |{ 0 irem(n1, 4) = 2| \ |{ / n1 \ | {1, ) |{ |, ) |{ |---- - 1| |, / |{ / n1 \ | / |{ \ 2 / n1 | ----- |{ |---- - 3/2| | ----- |{ 2 GAMMA(---- + 5/4) irem(n1, 4) = 2| n1 = 0 |{ \ 2 / / n1 \ / n1 \ | n1 = 0 |{ 4 | |{ 2 |---- + 1/4| |---- - 3/4|! irem(n1, 4) = 3| |{ | \{ \ 4 / \ 4 / / \{ 0 irem(n1, 4) = 3/ /{ 0 irem(n1, 4) = 0\ n - 1 |{ | ----- |{ / n1 \ / n1 \ n1 n1 | \ |{ |---- + 1/2| |---- - 1/4|! binomial(---- - 1/2, ---- - 1/4) irem(n1, 4) = 1| ) |{ \ 2 / \ 4 / 2 4 |, / |{ | ----- |{ 0 irem(n1, 4) = 2| n1 = 0 |{ | \{ 0 irem(n1, 4) = 3/ /{ / n1 \ \ |{ |----| | n - 1 |{ \ 2 / n1 | ----- |{ 2 GAMMA(---- + 5/4) irem(n1, 4) = 0| \ |{ 4 | ) |{ |} / |{ 0 irem(n1, 4) = 1| ----- |{ | n1 = 0 |{ 0 irem(n1, 4) = 2| |{ | \{ 0 irem(n1, 4) = 3/ "A108958" n (2 n + 1) binomial(2 n, n) {2 , --------------------------} n + 1 "A108999" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 3 2 (2 (2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) - (32 n + 30 n + 5) (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------------, n (2 n - 1) (10 n + 7) (n + 1) LegendreP(n + 1, 3) + (-6 n - 3) LegendreP(n, 3), (n + 1) LegendreQ(n + 1, 3) + (-6 n - 3) LegendreQ(n, 3)} "A109033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A109034" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A109078" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { n2 \ \ | | { 12 binomial(n2, ----) (n2 + 1) | | | | { 2 | | | | { ------------------------------ n2::even| | | | { n2 + 2 | | | | { | | | | { n2 | | | | { 2 binomial(n2 + 1, ---- + 1/2) (3 n2 + 5) | | |n - 1 |n1 - 1 { 2 | | |----- |----- { ----------------------------------------- n2::odd | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { n2 + 3 | | {(-1/2 I 2 ) , (1/2 I 2 ) , (-1/2 I 2 ) | ) (-1) 2 | ) -----------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { 4 (3 n2 + 5) | | | | { 1/2 ------------------------------------ n2::even| | | | { n2 | | | | { (n2 + 1) (n2 + 3) binomial(n2, ----) | | | | { 2 | | | | { | | | | { (2 n2 - 2) | | | | { 3 2 (n2 + 1) | | | | { ---------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) ----------------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A109081" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A109141" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A109188" LREtools/SearchTable: "SearchTable successful" n {(-1) (n + 1) hypergeom([1/2, -n], [1], 4)} "A109190" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / | 1/2 n 1/2 n 1/2 n | {(1 - 5 ) , (5 + 1) , (1 - 5 ) | | | \ n - 1 /n1 - 1 \ ----- |----- n2 1/2 (-n2 - 1) | \ 1/2 n1 1/2 (-n1 - 1) | \ (-1) (5 + 1) (hypergeom([1/2, -n2 - 1], [1], 4) - 3 hypergeom([1/2, -n2], [1], 4))| ) (5 + 1) (1 - 5 ) | ) ------------------------------------------------------------------------------------------------| / | / n2 + 2 | ----- |----- | n1 = 0 \n2 = 0 / \ | | |} | | / "A109192" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 2 _Z - 3 _Z - 1 "A109194" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n (-1) ((2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) {(-1) , 3 , --------------------------------------------------------------------------------------------} n + 2 "A109196" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n (-1) ((2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) {(-1) , 3 , --------------------------------------------------------------------------------------------} n + 2 "A109245" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 n 3 2 n 3 2 n {(-1/2) , RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A109262" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (5 n2 + 4)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A109263" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) n2|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -----------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A109388" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { n { |---| { 3 (2 n + 7) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) { \ 3 / { -------------------------------------------- irem(n, 3) = 0 { 54 2 (n + 1) binomial(n, n/3) { (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 13/6) { ---------------------------------- irem(n, 3) = 0 { { 2 n + 3 { (n - 1) { {{ 3 3 (2 n + 5) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) , { /2 n \ , { ---------------------------------------------------- irem(n, 3) = 1 { |--- + 4/3| { (n + 2) GAMMA(n/3 + 4/3) GAMMA(n/3 + 11/6) { \ 3 / { { 3 2 binomial(n + 2, n/3 + 2/3) irem(n, 3) = 1 { n { { 2 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { /2 n \ { -------------------------------------- irem(n, 3) = 2 { |--- + 2/3| { GAMMA(n/3 + 1) GAMMA(n/3 + 3/2) { \ 3 / { 9 2 binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 { n { 3 3 (2 n + 5) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) { ---------------------------------------------- irem(n, 3) = 0 { (n + 2) GAMMA(n/3 + 4/3) GAMMA(n/3 + 11/6) { { (n - 1) { 18 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) } { --------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 1) GAMMA(n/3 + 3/2) { { (n + 1) { 3 (2 n + 7) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) { -------------------------------------------------- irem(n, 3) = 2 { (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 13/6) "A109475" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A109575" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(2 RootOf(_Z - 6 _Z - 1, index = 1) - 12) n!, (2 RootOf(_Z - 6 _Z - 1, index = 2) - 12) n!, (2 RootOf(_Z - 6 _Z - 1, index = 3) - 12) n!} "A109576" n {(-2) n! (3 n + 1), n!} "A109581" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(RootOf(_Z - _Z + 1, index = 1) - 1) n!, (RootOf(_Z - _Z + 1, index = 2) - 1) n!, (RootOf(_Z - _Z + 1, index = 3) - 1) n!} "A109582" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n 3 2 n 3 2 n {(RootOf(_Z - _Z + 1, index = 1) - 1) n!, (RootOf(_Z - _Z + 1, index = 2) - 1) n!, (RootOf(_Z - _Z + 1, index = 3) - 1) n!} "A109742" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 | 3 2 | \ (-1) (2 n1 - 1) n1 (n1 - 1) (n1 + 1) | n! (2 n - 3 n - 2 n + 2) | ) -----------------------------------------------------------------------| | / 3 2 3 2 | 3 2 |----- (n1 + 1)! (2 (n1 + 1) - 3 (n1 + 1) - 2 n1) (2 n1 - 3 n1 - 2 n1 + 2)| n! (2 n - 3 n - 2 n + 2) \n1 = 0 / {--------------------------, -----------------------------------------------------------------------------------------------------------} (n - 1) (n - 2) n (n - 1) (n - 2) n "A109743" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 | \ (-1) (n1 + 2) (2 n1 + 3) (n1 + 1) | n! (2 n + 9 n + 10 n + 2) | ) ------------------------------------------------------------------------------| | / 3 2 3 2 | 3 2 |----- (n1 + 1)! (2 (n1 + 1) + 9 (n1 + 1) + 10 n1 + 12) (2 n1 + 9 n1 + 10 n1 + 2)| n! (2 n + 9 n + 10 n + 2) \n1 = 0 / {---------------------------, -------------------------------------------------------------------------------------------------------------------} n n "A109771" LREtools/SearchTable: "SearchTable successful" n n (-1) ((3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3)) (-1) ((3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3)) {-------------------------------------------------------------------, -------------------------------------------------------------------} (n - 1) n (n - 1) n "A109779" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2| | \ (n1 + 3) (n1 + 2) (n1 + 1) (n1!) | {(n + 1) n!, (n + 1) n! | ) ----------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | 2 2 | \ (n2 + 1) n2! || | (n1 + 3) (n1 + 2) (n1 + 1) (n1!) | ) ----------------------------------------|| |n - 1 | / 2 2|| |----- |----- (n2 + 2) (n2 + 4) (n2 + 3) ((n2 + 1)!) || | \ \n2 = 0 /| (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | 2 2 | \ (n2 + 1) n2! || | (n1 + 3) (n1 + 2) (n1 + 1) (n1!) | ) ---------------------------------------|| |n - 1 | / 2|| |----- |----- (n2 + 3) (n2 + 2) (n2 + 4) ((n2 + 1)!) || | \ \n2 = 0 /| (n + 1) n! | ) -----------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- 2 2||| | | | \ (n3 + 3) (n3 + 2) (n3 + 1) (n3!) ||| | | (n2 + 1) n2! | ) ----------------------------------||| | |n1 - 1 | / (n3 + 1)! ||| | |----- |----- ||| | 2 2 | \ \n3 = 0 /|| | (n1 + 3) (n1 + 2) (n1 + 1) (n1!) | ) --------------------------------------------------------|| |n - 1 | / 2 || |----- |----- (n2 + 3) (n2 + 2) (n2 + 4) ((n2 + 1)!) || | \ \n2 = 0 /| (n + 1) n! | ) ----------------------------------------------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A109780" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | 2 2 | \ (n2 + 1) n2! || | (n1 + 2) (n1 + 1) (n1!) | ) -------------------------------|| /n - 1 \ |n - 1 | / 2 2|| |----- 2 2| |----- |----- (n2 + 2) (n2 + 3) ((n2 + 1)!) || | \ (n1 + 2) (n1 + 1) (n1!) | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) -------------------------|, (n + 1) n! | ) ------------------------------------------------------------------|, | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / /n1 - 1 \\ | |----- || | 2 2 | \ (n2 + 1) n2! || | (n1 + 2) (n1 + 1) (n1!) | ) ------------------------------|| |n - 1 | / 2|| |----- |----- (n2 + 3) (n2 + 2) ((n2 + 1)!) || | \ \n2 = 0 /| (n + 1) n! | ) -----------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- 2 2||| | | | \ (n3 + 2) (n3 + 1) (n3!) ||| | | (n2 + 1) n2! | ) -------------------------||| | |n1 - 1 | / (n3 + 1)! ||| | |----- |----- ||| | 2 2 | \ \n3 = 0 /|| | (n1 + 2) (n1 + 1) (n1!) | ) -----------------------------------------------|| |n - 1 | / 2 || |----- |----- (n2 + 3) (n2 + 2) ((n2 + 1)!) || | \ \n2 = 0 /| (n + 1) n! | ) ----------------------------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A109792" memory used=51661.9MB, alloc=1783.5MB, time=359.43 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 1) n!, (n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A109851" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1!} / ----- n1 = 0 "A109980" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- (-n1 - 1) | |----- (-n1 - 1) | n n | \ 6 (LegendreP(n1 + 1, 3) - 3 LegendreP(n1, 3))| n | \ 6 (LegendreQ(n1 + 1, 3) - 3 LegendreQ(n1, 3))| {6 , 6 | ) ------------------------------------------------------|, 6 | ) ------------------------------------------------------|} | / n1 | | / n1 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A109984" LREtools/SearchTable: "SearchTable successful" {(n + 1) LegendreP(n + 1, 3) + (n - 3) LegendreP(n, 3), (n + 1) LegendreQ(n + 1, 3) + (n - 3) LegendreQ(n, 3)} "A110038" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A110099" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(3 + 2 2 ) , (-2 2 + 3) , LegendreP(n, 3), LegendreQ(n, 3)} "A110110" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::even { n + 1 {{ , { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 ((7 n + 13) LegendreP(n/2 + 1/2, 3) + (-n - 3) LegendreP(n/2 + 3/2, 3)) { ------------------------------------------------------------------------- n::even { n + 1 { , { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::even { n { , { 2 ((7 n + 13) LegendreP(n/2 + 1/2, 3) + (-n - 3) LegendreP(n/2 + 3/2, 3)) { ------------------------------------------------------------------------- n::odd { n + 1 { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::even { n { } { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::odd { n + 1 "A110122" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / |n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 3 5 | | 3 5 | | 3 5 | | \ {|7/2 - ------| , |7/2 + ------| , |7/2 - ------| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 3 5 | ||| | |----- |7/2 + ------| ((10 n2 + 17) LegendreP(n2 + 1, 3) + (-2 n2 - 3) LegendreP(n2, 3))||| | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |2 (7 + 3 5 ) (7 - 3 5 ) | ) ------------------------------------------------------------------------------------------|||, | | / (n2 + 2) (n2 + 3) ||| | |----- ||| \ \n2 = 0 /// / |n - 1 / 1/2\n |----- | 3 5 | | \ |7/2 - ------| | ) \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 3 5 | ||| | |----- |7/2 + ------| ((10 n2 + 17) LegendreQ(n2 + 1, 3) + (-2 n2 - 3) LegendreQ(n2, 3))||| | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |2 (7 + 3 5 ) (7 - 3 5 ) | ) ------------------------------------------------------------------------------------------|||} | | / (n2 + 2) (n2 + 3) ||| | |----- ||| \ \n2 = 0 /// "A110127" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n (n + 1) LegendreP(n + 1, 3) + (-5 n - 3) LegendreP(n, 3) (n + 1) LegendreQ(n + 1, 3) + (-5 n - 3) LegendreQ(n, 3) {(3 + 2 2 ) , (-2 2 + 3) , --------------------------------------------------------, --------------------------------------------------------} n n "A110144" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) { 6 2 (n/2)! { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { --------------- n::even {{ , { n } { (- n/2 + 1/2) { { 3 2 binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { (n/2 + 1/2) { 2 2 (n/2 + 1/2)! n::odd "A110145" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { (n/2) {binomial(2 n, n), { (n/2 - 1/2) , { (-1) binomial(n, n/2) n::even} { (-16) { { ---------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A110149" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A110166" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {5 , (2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)} "A110167" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 2 | n n | \ (16 n1 + 35 n1 + 13) hypergeom([-1/2, -n1 - 1], [1], -4) - (16 n1 + 27) (n1 + 1) hypergeom([-1/2, -n1], [1], -4)| {(-1/3) , (-1/3) | ) -----------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1/3) | \n1 = 0 / "A110170" LREtools/SearchTable: "SearchTable successful" (n + 1) LegendreP(n + 1, 3) + (-5 n - 3) LegendreP(n, 3) (n + 1) LegendreQ(n + 1, 3) + (-5 n - 3) LegendreQ(n, 3) {--------------------------------------------------------, --------------------------------------------------------} n n "A110184" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n (n + 1) LegendreP(n + 1, 3) + (-5 n - 3) LegendreP(n, 3) (n + 1) LegendreQ(n + 1, 3) + (-5 n - 3) LegendreQ(n, 3) {(3 + 2 2 ) , (-2 2 + 3) , --------------------------------------------------------, --------------------------------------------------------} n n "A110190" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (n + 6 n + 6 n + 8) LegendreP(n + 1, 3) + (-7 n - 26 n - 34 n - 24) LegendreP(n, 3) {---------------------------------------------------------------------------------------, (n + 2) n (n - 1) (n - 2) 3 2 3 2 (n + 6 n + 6 n + 8) LegendreQ(n + 1, 3) + (-7 n - 26 n - 34 n - 24) LegendreQ(n, 3) ---------------------------------------------------------------------------------------} (n + 2) n (n - 1) (n - 2) "A110198" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A110199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 /{ n1 \ ----- |{ | ----- |{ 4 | \ |{ n1 | \ |{ ------------------------------------ n1::even| {1, ) |{ 4 binomial(n1 - 1, ---- - 1/2) n1 |, ) |{ n1 |} / |{ 2 | / |{ (n1 + 1) (n1 + 3) binomial(n1, ----) | ----- |{ --------------------------------- n1::odd | ----- |{ 2 | n1 = 0 \{ (n1 + 1) (n1 + 3) / n1 = 0 |{ | \{ 0 n1::odd / "A110236" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110239" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110313" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A110318" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110320" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110322" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ | |----- / n2 \| | | n1 (-n1 - 1) | \ | (-1) || | | (-1) 2 (n1 + 1) | ) |- ------------------|| n1!| |n - 1 | / \ (n2 + 2) (n2 + 1)!/| | |----- |----- | | n n n | \ \n2 = 0 / | {(-1) n!, 2 n!, 2 n! | ) --------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A110328" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(1 - 2 ) n!, (1 + 2 ) n!} "A110334" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 %1 := _Z - 3 _Z + 3 _Z - _Z + 1 "A110335" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110347" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /3 n \ 2 3 n { |--- + 1| ((n/2)!) binomial(n, n/2) binomial(---, n/2) n::even { \ 2 / 2 {{ , { 2 3 n { 1/2 ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) n::odd { 2 { (-n) 2 3 n 3 n { 4 ((n/2)!) (3 n + 1) binomial(3 n, ---) binomial(---, n/2) n::even { 2 2 { } { (-2 n + 2) 2 3 n 3 n { 2 ((n/2 - 1/2)!) (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 2 "A110467" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- / (n1 + 1) \| n n | \ (n1 + 1) n1! | n | \ | 5 (n1 + 1) n1!|| {(n + 1) (4/5) n!, (n + 1) (4/5) n! | ) --------------------------------|, (n + 1) (4/5) n! | ) |1/4 ----------------------||} | / (n1 + 1) | | / \ (n1 + 2) (n1 + 1)! /| |----- (n1 + 2) (4/5) (n1 + 1)!| |----- | \n1 = 0 / \n1 = 0 / "A110469" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- / (n1 + 1) \| n n | \ (n1 + 1) n1! | n | \ | 3 (n1 + 1) n1!|| {(n + 1) (2/3) n!, (n + 1) (2/3) n! | ) -----------------------------------------|, (n + 1) (2/3) n! | ) |1/2 ----------------------||, | / (n1 + 1) | | / \ (n1 + 2) (n1 + 1)! /| |----- (n1 + 3) (n1 + 2) (2/3) (n1 + 1)!| |----- | \n1 = 0 / \n1 = 0 / n (n + 1) (-1) n! ----------------} n + 2 "A110491" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 2 { 4 ((n/2)!) { 2 ((n/2)!) binomial(n, n/2) n::even { ------------ n::even { {{ n , { 2 2 } { { ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) { (2 n - 2) 2 { ------------------------------------------- n::odd { 2 2 ((n/2 - 1/2)!) n::odd { n "A110505" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { (-n1) n1 2 // n1 \ \2 \ | { 4 binomial(n1, ----) ||----|!| (2 n1 + 2) n1::even| | { 2 \\ 2 / / | | { | |n - 1 { (-2 n1 + 2) n1 2 // n1 \ \2 | |----- { -2 n1 binomial(n1 - 1, ---- - 1/2) ||---- - 1/2|!| (n1 + 2) n1::odd | | \ { 2 \\ 2 / / | {n! | ) ------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { // n1 \ \2 \ | { ||----|!| (n1 + 2) | | { \\ 2 / / | | { - ------------------- n1::even| | { n1 | | { | | { // n1 \ \2 | | { 2 ||---- + 1/2|!| | |n - 1 { \\ 2 / / | |----- { ------------------ n1::odd | | \ { n1 + 1 | n! | ) ---------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A110508" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z + 2 _Z + 1, index = 1) , RootOf(_Z + 2 _Z + 1, index = 2) , RootOf(_Z + 2 _Z + 1, index = 3) } "A110520" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 3 binomial(2 n1, n1)| {(-4) , (-4) | ) ----------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4) | \n1 = 0 / "A110521" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(2 _Z + 2 _Z + 1, index = 1) , RootOf(2 _Z + 2 _Z + 1, index = 2) , RootOf(2 _Z + 2 _Z + 1, index = 3) } "A110525" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(3 _Z - _Z + 1, index = 1) , RootOf(3 _Z - _Z + 1, index = 2) , RootOf(3 _Z - _Z + 1, index = 3) } "A110695" memory used=52290.9MB, alloc=1847.5MB, time=363.88 2 3 {1, binomial(2 n, n) , binomial(2 n, n) , binomial(2 n, n)} "A110696" 2 3 {1, binomial(2 n, n) , binomial(2 n, n) , binomial(2 n, n)} "A110697" 2 3 {1, binomial(2 n, n) , binomial(2 n, n) , binomial(2 n, n)} "A110698" 2 3 {1, binomial(2 n, n) , binomial(2 n, n) , binomial(2 n, n)} "A110706" LREtools/SearchTable: "SearchTable successful" (3 n + 5) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (3 n + 2) hypergeom([-n, -n, -n], [1, 1], -1) {---------------------------------------------------------------------------------------------------------} n + 2 "A110707" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + hypergeom([-n, -n, -n], [1, 1], -1)} "A110711" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (n - 2) hypergeom([-n, -n, -n], [1, 1], -1) {-----------------------------------------------------------------------------------------------------} n + 2 "A110822" LREtools/SearchTable: "SearchTable successful" n (-1) ((15 n + 15) LegendreP(n + 1, 15) + (-449 n - 225) LegendreP(n, 15)) {--------------------------------------------------------------------------, (n - 1) n n (-1) ((15 n + 15) LegendreQ(n + 1, 15) + (-449 n - 225) LegendreQ(n, 15)) --------------------------------------------------------------------------} (n - 1) n "A110824" LREtools/SearchTable: "SearchTable successful" n (-1) ((31 n + 31) LegendreQ(n + 1, 31) + (-1921 n - 961) LegendreQ(n, 31)) {---------------------------------------------------------------------------, (n - 1) n n (-1) ((1921 n + 961) LegendreP(n, 31) + (-31 n - 31) LegendreP(n + 1, 31)) - ---------------------------------------------------------------------------} (n - 1) n "A110826" LREtools/SearchTable: "SearchTable successful" n (-1) ((63 n + 63) LegendreP(n + 1, 63) + (-7937 n - 3969) LegendreP(n, 63)) {----------------------------------------------------------------------------, (n - 1) n n (-1) ((63 n + 63) LegendreQ(n + 1, 63) + (-7937 n - 3969) LegendreQ(n, 63)) ----------------------------------------------------------------------------} (n - 1) n "A110828" LREtools/SearchTable: "SearchTable successful" n (-1) ((127 n + 127) LegendreP(n + 1, 127) + (-32257 n - 16129) LegendreP(n, 127)) {----------------------------------------------------------------------------------, (n - 1) n n (-1) ((127 n + 127) LegendreQ(n + 1, 127) + (-32257 n - 16129) LegendreQ(n, 127)) ----------------------------------------------------------------------------------} (n - 1) n "A110830" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A110886" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A111053" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A111063" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) | n! | ) ---------| | / (n1 + 1)!| |----- | n! \n1 = 0 / {----, ---------------------} n n "A111139" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\n1\ / / 1/2 \n1\ |n - 1 | 5 | | |n - 1 |5 | | |----- |1/2 - ----| | |----- |---- + 1/2| | | \ \ 2 / | | \ \ 2 / | {n! | ) --------------|, n! | ) --------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A111140" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n! (LaguerreL(n + 1, -1) - 2 LaguerreL(n, -1)) {----------------------------------------------} n "A111160" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) n {----------------, (-1) n + 1 ((8 n + 4) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (3 n + 3) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/((5 n + 3) (n + 1))} "A111177" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 2 | \ n1 + 5 n1 + 10 | {n! (n + 1) (n + n + 2), n! (n + 1) (n + n + 2) | ) ------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! (n1 + 2) ((n1 + 1) + n1 + 3) (n1 + n1 + 2)| \n1 = 0 / "A111279" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2) (n2 + 9)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A111308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (-n) {{ , { 2 (n + 1) (n + 3) binomial(n, n/2) (n/2)! n::even} { 1/4 (n/2 - 1/2)! (n + 1) (n + 3) n::odd { { 0 n::odd "A111394" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (- n/2) /3 n \ 2 3 n { 3 |--- + 1| ((n/2)!) binomial(n, n/2) binomial(---, n/2) n::even { \ 2 / 2 {{ , { (- n/2 - 1/2) 2 3 n { 3 ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) n::odd { 2 { (- n/2) 2 3 n 3 n { 48 ((n/2)!) (6 n + 2) binomial(3 n, ---) binomial(---, n/2) n::even { 2 2 { } { (- n/2 + 1/2) 2 3 n 3 n { 48 ((n/2 - 1/2)!) (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 2 "A111424" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (2 n1 + 3) (2 n1 + 1) n1! binomial(2 n1, n1)} / ----- n1 = 0 "A111601" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 2 { 2 ((n/2)!) (n/4 + 1/4) n::even { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { } { (n - 1) 2 2 { (-n - 1) 2 2 { 1/4 2 (n + 1) ((n/2 - 1/2)!) n::odd { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 1) n::odd "A111602" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 { 1/2 2 (n + 2) ((n/2)!) (n + 1) n::even {{ , { (n + 1) 2 { 1/4 2 ((n/2 + 1/2)!) (n + 2) (2 n + 3) n::odd { (-n) 2 2 2 { 1/2 2 (n + 1) binomial(n, n/2) ((n/2)!) (n + 2) (2 n + 3) n::even { } { (-n + 1) 2 2 2 2 { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 1) n::odd "A111752" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------} n "A111753" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------} n "A111777" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 { 1/16 2 (n + 2) ((n/2)!) (n + 1) (n + 3) (4 n + 11) n::even {{ , { (n - 1) 2 2 2 { 1/16 2 (n + 1) (n + 3) ((n/2 - 1/2)!) (n + 2) (4 n + 5) n::odd { (-n) 2 2 2 2 { 1/2 2 (n + 1) (n + 3) binomial(n, n/2) ((n/2)!) (n + 2) (4 n + 5) n::even { } { (-n - 1) 2 2 2 { 1/2 2 (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 1) (n + 3) (4 n + 11) n::odd "A111778" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 2 { 1/4 2 (n + 2) (n + 4) ((n/2)!) (n + 1) (n + 3) (2 n + 5) n::even {{ , { (n + 1) 2 2 2 { 1/16 2 (n + 3) ((n/2 + 1/2)!) (n + 2) (n + 4) (8 n + 40 n + 35) n::odd { (-n) 2 2 2 2 2 { 1/4 2 (n + 1) (n + 3) binomial(n, n/2) ((n/2)!) (n + 2) (n + 4) (8 n + 40 n + 35) n::even { } { (-n + 1) 2 2 2 2 2 { 2 n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 1) (n + 3) (2 n + 5) n::odd "A111779" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 2 2 { 1/64 2 (n + 2) (n + 4) ((n/2)!) (n + 1) (n + 3) (n + 5) (16 n + 116 n + 183) n::even {{ , { (n - 1) 2 2 2 2 2 { 1/64 2 (n + 1) (n + 3) (n + 5) ((n/2 - 1/2)!) (n + 2) (n + 4) (16 n + 76 n + 63) n::odd { (-n) 2 2 2 2 2 2 { 1/4 2 (n + 1) (n + 3) (n + 5) binomial(n, n/2) ((n/2)!) (n + 2) (n + 4) (16 n + 76 n + 63) n::even { } { (-n - 1) 2 2 2 2 2 { 1/4 2 (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 1) (n + 3) (n + 5) (16 n + 116 n + 183) n::odd "A111780" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 2 2 2 { 1/32 2 (n + 2) (n + 4) (n + 6) ((n/2)!) (n + 1) (n + 3) (n + 5) (16 n + 112 n + 169) n::even {{ , { { (n + 1) 2 2 2 2 { { 1/64 2 (n + 3) (n + 5) ((n/2 + 1/2)!) (n + 2) (n + 4) (n + 6) (2 n + 7) (16 n + 112 n + 99) n::odd (-n) 2 2 2 2 2 2 1/8 2 (n + 1) (n + 3) (n + 5) binomial(n, n/2) ((n/2)!) (n + 2) (n + 4) (n + 6) (2 n + 7) (16 n + 112 n + 99) , n::even (-n + 1) 2 2 2 2 2 2 2 } 1/4 2 n (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 1) (n + 3) (n + 5) (16 n + 112 n + 169) , n::odd "A111781" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 2 2 3 2 { 1/256 2 (n + 2) (n + 4) (n + 6) ((n/2)!) (n + 1) (n + 3) (n + 5) (n + 7) (64 n + 880 n + 3656 n + 4409) n::even {{ , { { (n - 1) 2 2 2 2 2 3 2 { { 1/256 2 (n + 1) (n + 3) (n + 5) (n + 7) ((n/2 - 1/2)!) (n + 2) (n + 4) (n + 6) (64 n + 656 n + 1864 n + 1287) n::odd (-n) 2 2 2 2 2 2 3 2 1/8 2 (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) ((n/2)!) (n + 2) (n + 4) (n + 6) (64 n + 656 n + 1864 n + 1287) , n::even 1/8 (-n - 1) 2 2 2 2 2 3 2 2 (n + 2) (n + 4) (n + 6) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 1) (n + 3) (n + 5) (n + 7) (64 n + 880 n + 3656 n + 4409) } , n::odd "A111882" LREtools/SolveLRE: "Reduced the order of" E^3+(n+1)*E^2-(n+1)*(n+2)*E-(n+2)*(n+1)^2 "to two: Symmetric square" E^2-E+n+1 LREtools/SearchTable: "SearchTable successful" 1/2 2 (-n) 2 {2 HermiteH(n, ----) } 2 "A111883" LREtools/SolveLRE: "Reduced the order of" E^3+(-n-3)*E^2-(n+2)*(n+3)*E+(n+2)*(n+1)^2 "to two: Symmetric square" E^2-E-n-1 LREtools/SearchTable: "SearchTable successful" n 1/2 2 {(-1/2) HermiteH(n, 1/2 I 2 ) } "A111884" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) LaguerreL(n + 1, 1) - n LaguerreL(n, 1)) n! {----------------------------------------------------------} n "A111942" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { (-n) 2 2 { 2 ((n/2)!) { -4 binomial(n, n/2) ((n/2)!) n::even { ----------- n::even { {{ n , { (-2 n - 2) 2 2 } { { 2 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) { 2 { --------------------------------------------------------- n::odd { -((n/2 - 1/2)!) n::odd { n "A111961" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / | 1/2 n 1/2 n 1/2 n | {(1 - 5 ) , (5 + 1) , (1 - 5 ) | | | \ n - 1 /n1 - 1 \ ----- |----- n2 1/2 (-n2 - 1) | \ 1/2 n1 1/2 (-n1 - 1) | \ (-1) (5 + 1) (hypergeom([1/2, -n2 - 1], [1], 4) - 3 hypergeom([1/2, -n2], [1], 4))| ) (5 + 1) (1 - 5 ) | ) ------------------------------------------------------------------------------------------------| / | / n2 + 2 | ----- |----- | n1 = 0 \n2 = 0 / \ | | |} | | / "A111962" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=52917.2MB, alloc=1847.5MB, time=368.11 n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 2 _Z - 3 _Z - 1 "A111964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z + 3 _Z + 2 _Z - 1 "A111966" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(3 - 5 ) , (3 + 5 ) , (3 - 5 ) | ) (3 + 5 ) (3 - 5 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) || | \ (3 + 5 ) ((4 n2 + 1) hypergeom([-1/2, -n2 - 1], [1], -4) + (-4 n2 - 3) hypergeom([-1/2, -n2], [1], -4))|| | ) ------------------------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A111968" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A111993" LREtools/SearchTable: "SearchTable successful" 2 2 (17 n + 21 n + 1) LegendreP(n + 1, 3) + (-3 n - 3 n - 3) LegendreP(n, 3) {--------------------------------------------------------------------------, (n + 5) (n + 4) n 2 2 (17 n + 21 n + 1) LegendreQ(n + 1, 3) + (-3 n - 3 n - 3) LegendreQ(n, 3) --------------------------------------------------------------------------} (n + 5) (n + 4) n "A111994" LREtools/SearchTable: "SearchTable successful" 5 4 3 2 5 4 3 2 (58 n + 553 n + 1866 n + 2633 n + 1322 n + 30) LegendreP(n + 1, 3) + (-10 n - 91 n - 290 n - 395 n - 246 n - 90) LegendreP(n, 3) {----------------------------------------------------------------------------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) n 5 4 3 2 5 4 3 2 (58 n + 553 n + 1866 n + 2633 n + 1322 n + 30) LegendreQ(n + 1, 3) + (-10 n - 91 n - 290 n - 395 n - 246 n - 90) LegendreQ(n, 3) ----------------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) n "A111995" LREtools/SearchTable: "SearchTable successful" 6 5 4 3 2 {((99 n + 1335 n + 6825 n + 16461 n + 18654 n + 7962 n + 90) LegendreP(n + 1, 3) 6 5 4 3 2 + (-17 n - 221 n - 1075 n - 2431 n - 2586 n - 1206 n - 270) LegendreP(n, 3))/((n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) n), ( 6 5 4 3 2 (99 n + 1335 n + 6825 n + 16461 n + 18654 n + 7962 n + 90) LegendreQ(n + 1, 3) 6 5 4 3 2 + (-17 n - 221 n - 1075 n - 2431 n - 2586 n - 1206 n - 270) LegendreQ(n, 3))/((n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) n)} "A111996" LREtools/SearchTable: "SearchTable successful" 7 6 5 4 3 2 {((169 n + 3056 n + 21922 n + 79526 n + 153187 n + 147890 n + 55824 n + 315) LegendreP(n + 1, 3) 7 6 5 4 3 2 + (-29 n - 510 n - 3524 n - 12156 n - 21917 n - 19626 n - 7452 n - 945) LegendreP(n, 3))/((n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) 7 6 5 4 3 2 (n + 2) n), ((169 n + 3056 n + 21922 n + 79526 n + 153187 n + 147890 n + 55824 n + 315) LegendreQ(n + 1, 3) 7 6 5 4 3 2 + (-29 n - 510 n - 3524 n - 12156 n - 21917 n - 19626 n - 7452 n - 945) LegendreQ(n, 3))/((n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) n)} "A111997" LREtools/SearchTable: "SearchTable successful" 6 5 4 3 2 {((577 n + 8267 n + 42695 n + 101041 n + 110334 n + 44634 n + 126) LegendreP(n + 1, 3) 6 5 4 3 2 + (-99 n - 1369 n - 6693 n - 14651 n - 14442 n - 5382 n - 378) LegendreP(n, 3))/((n + 9) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) n), ( 6 5 4 3 2 (577 n + 8267 n + 42695 n + 101041 n + 110334 n + 44634 n + 126) LegendreQ(n + 1, 3) 6 5 4 3 2 + (-99 n - 1369 n - 6693 n - 14651 n - 14442 n - 5382 n - 378) LegendreQ(n, 3))/((n + 9) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) n)} "A111998" LREtools/SearchTable: "SearchTable successful" 7 6 5 4 3 2 {((1970 n + 37861 n + 285908 n + 1077145 n + 2119040 n + 2054794 n + 765852 n + 1080) LegendreP(n + 1, 3) 7 6 5 4 3 2 + (-338 n - 6327 n - 46064 n - 164835 n - 301592 n - 265518 n - 90036 n - 3240) LegendreP(n, 3))/((n + 10) (n + 9) (n + 8) (n + 6) (n + 5) 7 6 5 4 3 2 (n + 4) (n + 2) n), ((1970 n + 37861 n + 285908 n + 1077145 n + 2119040 n + 2054794 n + 765852 n + 1080) LegendreQ(n + 1, 3) 7 6 5 4 3 2 + (-338 n - 6327 n - 46064 n - 164835 n - 301592 n - 265518 n - 90036 n - 3240) LegendreQ(n, 3))/((n + 10) (n + 9) (n + 8) (n + 6) (n + 5) (n + 4) (n + 2) n)} "A112019" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A112028" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { 0 n::even { { (n/2 - 1/2) {{ 4 (-1) , { ----------------------------------------------------------------------------- n::odd { 3 n { binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { (n/2) 2 { (-1) binomial(n, n/2) { ----------------------------------------------- n::even { 3 n 3 n } { binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { { 0 n::odd "A112029" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 1) binomial(2 n1, n1) (21 n1 + 29) (n1 + 3/2) binomial(2 n1 + 2, n1 + 1) ) ---------------------------------------------------------------------------------- / 2 ----- (n1 + 1) (4 n1 + 6) 1 n1 = 0 {1/2 --------------------------, 1/2 -----------------------------------------------------------------------------------------} (n + 1/2) binomial(2 n, n) (n + 1/2) binomial(2 n, n) "A112241" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A112242" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A112243" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A112293" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A112308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (13 n1 + 25) {1, ) -----------------------------------------------------} / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 1) ----- n1 = 0 "A112328" n (2 n + 1) binomial(2 n, n) (4 n + 7) {4 , ------------------------------------} n + 1 "A112368" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) (n1 + 1) 2 n1!} / ----- n1 = 0 "A112369" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) (n1 + 1) 2 n1!} / ----- n1 = 0 "A112370" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) (n1 + 1) 3 n1!} / ----- n1 = 0 "A112478" LREtools/SearchTable: "SearchTable successful" n n (-1) ((3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3)) (-1) ((3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3)) {-------------------------------------------------------------------, -------------------------------------------------------------------} (n - 1) n (n - 1) n "A112483" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A112488" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 4) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A112520" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-1/2) , (-1/2) | ) -----------------------|, | / (n1 + 1)| |----- (n1 + 1) (-1/2) | \n1 = 0 / /n - 1 \ |----- | n | \ ((10 n1 + 5) hypergeom([1/2, -n1 - 1], [-2 n1 - 2], -4) + (14 n1 + 3) hypergeom([1/2, -n1], [-2 n1], -4)) binomial(2 n1, n1)| (-1/2) | ) ----------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (4 n1 + 1) (n1 + 1) (-1/2) | \n1 = 0 / "A112521" LREtools/SearchTable: "SearchTable successful" (2 n + 1) n binomial(2 n, n) (5 hypergeom([1/2, -n - 1], [-2 n - 2], -4) - hypergeom([1/2, -n], [-2 n], -4)) {------------------------------------------------------------------------------------------------------------} (4 n + 1) (4 n - 1) "A112556" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1)| {1, (-1/2) , (-1/2) | ) ------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A112657" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) + 3 hypergeom([1/2, -n1], [1], 4))| {(7/2) , (7/2) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (7/2) | \n1 = 0 / "A112696" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 2 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112697" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 3 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112698" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 4 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112699" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 5 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 6 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112701" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 7 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112702" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 8 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112703" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 9 binomial(2 n1, n1) {1, ) ---------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112704" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) 10 binomial(2 n1, n1) {1, ) ----------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112710" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-3) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112711" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-4) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A112849" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 1 "A112850" {(2 n + 1) binomial(2 n, n), n + 1} "A112951" memory used=53606.2MB, alloc=1879.5MB, time=372.69 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3, 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / / |n - 1 | / 1/2\n |----- | | 3 17 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {1, %1, |11/4 + -------| , %1 | ) |-4 (-1) (-11 + 3 17 ) (11 + 3 17 ) \ 4 / | / | |----- | \n1 = 0 \ / / 1/2\(-n2 - 1) \\\ |n1 - 1 | 3 17 | ||| |----- |11/4 + -------| ((29 n2 + 61) LegendreP(n2 + 1, 3) + (-7 n2 - 15) LegendreP(n2, 3))||| | \ \ 4 / ||| | ) ---------------------------------------------------------------------------------------------|||, | / (n2 + 2) (n2 + 3) ||| |----- ||| \n2 = 0 /// / / / / 1/2\(-n2 - 1) \\\ / |n - 1 | |n1 - 1 | 3 17 | ||| | |----- | |----- |11/4 + -------| (2 n2 + 1) binomial(2 n2, n2) (n2 + 9)||| | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 4 / ||| | %1 | ) |-4 (-1) (-11 + 3 17 ) (11 + 3 17 ) | ) ----------------------------------------------------------------|||, %1 | | / | | / (n2 + 3) (n2 + 2) (n2 + 1) ||| | |----- | |----- ||| | \n1 = 0 \ \n2 = 0 /// \ / n - 1 | ----- | \ | n1 1/2 (-n1 - 1) 1/2 n1 ) |-4 (-1) (-11 + 3 17 ) (11 + 3 17 ) / | ----- | n1 = 0 \ / / / 1/2\(-n2 - 1) \\\\ |n1 - 1 | | 3 17 | |||| |----- | |11/4 + -------| ((7 n2 + 15) LegendreQ(n2, 3) + (-29 n2 - 61) LegendreQ(n2 + 1, 3))|||| | \ | \ 4 / |||| | ) |- ---------------------------------------------------------------------------------------------||||} | / \ (n2 + 2) (n2 + 3) /||| |----- ||| \n2 = 0 /// / 1/2\n | 3 17 | %1 := |11/4 - -------| \ 4 / "A113179" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A113180" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A113235" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A113236" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A113237" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A113247" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {n!, { (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A113264" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-5) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113265" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-6) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113266" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-7) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113267" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-8) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113268" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-9) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113269" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (2 n1 + 1) (-10) binomial(2 n1, n1) {1, ) -------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A113281" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A113289" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(1/2 I) (n + 1) HermiteH(n, 1/2 I)} "A113291" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-1/2 I) (HermiteH(n1 + 1, 1/2 I) - HermiteH(n1, 1/2 I) I)|| {(-1) , (-1) | ) |- ------------------------------------------------------------||} | / \ n1 /| |----- | \n1 = 0 / "A113292" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ n1 | {(-1) , (-1) | ) (-1/2 I) (HermiteH(n1 + 1, 1/2 I) - HermiteH(n1, 1/2 I) I) I|} | / | |----- | \n1 = 0 / "A113337" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A113409" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A113485" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {2 , (-1/2 I) ((n + 2) HermiteH(n + 1, I) + 2 I (n + 1) HermiteH(n, I))} "A113549" n n {(n + 1) n!, (n + 1) (-1) n!, (2 n + 1) n! binomial(2 n, n), (2 n + 1) (-1) n! binomial(2 n, n)} "A113550" n n {(n + 1) n!, (n + 1) (-1) n!, (2 n + 1) n! binomial(2 n, n), (2 n + 1) (-1) n! binomial(2 n, n)} "A113551" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { {{ (n - 1) 2 3 n , { 1/4 2 ((n/2 - 1/2)!) (3 n - 1) (3 n + 1) binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 { (-n) 2 3 n 3 n { 2 ((n/2)!) (3 n + 1) binomial(3 n, ---) binomial(---, n/2) n::even { 2 2 } { { 0 n::odd "A113682" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) hypergeom([1/2, -n - 1], [1], 4)} "A113775" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A113873" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/3) { (n/3) { (-1) %4 irem(n, 3) = 0 { (-1) %3 irem(n, 3) = 0 { { {{ (n/3 - 1/3) , { (n/3 - 1/3) , { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 1 { (-1) %4 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 - 2/3) { (-1) %3 irem(n, 3) = 2 { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 2 { (n/3) { (n/3) { (-1) %2 irem(n, 3) = 0 { (-1) %1 irem(n, 3) = 0 { { { (n/3 - 1/3) , { (n/3 - 1/3) , { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 1 { (-1) %2 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 - 2/3) { (-1) %1 irem(n, 3) = 2 { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 2 { (n/3) { (n/3) { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 0 { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 0 { { { (n/3 + 2/3) , { (n/3 + 2/3) } { (-1) %3 irem(n, 3) = 1 { (-1) %1 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 + 1/3) { (-1) %4 irem(n, 3) = 2 { (-1) %2 irem(n, 3) = 2 %1 := -3 BesselK(n/3 + 7/6, -1/2) - 3 BesselK(1/6 + n/3, -1/2) %2 := -3 BesselK(n/3 + 5/6, -1/2) + 3 BesselK(n/3 - 1/6, -1/2) %3 := -3 BesselI(n/3 + 7/6, 1/2) - 3 BesselI(1/6 + n/3, 1/2) %4 := -3 BesselI(n/3 + 5/6, 1/2) + 3 BesselI(n/3 - 1/6, 1/2) "A113874" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/3) { (n/3) { (-1) %4 irem(n, 3) = 0 { (-1) %3 irem(n, 3) = 0 { { {{ (n/3 - 1/3) , { (n/3 - 1/3) , { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 1 { (-1) %4 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 - 2/3) { (-1) %3 irem(n, 3) = 2 { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 2 { (n/3) { (n/3) { (-1) %2 irem(n, 3) = 0 { (-1) %1 irem(n, 3) = 0 { { { (n/3 - 1/3) , { (n/3 - 1/3) , { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 1 { (-1) %2 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 - 2/3) { (-1) %1 irem(n, 3) = 2 { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 2 { (n/3) { (n/3) { -6 (-1) BesselI(n/3 + 1/2, 1/2) irem(n, 3) = 0 { -6 (-1) BesselK(n/3 + 1/2, -1/2) irem(n, 3) = 0 { { { (n/3 + 2/3) , { (n/3 + 2/3) } { (-1) %3 irem(n, 3) = 1 { (-1) %1 irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 + 1/3) { (-1) %4 irem(n, 3) = 2 { (-1) %2 irem(n, 3) = 2 %1 := -3 BesselK(n/3 + 7/6, -1/2) - 3 BesselK(1/6 + n/3, -1/2) %2 := -3 BesselK(n/3 + 5/6, -1/2) + 3 BesselK(n/3 - 1/6, -1/2) %3 := -3 BesselI(n/3 + 7/6, 1/2) - 3 BesselI(1/6 + n/3, 1/2) %4 := -3 BesselI(n/3 + 5/6, 1/2) + 3 BesselI(n/3 - 1/6, 1/2) "A113956" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - 4 _Z - 1, index = 1) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 2) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 3) } "A114121" n {4 , binomial(2 n, n)} "A114160" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 binomial(2 n1, n1) n1! \| {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A114161" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 binomial(2 n1, n1) n1! \| {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A114190" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z + 2 _Z - 1, index = 1) , RootOf(_Z + 2 _Z - 1, index = 2) , RootOf(_Z + 2 _Z - 1, index = 3) } "A114191" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (-2) binomial(2 n1, n1) {1, ) -------------------------} / n1 + 1 ----- n1 = 0 "A114194" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- n2 || n n | \ (n1 + 1) | \ (2 n2 + 1) (-2) binomial(2 n2, n2) (17 n2 + 27)|| {1, (1/3) , (1/3) | ) 3 | ) -------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A114198" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" memory used=54268.9MB, alloc=1879.5MB, time=377.34 / 1/2\n / 1/2\n / 1/2 \n | 5 | 1/2 | 5 | 1/2 |5 | 1/2 {|1/2 - ----| LegendreP(n, -2 - 5 ), |1/2 - ----| LegendreQ(n, -2 - 5 ), |---- + 1/2| LegendreP(n, -2 + 5 ), \ 2 / \ 2 / \ 2 / / 1/2 \n |5 | 1/2 |---- + 1/2| LegendreQ(n, -2 + 5 )} \ 2 / "A114277" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) ---------------------------------------------------} / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 1) ----- n1 = 0 "A114311" {(n - 1) n, n!} "A114347" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" /{ 0 irem(n1, 3) = 0\ /{ 0 irem(n1, 3) = 0\ n - 1 |{ | n - 1 |{ | ----- |{ 0 irem(n1, 3) = 1| ----- |{ / n1 \ | \ |{ | \ |{ |---- - 1/3| | {1, ) |{ / n1 \ |, ) |{ \ 3 / n1 |, / |{ |---- - 2/3| | / |{ 3 GAMMA(---- + 4/3) irem(n1, 3) = 1| ----- |{ / n1 \ \ 3 / / n1 \ | ----- |{ 3 | n1 = 0 |{ |---- + 1/3| 3 |---- - 2/3|! irem(n1, 3) = 2| n1 = 0 |{ | \{ \ 3 / \ 3 / / \{ 0 irem(n1, 3) = 2/ /{ / n1 \ \ n - 1 |{ |----| | ----- |{ \ 3 / n1 | \ |{ 3 GAMMA(---- + 4/3) irem(n1, 3) = 0| ) |{ 3 |} / |{ | ----- |{ 0 irem(n1, 3) = 1| n1 = 0 |{ | \{ 0 irem(n1, 3) = 2/ "A114464" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A114465" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A114487" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + _Z + 1, index = 1) , RootOf(_Z + _Z + 1, index = 2) , RootOf(_Z + _Z + 1, index = 3) } "A114495" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (15 n1 + 32) | {(-1/2) (9 n + 8), (-1/2) (9 n + 8) | ) ------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 4) (n1 + 3) (n1 + 2) (-1/2) (9 n1 + 17) (18 n1 + 16)| \n1 = 0 / "A114496" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, 2 n + 1], [1], -1)} "A114500" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 3 %1 := _Z + _Z + 1 "A114515" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) | {(-1/2) n, (-1/2) n | ) --------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1/2) (n1 + 1)| \n1 = 0 / "A114582" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 n 3 n 3 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A114584" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A114587" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 11) | {(-1/2) (3 n + 7), (-1/2) (3 n + 7) | ) -----------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 5) (n1 + 4) (n1 + 1) (-1/2) (3 n1 + 10) (6 n1 + 14)| \n1 = 0 / "A114589" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- n n n | \ | \ (n2 + 1) {(-1/2 - 1/2 I) , (-1/2 + 1/2 I) , (-1/2 - 1/2 I) | ) (-1 + I) exp(-1/2 I n1 Pi) | ) (- (1 + I) ( | / | / |----- |----- \n1 = 0 \n2 = 0 4 3 2 (121 n2 + 1482 n2 + 6479 n2 + 11934 n2 + 7776) hypergeom([1/2, -n2 - 1], [1], 4) \\ || 3 2 || - 3 (41 n2 + 441 n2 + 1534 n2 + 1728) (n2 + 1) hypergeom([1/2, -n2], [1], 4))/((n2 + 2) (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 6)))||} || || // "A114590" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 7) | {(-1/2) (3 n + 5), (-1/2) (3 n + 5) | ) -------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 3) (n1 + 1) (-1/2) (3 n1 + 8) (6 n1 + 10)| \n1 = 0 / "A114627" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + 3 _Z + 2 _Z + 1, index = 1) , RootOf(_Z + 3 _Z + 2 _Z + 1, index = 2) , RootOf(_Z + 3 _Z + 2 _Z + 1, index = 3) } "A114633" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- / n2 \|| | n1 | \ | (-1) ||| | (n1 + 1) (-1) n1! | ) |- ------------------||| |n - 1 | / \ (n2 + 2) (n2 + 1)!/|| |----- |----- || n | \ \n2 = 0 /| {(n + 2) (n + 1) n!, (n + 2) (n + 1) (-1) n!, (n + 2) (n + 1) n! | ) ---------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A114693" LREtools/SearchTable: "SearchTable successful" 2 2 (31 n + 93 n + 65) LegendreP(n + 1, 3) + (-5 n - 11 n - 3) LegendreP(n, 3) {----------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) 2 2 (31 n + 93 n + 65) LegendreQ(n + 1, 3) + (-5 n - 11 n - 3) LegendreQ(n, 3) ----------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A114710" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ 3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3)| n | \ 3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3)| {(-2/3) , (-2/3) | ) -----------------------------------------|, (-2/3) | ) -----------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (n1 + 2) (-2/3) | |----- (n1 + 2) (-2/3) | \n1 = 0 / \n1 = 0 / "A114713" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A114799" LREtools/SolveLRE: "Absolute Factorization reduced the order from 7 to 1 (Liouvillian solutions)" { 0 irem(n, 7) = 0 { 0 irem(n, 7) = 0 { { { 0 irem(n, 7) = 1 { 0 irem(n, 7) = 1 { { { 0 irem(n, 7) = 2 { 0 irem(n, 7) = 2 { { {{ 0 irem(n, 7) = 3, { 0 irem(n, 7) = 3, { { { 0 irem(n, 7) = 4 { 0 irem(n, 7) = 4 { { { 0 irem(n, 7) = 5 { (n/7 - 5/7) { { 7 GAMMA(n/7 + 1) irem(n, 7) = 5 { (n/7 - 6/7) { { 7 GAMMA(n/7 + 1) irem(n, 7) = 6 { 0 irem(n, 7) = 6 { 0 irem(n, 7) = 0 { 0 irem(n, 7) = 0 { { { 0 irem(n, 7) = 1 { 0 irem(n, 7) = 1 { { { 0 irem(n, 7) = 2 { 0 irem(n, 7) = 2 { { { 0 irem(n, 7) = 3, { (n/7 - 3/7) , { { 7 GAMMA(n/7 + 1) irem(n, 7) = 3 { (n/7 - 4/7) { { 7 GAMMA(n/7 + 1) irem(n, 7) = 4 { 0 irem(n, 7) = 4 { { { 0 irem(n, 7) = 5 { 0 irem(n, 7) = 5 { { { 0 irem(n, 7) = 6 { 0 irem(n, 7) = 6 { 0 irem(n, 7) = 0 { 0 irem(n, 7) = 0 { (n/7) { { { 7 (n/7)! irem(n, 7) = 0 { 0 irem(n, 7) = 1 { (n/7 - 1/7) { { { 7 GAMMA(n/7 + 1) irem(n, 7) = 1 { 0 irem(n, 7) = 1 { (n/7 - 2/7) { { { 7 GAMMA(n/7 + 1) irem(n, 7) = 2 { 0 irem(n, 7) = 2 { 0 irem(n, 7) = 2 { , { , { } { 0 irem(n, 7) = 3 { 0 irem(n, 7) = 3 { 0 irem(n, 7) = 3 { { { { 0 irem(n, 7) = 4 { 0 irem(n, 7) = 4 { 0 irem(n, 7) = 4 { { { { 0 irem(n, 7) = 5 { 0 irem(n, 7) = 5 { 0 irem(n, 7) = 5 { { { { 0 irem(n, 7) = 6 { 0 irem(n, 7) = 6 { 0 irem(n, 7) = 6 "A114800" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 1 (Liouvillian solutions)" { 0 irem(n, 8) = 0 { 0 irem(n, 8) = 0 { { { 0 irem(n, 8) = 1 { 0 irem(n, 8) = 1 { { { 0 irem(n, 8) = 2 { 0 irem(n, 8) = 2 { { { 0 irem(n, 8) = 3 { 0 irem(n, 8) = 3 { {{ , { 0 irem(n, 8) = 4, { 0 irem(n, 8) = 4 { { { 0 irem(n, 8) = 5 { 0 irem(n, 8) = 5 { { { /3 n \ { 0 irem(n, 8) = 6 { |--- - 9/4| { { \ 8 / { (n/8 - 7/8) { 2 GAMMA(n/8 + 1) irem(n, 8) = 6 { 8 GAMMA(n/8 + 1) irem(n, 8) = 7 { { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 0 { 0 irem(n, 8) = 0 { { { 0 irem(n, 8) = 1 { 0 irem(n, 8) = 1 { { { 0 irem(n, 8) = 2 { 0 irem(n, 8) = 2 { { { 0 irem(n, 8) = 3 { 0 irem(n, 8) = 3 { , { , { 0 irem(n, 8) = 4 { (n/8 - 1/2) { { 1/4 n 2 (n/8 - 1/2)! binomial(n/4 - 1, n/8 - 1/2) irem(n, 8) = 4 { (n/8 - 5/8) { { 8 GAMMA(n/8 + 1) irem(n, 8) = 5 { 0 irem(n, 8) = 5 { { { 0 irem(n, 8) = 6 { 0 irem(n, 8) = 6 { { { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 0 { 0 irem(n, 8) = 0 { { { 0 irem(n, 8) = 1 { 0 irem(n, 8) = 1 { { { 0 irem(n, 8) = 2 { /3 n \ { { |--- - 3/4| { /3 n \ { \ 8 / { |--- - 9/8| { 2 GAMMA(n/8 + 1) irem(n, 8) = 2 { \ 8 / , { , { 2 GAMMA(n/8 + 1) irem(n, 8) = 3 { 0 irem(n, 8) = 3 { { { 0 irem(n, 8) = 4 { 0 irem(n, 8) = 4 { { { 0 irem(n, 8) = 5 { 0 irem(n, 8) = 5 { { { 0 irem(n, 8) = 6 { 0 irem(n, 8) = 6 { { { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 0 { /3 n\ { { |---| { /3 n \ { \ 8 / { |--- - 3/8| { 2 (n/8)! irem(n, 8) = 0 { \ 8 / { { 2 GAMMA(n/8 + 1) irem(n, 8) = 1 { 0 irem(n, 8) = 1 { { { 0 irem(n, 8) = 2 { 0 irem(n, 8) = 2 { , { } { 0 irem(n, 8) = 3 { 0 irem(n, 8) = 3 { { { 0 irem(n, 8) = 4 { 0 irem(n, 8) = 4 { { { 0 irem(n, 8) = 5 { 0 irem(n, 8) = 5 { { { 0 irem(n, 8) = 6 { 0 irem(n, 8) = 6 { { { 0 irem(n, 8) = 7 { 0 irem(n, 8) = 7 "A114805" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" /{ 0 irem(n1, 5) = 0\ /{ 0 irem(n1, 5) = 0\ |{ | |{ | |{ 0 irem(n1, 5) = 1| |{ 0 irem(n1, 5) = 1| n - 1 |{ | n - 1 |{ | ----- |{ 0 irem(n1, 5) = 2| ----- |{ 0 irem(n1, 5) = 2| \ |{ | \ |{ | {1, ) |{ 0 irem(n1, 5) = 3|, ) |{ / n1 \ |, / |{ | / |{ |---- - 3/5| | ----- |{ / n1 \ | ----- |{ \ 5 / n1 | n1 = 0 |{ |---- - 4/5| | n1 = 0 |{ 5 GAMMA(---- + 6/5) irem(n1, 5) = 3| |{ / n1 \ \ 5 / / n1 \ | |{ 5 | |{ |---- + 1/5| 5 |---- - 4/5|! irem(n1, 5) = 4| |{ | \{ \ 5 / \ 5 / / \{ 0 irem(n1, 5) = 4/ /{ 0 irem(n1, 5) = 0\ /{ 0 irem(n1, 5) = 0\ |{ | |{ | |{ 0 irem(n1, 5) = 1| |{ / n1 \ | n - 1 |{ | n - 1 |{ |---- - 1/5| | ----- |{ / n1 \ | ----- |{ \ 5 / n1 | \ |{ |---- - 2/5| | \ |{ 5 GAMMA(---- + 6/5) irem(n1, 5) = 1| ) |{ \ 5 / n1 |, ) |{ 5 |, / |{ 5 GAMMA(---- + 6/5) irem(n1, 5) = 2| / |{ | ----- |{ 5 | ----- |{ 0 irem(n1, 5) = 2| n1 = 0 |{ | n1 = 0 |{ | |{ 0 irem(n1, 5) = 3| |{ 0 irem(n1, 5) = 3| |{ | |{ | \{ 0 irem(n1, 5) = 4/ \{ 0 irem(n1, 5) = 4/ /{ / n1 \ \ |{ |----| | |{ \ 5 / n1 | n - 1 |{ 5 GAMMA(---- + 6/5) irem(n1, 5) = 0| ----- |{ 5 | \ |{ | ) |{ 0 irem(n1, 5) = 1|} / |{ | ----- |{ 0 irem(n1, 5) = 2| n1 = 0 |{ | |{ 0 irem(n1, 5) = 3| |{ | \{ 0 irem(n1, 5) = 4/ "A114806" LREtools/SolveLRE: "Absolute Factorization reduced the order from 9 to 1 (Liouvillian solutions)" { 0 irem(n, 9) = 0 { 0 irem(n, 9) = 0 { { { 0 irem(n, 9) = 1 { 0 irem(n, 9) = 1 { { { 0 irem(n, 9) = 2 { 0 irem(n, 9) = 2 { { { 0 irem(n, 9) = 3 { 0 irem(n, 9) = 3 { { {{ 0 irem(n, 9) = 4, { 0 irem(n, 9) = 4, { { { 0 irem(n, 9) = 5 { 0 irem(n, 9) = 5 { { { 0 irem(n, 9) = 6 { 0 irem(n, 9) = 6 { { { 0 irem(n, 9) = 7 { (n/9 - 7/9) { { 9 GAMMA(n/9 + 1) irem(n, 9) = 7 { (n/9 - 8/9) { { 9 GAMMA(n/9 + 1) irem(n, 9) = 8 { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 0 { { 0 irem(n, 9) = 0 { 0 irem(n, 9) = 1 { { { 0 irem(n, 9) = 1 { 0 irem(n, 9) = 2 { { { 0 irem(n, 9) = 2 { 0 irem(n, 9) = 3 { { { 0 irem(n, 9) = 3 { 0 irem(n, 9) = 4 { { , { 0 irem(n, 9) = 4, { 0 irem(n, 9) = 5 { { { (n/9 - 5/9) { /2 n \ { 9 GAMMA(n/9 + 1) irem(n, 9) = 5 { |--- - 4/3| { { \ 9 / { 0 irem(n, 9) = 6 { 3 GAMMA(n/9 + 1) irem(n, 9) = 6 { { { 0 irem(n, 9) = 7 { 0 irem(n, 9) = 7 { { { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 0 { 0 irem(n, 9) = 0 { { { 0 irem(n, 9) = 1 { 0 irem(n, 9) = 1 { { { 0 irem(n, 9) = 2 { 0 irem(n, 9) = 2 { { { 0 irem(n, 9) = 3 { /2 n \ { { |--- - 2/3| { /2 n \ { \ 9 / { |--- - 8/9| , { 3 GAMMA(n/9 + 1) irem(n, 9) = 3, { \ 9 / { { 3 GAMMA(n/9 + 1) irem(n, 9) = 4 { 0 irem(n, 9) = 4 { { { 0 irem(n, 9) = 5 { 0 irem(n, 9) = 5 { { { 0 irem(n, 9) = 6 { 0 irem(n, 9) = 6 { { { 0 irem(n, 9) = 7 { 0 irem(n, 9) = 7 { { { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 0 { 0 irem(n, 9) = 0 { /2 n\ { { { |---| { 0 irem(n, 9) = 1 { /2 n \ { \ 9 / { { |--- - 2/9| { 3 (n/9)! irem(n, 9) = 0 { /2 n \ { \ 9 / { { |--- - 4/9| { 3 GAMMA(n/9 + 1) irem(n, 9) = 1 { 0 irem(n, 9) = 1 { \ 9 / { { { 3 GAMMA(n/9 + 1) irem(n, 9) = 2 { 0 irem(n, 9) = 2 { 0 irem(n, 9) = 2 { { { { 0 irem(n, 9) = 3, { 0 irem(n, 9) = 3, { 0 irem(n, 9) = 3} { { { { 0 irem(n, 9) = 4 { 0 irem(n, 9) = 4 { 0 irem(n, 9) = 4 { { { { 0 irem(n, 9) = 5 { 0 irem(n, 9) = 5 { 0 irem(n, 9) = 5 { { { { 0 irem(n, 9) = 6 { 0 irem(n, 9) = 6 { 0 irem(n, 9) = 6 { { { { 0 irem(n, 9) = 7 { 0 irem(n, 9) = 7 { 0 irem(n, 9) = 7 { { { { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 8 { 0 irem(n, 9) = 8 "A114851" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A114870" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A114938" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselI(n + 1/2, 1), (-1) n! BesselK(n + 1/2, -1)} "A114997" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A115081" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A115082" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A115112" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 1 "A115128" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (13 n1 + 75 n1 + 110) {1, n, ) ---------------------------------------------------------------} / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A115136" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {1, (-1) , (-1) | ) |- ----------------------------------------||} | / \ (n1 + 2) (n1 + 3) /| |----- | \n1 = 0 / "A115137" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 n1 binomial(2 n1, n1)|| {(-1) , (-1) | ) |- --------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A115138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (n1 - 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- --------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A115150" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 2) (n1 + 3) (n1 + 4) /| |----- | \n1 = 0 / "A115151" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 3) (n1 + 4) (n1 + 5) /| |----- | \n1 = 0 / "A115152" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- --------------------------------------------------------------||} | / \ (n1 + 3) (n1 + 4) (n1 + 5) (n1 + 6) /| |----- | \n1 = 0 / "A115153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- --------------------------------------------------------------||} | / \ (n1 + 4) (n1 + 5) (n1 + 6) (n1 + 7) /| |----- | \n1 = 0 / "A115178" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" memory used=54921.5MB, alloc=1879.5MB, time=382.02 LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A115187" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) binomial(2 n1, n1)|| {1, (-1/2) , (-1/2) | ) |- ------------------------------------------||} | / \ (n1 + 2) (n1 + 3) /| |----- | \n1 = 0 / "A115188" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- ------------------------------------------||} | / \ (n1 + 2) (n1 + 3) /| |----- | \n1 = 0 / "A115189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- -----------------------------------------------------||} | / \ (n1 + 2) (n1 + 3) (n1 + 4) /| |----- | \n1 = 0 / "A115190" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- -----------------------------------------------------||} | / \ (n1 + 3) (n1 + 4) (n1 + 5) /| |----- | \n1 = 0 / "A115191" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- ----------------------------------------------------------------||} | / \ (n1 + 3) (n1 + 4) (n1 + 5) (n1 + 6) /| |----- | \n1 = 0 / "A115192" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- ----------------------------------------------------------------||} | / \ (n1 + 4) (n1 + 5) (n1 + 6) (n1 + 7) /| |----- | \n1 = 0 / "A115194" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (5 n1 + 8) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- --------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) /| |----- | \n1 = 0 / "A115197" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) (9 n + 5), (-1) (9 n + 5) | ) |- ----------------------------------------||} | / \ (n1 + 1) (9 n1 + 5) (9 n1 + 14) /| |----- | \n1 = 0 / "A115202" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (13 n1 + 25) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 3) (n1 + 4) (n1 + 5) /| |----- | \n1 = 0 / "A115203" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) (4 n1 + 9) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -------------------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 3) (n1 + 4) (n1 + 5) (n1 + 6) /| |----- | \n1 = 0 / "A115204" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) (19 n1 + 49) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 4) (n1 + 5) (n1 + 6) (n1 + 7) /| |----- | \n1 = 0 / "A115246" (2 n + 1) binomial(2 n, n) (3 n + 1) (3 n + 2) binomial(3 n, n) {1, --------------------------, ------------------------------------} n + 1 (n + 1) (2 n + 1) "A115256" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A115257" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 1) binomial(2 n1, n1) {1, ) -------------------------------} / 2 ----- (n1 + 1) n1 = 0 "A115326" LREtools/SolveLRE: "Reduced the order of" E^3+(-2*n-5)*E^2-2*(2*n+5)*(n+2)*E+8*(n+2)*(n+1)^2 "to two: Symmetric square" E^2-E-2*n-2 LREtools/SearchTable: "SearchTable successful" n 2 {(-1) HermiteH(n, 1/2 I) } "A115327" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 6 ) HermiteH(n, 1/6 I 6 )} "A115329" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-I 2 ) HermiteH(n, 1/4 I 2 )} "A115331" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 10 ) HermiteH(n, 1/10 I 10 )} "A115344" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A115399" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A115425" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { (- n/2) { 2 (3 n + 6) binomial(n/2, n/4) (n/4)! irem(n, 4) = 0 { (n/4 + 1) (n/4)! irem(n, 4) = 0 { { { (- n/2 + 1/2) { 2 (n/4 + 3/4)! irem(n, 4) = 1 { 2 (4 n + 4) binomial(n/2 - 1/2, n/4 - 1/4) (n/4 - 1/4)! irem(n, 4) = 1 {{ , { , { 3 (n/4 + 1/2)! irem(n, 4) = 2 { (- n/2 + 1) { { 1/4 2 n (n + 4) binomial(n/2 - 1, n/4 - 1/2) (n/4 - 1/2)! irem(n, 4) = 2 { 4 (n/4 + 1/4)! irem(n, 4) = 3 { { (- n/2 - 1/2) { 2 (2 n + 6) binomial(n/2 + 1/2, n/4 + 1/4) (n/4 + 1/4)! irem(n, 4) = 3 { 16 GAMMA(n/4 + 9/4) { 8 GAMMA(n/4 + 11/4) { ------------------- irem(n, 4) = 0 { ------------------- irem(n, 4) = 0 { n + 5 { n + 7 { { { GAMMA(n/4 + 2) irem(n, 4) = 1 { 12 GAMMA(n/4 + 5/2) { { ------------------- irem(n, 4) = 1 { 8 GAMMA(n/4 + 11/4) , { n + 6 } { ------------------- irem(n, 4) = 2 { { n + 7 { 16 GAMMA(n/4 + 9/4) { { ------------------- irem(n, 4) = 2 { 12 GAMMA(n/4 + 5/2) { n + 5 { ------------------- irem(n, 4) = 3 { { n + 6 { GAMMA(n/4 + 2) irem(n, 4) = 3 "A115752" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 n n { {3 , (-1) hypergeom([1/2, -n - 1], [1], 4), { 2 n , { 9 binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) (n - 1) n { 3 { --------------------------------------------------------------------- irem(n, 3) = 2 { 2 { (n + 1) { 0 irem(n, 3) = 0 { n { { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { (n - 1) { ---------------------------------- irem(n, 3) = 0 { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 2 { ---------------------------------------- irem(n, 3) = 1, { GAMMA(n/3 + 4/3) } { 2 { { GAMMA(n/3 + 4/3) { 0 irem(n, 3) = 1 { { { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A115864" LREtools/SearchTable: "SearchTable successful" n n {4 LegendreP(n, 2), 4 LegendreQ(n, 2)} "A115865" LREtools/SearchTable: "SearchTable successful" n n {6 LegendreP(n, 2), 6 LegendreQ(n, 2)} "A115866" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" /n - 1 \ |----- | n n | \ 1 | {(-2) , (-2) | ) ------------------------------------------|} | / (n1 + 1)| |----- (n1 + 1/2) binomial(2 n1, n1) (-2) | \n1 = 0 / "A115962" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A115967" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / |n - 1 / 1/2\n / 1/2\n / 1/2\n |----- / 1/2\n1 / 1/2\(-n1 - 1) | 2 13 | | 2 13 | | 2 13 | | \ | 2 13 | | 2 13 | {|4/3 - -------| , |4/3 + -------| , |4/3 - -------| | ) |4/3 + -------| |4/3 - -------| \ 3 / \ 3 / \ 3 / | / \ 3 / \ 3 / |----- \n1 = 0 / / 1/2\(-n2 - 1) \\ |n1 - 1 n2 | 2 13 | || |----- (-1) |4/3 + -------| (hypergeom([1/2, -n2 - 1], [1], 4) - 3 hypergeom([1/2, -n2], [1], 4))|| | \ \ 3 / || | ) -----------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A115968" memory used=55703.7MB, alloc=2007.5MB, time=386.58 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 5, 6, 7 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := 3 _Z - 12 _Z - 10 _Z + 12 _Z - 9 "A115969" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n | 2 33 | | 2 33 | {|4 - -------| , |4 + -------| , \ 3 / \ 3 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 2 33 | ||| / 1/2\n |----- | |----- |4 + -------| (3 LegendreP(n2 + 1, 3) - LegendreP(n2, 3))||| | 2 33 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 3 / ||| |4 - -------| | ) |3/2 (6 + 33 ) (6 - 33 ) | ) ------------------------------------------------------------------|||, \ 3 / | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 2 33 | ||| / 1/2\n |----- | |----- |4 + -------| (3 LegendreQ(n2 + 1, 3) - LegendreQ(n2, 3))||| | 2 33 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 3 / ||| |4 - -------| | ) |3/2 (6 + 33 ) (6 - 33 ) | ) ------------------------------------------------------------------|||} \ 3 / | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A115970" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(64/7) , (64/7) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (64/7) | \n1 = 0 / "A116078" n {4 , binomial(2 n, n) (n + 3)} "A116091" LREtools/SearchTable: "SearchTable successful" n n {(-4) LegendreP(n, 1/2), (-4) LegendreQ(n, 1/2)} "A116092" LREtools/SearchTable: "SearchTable successful" n n {(-8) LegendreP(n, 1/2), (-8) LegendreQ(n, 1/2)} "A116093" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 1/2 n 3 1/2 n 3 {(-2 3 ) LegendreP(n, ----), (-2 3 ) LegendreQ(n, ----)} 3 3 "A116218" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {-(-2 2 ) n! (n + 1) ((2 n + 1) 2 BesselI(n + 1/2, 2 ) - 2 BesselI(n - 1/2, 2 )), 1/2 n 1/2 1/2 1/2 -(-2 2 ) n! (n + 1) ((2 n + 1) 2 BesselK(n + 1/2, -2 ) - 2 BesselK(n - 1/2, -2 ))} "A116219" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A116220" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A116221" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A116363" LREtools/SearchTable: "SearchTable successful" {LegendreP(n + 1, 3) - 7 LegendreP(n, 3), LegendreQ(n + 1, 3) - 7 LegendreQ(n, 3)} "A116364" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 9 n + 13 {------------------------------------, (2 n + 3) (2 n + 1) binomial(2 n, n) /n - 1 \ |----- 2 2 2 | | \ (2 n1 + 1) binomial(2 n1, n1) (105 n1 + 319 n1 + 238) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1)| (9 n + 13) | ) ----------------------------------------------------------------------------------------------| | / 2 2 | |----- (n1 + 1) (n1 + 2) (9 n1 + 22) (9 n1 + 13) | \n1 = 0 / ------------------------------------------------------------------------------------------------------------------} (2 n + 3) (2 n + 1) binomial(2 n, n) "A116383" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { n2 \\\ | | | { 4 ||| | | | { --------------------------- n2::even||| | | | { n2 ||| | | | { (n2 + 1) binomial(n2, ----) ||| | | | { 2 ||| | | | { ||| | | | { (2 n2 - 2) ||| | | | { 4 2 ||| | | | { - ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) ------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { 8 binomial(n2, ----) ||| | | | { 2 ||| | | | { - -------------------- n2::even||| | | | { n2 + 2 ||| | | | { ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { 2 binomial(n2 + 1, ---- + 1/2) n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { 2 ||| (5 ) | ) |1/5 5 (-1) | ) ------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A116384" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /{ 0 n1::even\\\ |----- | |{ ||| n n n | \ | n1 |{ n1 ||| {(-1) , 2 , (-1) | ) |-(-1) |{ 2 binomial(n1 - 1, ---- - 1/2) n1 |||, | / | |{ 2 ||| |----- | |{ --------------------------------- n1::odd ||| \n1 = 0 \ \{ n1 + 1 /// /n - 1 / /{ n1 \\\ |----- | |{ 4 ||| n | \ | n1 |{ --------------------------- n1::even||| (-1) | ) |-(-1) |{ n1 |||} | / | |{ (n1 + 1) binomial(n1, ----) ||| |----- | |{ 2 ||| |n1 = 0 | |{ ||| \ \ \{ 0 n1::odd /// "A116385" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 4 binomial(n, n/2) (n + 1) n { 1/2 -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n - 1) { 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------------ n::odd { n + 3 { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A116387" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1 - 5 ) , (5 + 1) , (1 - 5 ) | ) (5 + 1) (1 - 5 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) || | \ (-1) (5 + 1) ((n2 + 1) hypergeom([1/2, -n2 - 1], [1], 4) + (n2 + 5) hypergeom([1/2, -n2], [1], 4))|| | ) ----------------------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A116388" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 3 _Z - 1, index = 1) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 2) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 3) } "A116390" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / / 1/2\n / 1/2\n / 1/2\n |----- | | 2 13 | | 2 13 | | 2 13 | | \ | {|1/3 - -------| , |1/3 + -------| , |1/3 - -------| | ) | \ 3 / \ 3 / \ 3 / | / | |----- | |n1 = 0 | \ \ /n1 - 1 /{ 0 n2::even\\\ |----- / 1/2\(-n2 - 1) |{ ||| n1 1/2 (-n1 - 1) 1/2 n1 | \ | 2 13 | |{ (2 n2 - 2) ||| -3 (-1) (-1 + 2 13 ) (1 + 2 13 ) | ) |1/3 + -------| |{ 2 ||| | / \ 3 / |{ ---------------------------------------- n2::odd ||| |----- |{ n2 ||| |n2 = 0 |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| \ \{ 2 /// \ | / 1/2\n | | 2 13 | |, |1/3 - -------| | \ 3 / | | / /n - 1 / /n1 - 1 /{ n2 \\\\ |----- | |----- / 1/2\(-n2 - 1) |{ 2 binomial(n2, ----) |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ | 2 13 | |{ 2 |||| | ) |-3 (-1) (-1 + 2 13 ) (1 + 2 13 ) | ) |1/3 + -------| |{ -------------------- n2::even||||} | / | | / \ 3 / |{ n2 + 2 |||| |----- | |----- |{ |||| \n1 = 0 \ \n2 = 0 \{ 0 n2::odd //// "A116391" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n 1/2 n 1/2 n {(-1) , (5 ) , (-5 ) , /n - 1 / /n1 - 1 / / n2 \ /n2 - 1 / /{ 0 n3::even\\\\\\\ |----- | |----- | |- ---- - 1/2| |----- | |{ ||||||| 1/2 n | \ | 1/2 n1 | \ | \ 2 / | \ | n3 |{ n3 ||||||| (5 ) | ) |1/5 5 (-1) | ) |-5 | ) |-(-1) |{ 4 binomial(n3 - 1, ---- - 1/2) n3 |||||||, | / | | / | | / | |{ 2 ||||||| |----- | |----- | |----- | |{ --------------------------------- n3::odd ||||||| \n1 = 0 \ \n2 = 0 \ \n3 = 0 \ \{ (n3 + 1) (n3 + 3) /////// /n - 1 / /n1 - 1 / / n2 \ /n2 - 1 / /{ n3 \\\\\\\ |----- | |----- | |- ---- - 1/2| |----- | |{ 4 ||||||| 1/2 n | \ | 1/2 n1 | \ | \ 2 / | \ | n3 |{ ------------------------------------ n3::even||||||| (5 ) | ) |1/5 5 (-1) | ) |-5 | ) |-(-1) |{ n3 |||||||} | / | | / | | / | |{ (n3 + 1) (n3 + 3) binomial(n3, ----) ||||||| |----- | |----- | |----- | |{ 2 ||||||| |n1 = 0 | |n2 = 0 | |n3 = 0 | |{ ||||||| \ \ \ \ \ \ \{ 0 n3::odd /////// "A116394" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 7 _Z - 8 _Z - 3 "A116396" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - 4 _Z - 1, index = 1) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 2) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 3) } "A116400" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 8 binomial(n, n/2) (n + 1) n { 1/2 -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n - 1) { 2 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------------ n::odd { n + 3 { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A116406" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { ------------------ n::even n { n binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n + 2) { binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A116408" LREtools/SearchTable: "SearchTable successful" n 2 3 2 (-1) ((n + 1) (n + 9 n + 6) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 30 n + 55 n + 6) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A116409" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n \ n1 {1, 3 , ) (-1) hypergeom([1/2, -n1 - 1], [1], 4)} / ----- n1 = 0 "A116410" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) hypergeom([1/2, -n], [1], 4)} "A116411" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A116447" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (-n) {n, (-1) n, { , { 2 binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2)! n::odd { { 0 n::odd "A116466" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { n2 \ \ | | { 4 4 | | | | { --------------------- n2::even| | | | { n2 | | | | { n2 binomial(n2, ----) | | | | { 2 | | | | { | | | | { (2 n2 + 2) | | | | { 2 2 | | | | { ------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { (n2 + 1) binomial(n2 + 1, ---- + 1/2) | | n 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { 2 | | {1, (-1) , (-1/2 I 2 ) , (1/2 I 2 ) , (-1/2 I 2 ) | ) (-1) 2 | ) -------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { binomial(n2, ----) n2::even| | | | { 2 | | | | { | | |n - 1 |n1 - 1 { n2 | | |----- |----- { 2 binomial(n2 - 1, ---- - 1/2) n2::odd | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) ------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A116467" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 binomial(2 n1, n1) | {1, (-1/2) (9 n - 13), (-1/2) (9 n - 13) | ) -------------------------------------------------|} | / (n1 + 1) | |----- (2 n1 - 1) (-1/2) (9 n1 - 4) (18 n1 - 26)| \n1 = 0 / "A116637" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { 3 binomial(---, n/2) { 2 { -------------------- n::even { n + 1 {{ , { 3 n { 4 binomial(--- + 3/2, n/2 + 1/2) { 2 { -------------------------------- n::odd { 3 n + 1 { (-n) 3 n 3 n { 8 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { (-2 n + 2) 3 n 3 n { 6 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { n (n + 1) binomial(n - 1, n/2 - 1/2) "A116723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- | ----- \ | \ (n2 + 2) (n2 - 1)| \ {1, ) n1! | ) -----------------|, ) n1!} / | / (n2 + 1)! | / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A116862" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (11 n1 + 15) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- -------------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (n1 + 3) /| |----- | \n1 = 0 / "A116867" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) binomial(2 n1, n1)|| {(-1/2) (25 n + 13), (-1/2) (25 n + 13) | ) |- ------------------------------------------||} | / \ (n1 + 1) (25 n1 + 38) (50 n1 + 26) /| |----- | \n1 = 0 / "A116873" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | (-n1 - 1) | \ | (-1) 6 binomial(2 n2, n2)|||| {(-3) , (-1) , (-3) | ) |-3 | ) |- -----------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116874" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | (-n1) | \ | (-1) 12 binomial(2 n2, n2)|||| {(-4) , (-2/3) , (-4) | ) |-1/4 6 | ) |-3/2 ------------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116875" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | n1 | \ | (-1) 2 binomial(2 n2, n2)|||| {(-5) , (-1/2) , (-1/2) | ) |-2 10 | ) |-1/5 -----------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116876" memory used=56381.9MB, alloc=1975.5MB, time=391.40 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | n1 | \ | (-1) 2 binomial(2 n2, n2)|||| {(-6) , (-2/5) , (-2/5) | ) |-5/2 15 | ) |-1/6 -----------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116877" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | n1 | \ | (-1) 2 binomial(2 n2, n2)|||| {(-7) , (-1/3) , (-1/3) | ) |-3 21 | ) |-1/7 -----------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116878" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | n1 | \ | (-1) 2 binomial(2 n2, n2)|||| {(-8) , (-2/7) , (-2/7) | ) |-7/2 28 | ) |-1/8 -----------------------------||||} | / | | / \ n2 + 1 /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A116879" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (3 n1 + 5) binomial(2 n1, n1)|| {(-1) (3 n + 7), (-1) (3 n + 7) | ) |- --------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 3) (3 n1 + 7) (3 n1 + 10) /| |----- | \n1 = 0 / "A116881" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (3 n1 + 2) binomial(2 n1, n1)|| {(-1) (3 n + 1), (-1) (3 n + 1) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 1) (3 n1 + 4) /| |----- | \n1 = 0 / "A116914" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 5) | {(-1/2) (3 n + 4), (-1/2) (3 n + 4) | ) ------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) (3 n1 + 7) (6 n1 + 8)| \n1 = 0 / "A117106" LREtools/SearchTable: "SearchTable successful" 3 2 {(2 (2 n + 1) (55 n + 215 n + 200 n - 2) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) 3 2 / 2 2 2 + (n + 1) (5 n + 15 n + 10 n + 6) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) / (n (n + 1) (n + 2) (n + 3) (n + 4) (n + 5))} / "A117186" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A117187" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 8 binomial(n, n/2) (n + 1) { 1/2 ------------------------ n::even { -------------------------- n::even { (n + 3) binomial(n, n/2) { n + 4 {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 2 (n + 1) { ------------------------------------ n::odd { ------------------------------------ n::odd { n + 3 { n (n + 4) binomial(n - 1, n/2 - 1/2) "A117376" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 n 3 2 n 3 2 n {(9/2) , RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A117397" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A117399" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1! (n1 + 3) n1} / ----- n1 = 0 "A117641" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (4 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-4 n1 - 3) hypergeom([-1/2, -n1], [1], -4)| {(-1/3) , (-1/3) | ) --------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1/3) | \n1 = 0 / "A117643" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A117813" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A117826" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { (- n/2) { 2 (3 n + 6) binomial(n/2, n/4) (n/4)! irem(n, 4) = 0 { (n/4 + 1) (n/4)! irem(n, 4) = 0 { { { (- n/2 + 1/2) { 2 (n/4 + 3/4)! irem(n, 4) = 1 { 2 (4 n + 4) binomial(n/2 - 1/2, n/4 - 1/4) (n/4 - 1/4)! irem(n, 4) = 1 {{ , { , { 3 (n/4 + 1/2)! irem(n, 4) = 2 { (- n/2 + 1) { { 1/4 2 n (n + 4) binomial(n/2 - 1, n/4 - 1/2) (n/4 - 1/2)! irem(n, 4) = 2 { 4 (n/4 + 1/4)! irem(n, 4) = 3 { { (- n/2 - 1/2) { 2 (2 n + 6) binomial(n/2 + 1/2, n/4 + 1/4) (n/4 + 1/4)! irem(n, 4) = 3 { 16 GAMMA(n/4 + 9/4) { 8 GAMMA(n/4 + 11/4) { ------------------- irem(n, 4) = 0 { ------------------- irem(n, 4) = 0 { n + 5 { n + 7 { { { GAMMA(n/4 + 2) irem(n, 4) = 1 { 12 GAMMA(n/4 + 5/2) { { ------------------- irem(n, 4) = 1 { 8 GAMMA(n/4 + 11/4) , { n + 6 } { ------------------- irem(n, 4) = 2 { { n + 7 { 16 GAMMA(n/4 + 9/4) { { ------------------- irem(n, 4) = 2 { 12 GAMMA(n/4 + 5/2) { n + 5 { ------------------- irem(n, 4) = 3 { { n + 6 { GAMMA(n/4 + 2) irem(n, 4) = 3 "A118093" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n n | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) (5 n1 + 16) binomial(2 n1, n1)|| {(-1) , 8 , (-1) | ) |- --------------------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) /| |----- | \n1 = 0 / "A118346" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A118376" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /3 n1\ /3 n1\ n - 1 |----| n - 1 |----| ----- \ 2 / 1/2 1/2 1/2 ----- \ 2 / 1/2 1/2 1/2 \ 2 (2 LegendreP(n1 + 1, 2 ) - LegendreP(n1, 2 )) \ 2 (2 LegendreQ(n1 + 1, 2 ) - LegendreQ(n1, 2 )) {1, ) ------------------------------------------------------------, ) ------------------------------------------------------------} / n1 + 2 / n1 + 2 ----- ----- n1 = 0 n1 = 0 "A118395" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A118447" n (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 2) {4 (n + 2), --------------------------------------------} n + 1 "A118448" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (7 n + 19) {(n + 2) (n + 3) 4 (8 n + 17), ---------------------------------------------------------} n + 1 "A118451" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" /n - 1 \ |----- n1 | n n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) 3 binomial(2 n1, n1) (23 n1 + 89)| {(-4) (n + 3), 12 (n + 3), (-4) (n + 3) | ) --------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-4) (n1 + 4) (n1 + 3) | \n1 = 0 / "A118589" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A118650" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A118802" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {n - 1, (n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-n + 3) hypergeom([-1/2, -n], [1], -8)} "A118934" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A118973" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(2 n1, n1) (5 n1 + 9 n1 + 2) | {(-1/2) , (-1/2) | ) -----------------------------------------|} | / (n1 + 1)| |----- (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | \n1 = 0 / "A118974" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ | ----- 2 | | \ binomial(2 n2, n2) (17 n2 + 37 n2 + 18)| | ) ----------------------------------------| |n - 1 / (n2 + 3) (n2 + 2) (n2 + 1) | |----- ----- | n n | \ n2 = 0 | {1, (-1/2) , (-1/2) | ) -----------------------------------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A119012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) n1 binomial(2 n1, n1) (3 n1 + 7) | {(-1/2) (3 n + 2), (-1/2) (3 n + 2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 4) (n1 + 3) (n1 + 2) (-1/2) (3 n1 + 5) (6 n1 + 4)| \n1 = 0 / "A119259" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (3 n1 + 2) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A119358" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { (n/2) {binomial(2 n, n), { (n/2 - 1/2) , { (-1) binomial(n, n/2) n::even} { (-16) { { ---------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A119363" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), (- 1/2 + 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), binomial(2 n, n)} "A119370" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A119371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-2/3) } "A119372" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-2/3) } "A119400" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A119692" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| binomial(2 n, n), |---- + 1/2| binomial(2 n, n)} \ 2 / \ 2 / "A119693" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| binomial(2 n, n), |---- + 1/2| binomial(2 n, n)} \ 2 / \ 2 / "A119694" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | |1/2 - ----| binomial(2 n, n) |---- + 1/2| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A119697" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | |1/2 - ----| binomial(2 n, n) n |---- + 1/2| binomial(2 n, n) n \ 2 / \ 2 / {--------------------------------, --------------------------------} n + 1 n + 1 "A119701" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | {n |1/2 - ----| binomial(2 n, n), n |---- + 1/2| binomial(2 n, n)} \ 2 / \ 2 / "A119702" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n 2 | 5 | 2 |5 | {n |1/2 - ----| binomial(2 n, n), n |---- + 1/2| binomial(2 n, n)} \ 2 / \ 2 / "A119703" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n 3 | 5 | 3 |5 | {n |1/2 - ----| binomial(2 n, n), n |---- + 1/2| binomial(2 n, n)} \ 2 / \ 2 / "A119913" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ n2 + 1 | {1, ) (n1 + 1) n1!, ) (n1 + 1) n1! | ) ---------|} / / | / (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A119967" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A119975" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 7 ) 7 (7 LegendreP(n + 1, 1/7 I 7 ) I - 3 LegendreP(n, 1/7 I 7 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreQ(n + 1, 1/7 I 7 ) I - 3 LegendreQ(n, 1/7 I 7 ))} "A119976" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) (-LegendreP(n, I) + LegendreP(n + 1, I) I), (-2 I) (-LegendreQ(n, I) + LegendreQ(n + 1, I) I)} "A120010" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A120012" memory used=57073.8MB, alloc=1975.5MB, time=396.16 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (n1 - 1) | {(9/2) (n - 4), (9/2) (n - 4) | ) ---------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (9/2) (n1 - 3) (2 n1 - 8)| \n1 = 0 / "A120017" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A120018" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A120278" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {1, (3 n + 8) | ) ---------------------------------------------------|, 3 n + 8} | / (n1 + 1) (n1 + 2) (3 n1 + 11) (3 n1 + 8) | |----- | \n1 = 0 / "A120279" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, n, ) ---------------------------------------------------} / (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A120304" binomial(2 n, n) {1, ----------------} n + 1 "A120305" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {1, (-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A120445" n 2 {n! (2 n + 3), (2 n + 1) (1/2) (n!) binomial(2 n, n)} "A120590" LREtools/SearchTable: "SearchTable successful" / 1/2\n |81 48 3 | {|-- - -------| \13 13 / / 1/2 1/2 \ | 512 288 3 1/2 512 288 3 | |(39 n + 52) hypergeom([5/6, - 2/3 - n], [5/3], --- + --------) + 27 (27 + 16 3 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], --- + --------)| \ 13 13 13 13 / / 1/2\ | 499 288 3 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- --- + --------|/(GAMMA(n + 1) GAMMA(n + 2/3))} \ 13 13 / "A120591" LREtools/SearchTable: "SearchTable successful" / 1/2\n |81 48 3 | {|-- - -------| \13 13 / / 1/2 1/2 \ | 512 288 3 1/2 512 288 3 | |(39 n + 52) hypergeom([5/6, - 2/3 - n], [5/3], --- + --------) + 27 (27 + 16 3 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], --- + --------)| \ 13 13 13 13 / / 1/2\ | 499 288 3 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- --- + --------|/(GAMMA(n + 1) GAMMA(n + 2/3))} \ 13 13 / "A120592" LREtools/SearchTable: "SearchTable successful" / 1/2\n |108 30 15 | {|--- - --------| \17 17 / / 1/2 1/2 \ | 250 60 15 1/2 250 60 15 | |(51 n + 68) hypergeom([5/6, - 2/3 - n], [5/3], --- + --------) + 18 (18 + 5 15 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], --- + --------)| \ 17 17 17 17 / / 1/2\ | 233 60 15 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- --- + --------|/(GAMMA(n + 1) GAMMA(n + 2/3))} \ 17 17 / "A120672" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {1, 2 , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A120765" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \ \\ |----- | |----- | |----- n2 (-n2 - 1)| || n n | \ n1 n1 | n | \ | n1 n1 | \ (-1) 2 | || {(-1) , (-1) | ) (-(-1) 2 n1!)|, (-1) | ) |-(-1) 2 | ) -----------------| n1!||} | / | | / | | / (n2 + 1)! | || |----- | |----- | |----- | || \n1 = 0 / \n1 = 0 \ \n2 = 0 / // "A120984" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A120985" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121079" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {2 n!, 4 n! LaguerreL(n, -1/4)} "A121080" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {2 n!, 4 n!, 4 n! LaguerreL(n, -1/4)} "A121285" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (3 n - 1) { ------------------ n::even { 3 n binomial(n, n/2) n::even n { n binomial(n, n/2) { {2 (n + 3), { , { binomial(n + 1, n/2 + 1/2) (n + 1) (3 n - 1) } { (2 n - 2) { 1/2 -------------------------------------------- n::odd { 6 2 { n { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A121320" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ / 1/2\n / 1/2 \n / 1/2\n |----- | |----- / 1/2 \(-n2 - 1) ||| | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | ||| {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| binomial(2 n2, n2)|||} \ 2 / \ 2 / \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A121323" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 1) BesselJ(n + 1/2, -1) + BesselJ(n - 1/2, -1)), (-1) ((2 n + 1) BesselY(n + 1/2, -1) + BesselY(n - 1/2, -1))} "A121351" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((3 n + 1) BesselJ(n + 1/3, -2/3) + BesselJ(n - 2/3, -2/3)), (-1) ((3 n + 1) BesselY(n + 1/3, -2/3) + BesselY(n - 2/3, -2/3))} "A121353" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n + 1/3, -2/3), (-1) BesselY(n + 1/3, -2/3)} "A121354" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((3 n - 1) BesselJ(n - 1/3, -2/3) + BesselJ(n - 4/3, -2/3)), (-1) ((3 n - 1) BesselY(n - 1/3, -2/3) + BesselY(n - 4/3, -2/3))} "A121498" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ (-1) 29 (2 n1 + 1) binomial(2 n1, n1)| {841 , 841 | ) --------------------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A121545" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(3 n1, n1) (7 n1 + 3) | {(-1/4) , (-1/4) | ) ----------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (2 n1 + 1) (-1/4) | \n1 = 0 / "A121553" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) n1! (n1 + 3 n1 + 4)| {(n + 1) n!, (n + 1) n! | ) -----------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A121555" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A121580" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) (n1 + 1) n1! (n1 + 3 n1 + 4)} / ----- n1 = 0 "A121582" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 2 | \ (n1 + 1) n1 n1! (n1 + 2 n1 - 1)| 2 {1, (n + n - 1) | ) ---------------------------------|, n + n - 1} | / 2 2 | |----- ((n1 + 1) + n1) (n1 + n1 - 1) | \n1 = 0 / "A121586" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) n1! | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A121629" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121633" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | | \ 1 | | \ (n1 + 1) n1! | {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A121636" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 2)| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A121638" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121696" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ | n | \ n1 | {(-1) , (-1) | ) (-(n1 + 1) n1!)|, (-1) | ) (-(-1) (n1 + 1) n1! (2 n1 + 5))|} | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / "A121723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 1) n! n, (n + 1) n! n | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A121724" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 /{ 0 n1::even\\ |----- |{ || n n | \ (-n1 - 1) |{ (3 n1 - 3) || {5 , 5 | ) 5 |{ 2 ||, | / |{ ---------------------------------------- n1::odd || |----- |{ n1 || |n1 = 0 |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // /n - 1 /{ n1 n1 \\ |----- |{ 2 2 binomial(n1, ----) || n | \ (-n1 - 1) |{ 2 || 5 | ) 5 |{ ------------------------ n1::even||} | / |{ n1 + 2 || |----- |{ || \n1 = 0 \{ 0 n1::odd // "A121725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 /{ 0 n1::even\\ |----- |{ || n n | \ (-n1 - 1) |{ (n1 - 1) || {10 , 10 | ) 10 |{ 12 ||, | / |{ ---------------------------------------- n1::odd || |----- |{ n1 || |n1 = 0 |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // /n - 1 /{ n1 n1 \\ |----- |{ 2 3 binomial(n1, ----) || n | \ (-n1 - 1) |{ 2 || 10 | ) 10 |{ ------------------------ n1::even||} | / |{ n1 + 2 || |----- |{ || \n1 = 0 \{ 0 n1::odd // "A121726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ 1 | {1, ) (n1 + 1) n1!, ) (n1 + 1) n1! | ) ---------|} / / | / (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A121735" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(n + 1) (- 1/2 - 1/2 I 3 ) n!, (n + 1) (- 1/2 + 1/2 I 3 ) n!} "A121746" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /{ // n1 \ \2 2 \\\ |----- | |{ 1/4 ||----|!| (n1 + 2) n1::even||| n n | \ | n1 |{ \\ 2 / / ||| {(-1) , (-1) | ) |-(-1) |{ |||, | / | |{ / n1 \ // n1 \ \2 ||| |----- | |{ |---- + 3/2| ||---- + 1/2|!| n1::odd ||| \n1 = 0 \ \{ \ 2 / \\ 2 / / /// /n - 1 / /{ (-n1) 2 n1 2 // n1 \ \2 \\\ |----- | |{ 2 4 (n1 + 1) binomial(n1, ----) ||----|!| (n1 + 3) n1::even||| n | \ | n1 |{ 2 \\ 2 / / ||| (-1) | ) |-(-1) |{ |||} | / | |{ (-2 n1 + 2) 2 2 n1 2 // n1 \ \2 ||| |----- | |{ 2 n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) ||---- - 1/2|!| n1::odd ||| \n1 = 0 \ \{ 2 \\ 2 / / /// "A121749" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1) (n/2)! ((n + 4) n BesselJ(n/2 + 1, -2) + (2 n + 4) BesselJ(n/2, -2)) n::even {{ , { (n/2 - 1/2) 2 { 1/2 (-1) (n/2 - 1/2)! (n + 1) ((n + 4 n - 1) BesselJ(n/2 + 1/2, -2) + (2 n + 6) BesselJ(n/2 - 1/2, -2)) n::odd { (n/2) { (-1) (n/2)! ((n + 4) n BesselY(n/2 + 1, -2) + (2 n + 4) BesselY(n/2, -2)) n::even { , { (n/2 - 1/2) 2 { 1/2 (-1) (n/2 - 1/2)! (n + 1) ((n + 4 n - 1) BesselY(n/2 + 1/2, -2) + (2 n + 6) BesselY(n/2 - 1/2, -2)) n::odd { (n/2) 2 { (-1/4) binomial(n, n/2) (n/2)! (n + 1) ((n + 4 n - 1) BesselJ(n/2 + 1/2, -2) + (2 n + 6) BesselJ(n/2 - 1/2, -2)) n::even { , { (n/2 + 1/2) { (-1/4) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::odd { (n/2) 2 { (-1/4) binomial(n, n/2) (n/2)! (n + 1) ((n + 4 n - 1) BesselY(n/2 + 1/2, -2) + (2 n + 6) BesselY(n/2 - 1/2, -2)) n::even { } { (n/2 + 1/2) { (-1/4) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::odd "A121777" 5 4 3 2 n 5 4 3 2 (n - 2) (n - 1) n binomial(2 n, n) (2 n + 3 n - 40 n + 41 n + 72 n - 144) {4 (8 n + 68 n + 82 n + 127 n + 1371 n + 1068), -----------------------------------------------------------------------------} (2 n - 1) (2 n - 3) (2 n - 5) "A121778" 2 3 2 n 2 n binomial(2 n, n) (n - 3) (2 n + 9 n - 32 n + 18) {4 (2 n + 9 n + 6), -----------------------------------------------------} (2 n - 3) (2 n - 1) "A121873" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121908" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121953" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121956" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121958" LREtools/SearchTable: "SearchTable not successful" {} "A121959" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A121965" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -2), (-1) BesselY(n, -2)} "A121966" LREtools/SearchTable: "SearchTable successful" 1/2 (- n/2) 2 {2 HermiteH(n, ----)} 2 "A121987" memory used=57741.2MB, alloc=1975.5MB, time=400.81 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {n - 2} "A121988" LREtools/SearchTable: "SearchTable successful" ((6 n + 3) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) + (-10 n - 4) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------} (2 n - 1) (4 n - 1) "A121989" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | | \ (1/2 I 2 ) (HermiteH(n1 + 1, 1/2 I 2 ) - 2 (n1 + 1) HermiteH(n1, 1/2 I 2 ) I)| {(n - 1) | ) ----------------------------------------------------------------------------------------|, n - 1} | / n1 (n1 - 1) | |----- | \n1 = 0 / "A122017" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) {{ (-1/2) (hypergeom([1/3, - n/2 - 1], [3/2], 3) - hypergeom([1/3, - n/2], [3/2], 3)) binomial(n, n/2) (n/2)! (-n - 3) , n::even { (n/2 - 1/2) //3 n \ / 3 n \ \ -2 (-1/2) ||--- + 3| hypergeom([1/3, - n/2 - 1/2], [3/2], 3) + |- --- - 2| hypergeom([1/3, - n/2 + 1/2], [3/2], 3)| \\ 2 / \ 2 / / binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n , n::odd, { (n/2) //3 n \ / 3 n \ \ { -2 (-2) ||--- + 3| hypergeom([1/3, - n/2 - 1/2], [3/2], 3) + |- --- - 2| hypergeom([1/3, - n/2 + 1/2], [3/2], 3)| (n/2)! n::even { \\ 2 / \ 2 / / { } { (n/2 + 1/2) { (-2) (hypergeom([1/3, - n/2 - 1], [3/2], 3) - hypergeom([1/3, - n/2], [3/2], 3)) (n/2 + 1/2)! (n + 3) { - ---------------------------------------------------------------------------------------------------------------- n::odd { n + 1 "A122018" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) {{ (-1/2) (hypergeom([1/3, - n/2 - 1], [3/2], 3) - hypergeom([1/3, - n/2], [3/2], 3)) binomial(n, n/2) (n/2)! (-n - 3) , n::even { (n/2 - 1/2) //3 n \ / 3 n \ \ -2 (-1/2) ||--- + 3| hypergeom([1/3, - n/2 - 1/2], [3/2], 3) + |- --- - 2| hypergeom([1/3, - n/2 + 1/2], [3/2], 3)| \\ 2 / \ 2 / / binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n , n::odd, { (n/2) //3 n \ / 3 n \ \ { -2 (-2) ||--- + 3| hypergeom([1/3, - n/2 - 1/2], [3/2], 3) + |- --- - 2| hypergeom([1/3, - n/2 + 1/2], [3/2], 3)| (n/2)! n::even { \\ 2 / \ 2 / / { } { (n/2 + 1/2) { (-2) (hypergeom([1/3, - n/2 - 1], [3/2], 3) - hypergeom([1/3, - n/2], [3/2], 3)) (n/2 + 1/2)! (n + 3) { - ---------------------------------------------------------------------------------------------------------------- n::odd { n + 1 "A122021" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122022" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122031" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A122033" LREtools/SearchTable: "SearchTable successful" HermiteH(n + 1, 1) - 2 HermiteH(n, 1) {-------------------------------------} n "A122044" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122048" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122049" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122050" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122057" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 1) (n + 2) n!, (n + 1) (n + 2) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A122092" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1! | 2 {(n - 3 n + 1) | ) ---------------------------------------|, n - 3 n + 1} | / 2 2 | |----- ((n1 + 1) - 3 n1 - 2) (n1 - 3 n1 + 1)| \n1 = 0 / "A122122" n {4 (n + 3), binomial(2 n, n) (4 n + 1)} "A122441" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ |3 (-1) 6 binomial(2 n1, n1)|| {(1/3) , (1/3) | ) |-------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A122446" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-2) } "A122447" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-3/4) } "A122448" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-3/4) } "A122578" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1) HermiteH(n/2 + 1/2, 1/2) n::even { {{ (n/2 - 1/2) , { (-1) (HermiteH(n/2 + 1, 1/2) - HermiteH(n/2, 1/2)) { - ------------------------------------------------------------- n::odd { n { n { (-1) HermiteH(n/2 + 1/2, -1/2) I n::even { { (n/2 - 1/2) (n/2) (n/2 + 1) , { (-1) ((-1) HermiteH(n/2, -1/2) - (-1) HermiteH(n/2 + 1, -1/2)) { --------------------------------------------------------------------------------------- n::odd { n { (n/2) { 2 (-1) (HermiteH(n/2 + 1, 1/2) - HermiteH(n/2, 1/2)) { - --------------------------------------------------------- n::even { n , { { (n/2 + 1/2) { 2 (-1) HermiteH(n/2 + 1/2, 1/2) n::odd { (n/2) (n/2) (n/2 + 1) { 2 (-1) ((-1) HermiteH(n/2, -1/2) - (-1) HermiteH(n/2 + 1, -1/2)) { ----------------------------------------------------------------------------------- n::even { n } { { n { -2 (-1) HermiteH(n/2 + 1/2, -1/2) n::odd "A122598" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (-2) (n/2)! (-n + 6) n::even { {{ (n/2 + 1/2) 2 , { (-2) (n/2 + 1/2)! (n - 12 n + 19) { 1/2 --------------------------------------------- n::odd { n + 1 { (n/2) 2 { (-1/2) binomial(n, n/2) (n/2)! (n - 12 n + 19) n::even { } { (n/2 - 1/2) { -2 n (-1/2) (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) (n - 6) n::odd "A122648" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" n { 0 n::even { (n/2)! binomial(n, n/2) n::even {(-I) HermiteH(n, I), { , { } { (n - 1) { 0 n::odd { 2 (n/2 - 1/2)! n::odd "A122649" n n {2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A122680" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A122693" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 2 { n 2 { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even { 1/4 2 n ((n/2)!) n::even {{ , { } { (-n - 1) 2 2 { (n - 1) 2 2 { 2 n binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) n::odd { 1/4 2 (n + 1) ((n/2 - 1/2)!) n::odd "A122737" LREtools/SearchTable: "SearchTable successful" (4 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n "A122749" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 4 3 2 { (n/2) 2 2 { ((n/2)!) (3/8 n + 6 n + 69/2 n + 84 n + 75) n::even { (-1) ((n/2)!) (1/2 n + 4 n + 7) n::even { { {{ 2 4 3 2 , { (n/2 + 1/2) 2 2 , { ((n/2 + 1/2)!) (3 n + 44 n + 174 n + 244 n + 87) { (-1) ((n/2 + 1/2)!) (n + 6 n + 7) { 1/4 ---------------------------------------------------- n::odd { ---------------------------------------------- n::odd { n + 1 { n + 1 { (-n) 2 2 4 3 2 { 2 4 binomial(n, n/2) ((n/2)!) (n + 1) (3 n + 44 n + 174 n + 244 n + 87) n::even { , { (-2 n + 2) 2 2 2 4 3 2 { 3 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 16 n + 92 n + 224 n + 200) n::odd { /-1\(n/2) 2 2 2 { 2 |--| binomial(n, n/2) ((n/2)!) (n + 6 n + 7) (n + 1) n::even { \16/ { } { /-1\(n/2 - 1/2) 2 2 2 2 { |--| binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n (n + 8 n + 14) n::odd { \16/ "A122752" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122849" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A122852" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) HermiteH(n + 1, 1/2 I)} "A122868" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 3 ) LegendreP(n, 3 I), (-I 3 ) LegendreQ(n, 3 I)} "A122871" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-2 I 2 ) (2 LegendreQ(n + 1, 1/2 I 2 ) I - 2 LegendreQ(n, 1/2 I 2 )) {-------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-2 I 2 ) (-2 LegendreP(n + 1, 1/2 I 2 ) I + 2 LegendreP(n, 1/2 I 2 )) - --------------------------------------------------------------------------------} n + 2 "A122877" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreP(n, 1/7 I 7 ) + 7 (4 n + 7) LegendreP(n + 1, 1/7 I 7 ) I) {-----------------------------------------------------------------------------------------------, (n + 2) (n + 3) 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreQ(n, 1/7 I 7 ) + 7 (4 n + 7) LegendreQ(n + 1, 1/7 I 7 ) I) -----------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A122880" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2\n | 5 | | 5 | (2 n + 1) binomial(2 n, n) {|3/2 - ----| , |3/2 + ----| , --------------------------} \ 2 / \ 2 / (n + 1) (n + 2) "A122898" LREtools/SearchTable: "SearchTable successful" n {3 ((8 n + 1) hypergeom([-1/2, -n - 1], [1], -4/3) + (-8 n - 5) hypergeom([-1/2, -n], [1], -4/3))} "A122920" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) | {(-1/2) , (-1/2) | ) -----------------------------------------|} | / (n1 + 1)| |----- (n1 + 4) (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A122932" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 1/2 n 1/2 {2 , (-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A122951" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n 2 (2 LegendreP(n + 1, 2) - LegendreP(n, 2)) 2 (2 LegendreQ(n + 1, 2) - LegendreQ(n, 2)) {--------------------------------------------, --------------------------------------------} n + 2 n + 2 "A122972" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) , (-1) | ) (-(-1) (n1 + 1) n1!)|} | / | |----- | \n1 = 0 / "A123024" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A123025" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n/2 - 1/2) 2 2 { (-4) GAMMA(n/2 - 1/2 - RootOf(4 _Z + 10 _Z + 5)) GAMMA(n/2 + 2 + RootOf(4 _Z + 10 _Z + 5)) n::odd { (n/2) 2 2 { (-4) GAMMA(n/2 - RootOf(4 _Z + 6 _Z + 1)) GAMMA(n/2 + 3/2 + RootOf(4 _Z + 6 _Z + 1)) n::even} { { 0 n::odd "A123026" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n - 1) 2 2 { 2 GAMMA(n/2 - 1/2 - RootOf(2 _Z + 4 _Z + 1)) GAMMA(n/2 + 3/2 + RootOf(2 _Z + 4 _Z + 1)) n::odd { n 2 2 { 2 GAMMA(n/2 - RootOf(4 _Z + 4 _Z - 1)) GAMMA(n/2 + 1 + RootOf(4 _Z + 4 _Z - 1)) n::even} { { 0 n::odd "A123028" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n/2 - 1/2) 2 2 { 12 GAMMA(n/2 + 3 + RootOf(12 _Z + 42 _Z + 37)) GAMMA(n/2 - 1/2 - RootOf(12 _Z + 42 _Z + 37)) n::odd { (n/2) 2 2 { 12 GAMMA(n/2 + 5/2 + RootOf(12 _Z + 30 _Z + 19)) GAMMA(n/2 - RootOf(12 _Z + 30 _Z + 19)) n::even} { { 0 n::odd "A123130" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 3) | n n | \ 3 2 (2 n1 + 1) binomial(2 n1, n1) n1!| {8 n!, 8 n! | ) --------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A123144" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { /2 n\ { |---| { |---| { \ 3 / { \ 3 / { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) irem(n, 3) = 0 { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) irem(n, 3) = 0 { { { /2 n \ { /2 n \ { |--- + 4/3| { |--- - 2/3| {{ \ 3 / , { \ 3 / , { 3 GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) irem(n, 3) = 1 { ---------------------------------------------- irem(n, 3) = 1 { { n + 2 { /2 n \ { { |--- + 2/3| { /2 n \ { \ 3 / { |--- + 2/3| { 3 GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) { \ 3 / { ---------------------------------------------- irem(n, 3) = 2 { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) irem(n, 3) = 2 { n + 2 { (- n/3) 2 2 n { 3 ((n/3)!) (n + 1) binomial(n, n/3) binomial(---, n/3) irem(n, 3) = 0 { 3 { { (1/3 - n/3) 2 2 n { 3 n ((n/3 - 1/3)!) (n + 1) binomial(n - 1, n/3 - 1/3) binomial(--- - 2/3, n/3 - 1/3) irem(n, 3) = 1} { 3 { { (- n/3 + 2/3) 2 2 n { 3 n ((n/3 - 2/3)!) (n - 1) binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) irem(n, 3) = 2 { 3 "A123151" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { {{ 0 irem(n, 3) = 1, { { (n/3 - 2/3) 2 2 { (9/2) GAMMA(n/3 + 3 + RootOf(9 _Z + 33 _Z + 32)) GAMMA(n/3 - 2/3 - RootOf(9 _Z + 33 _Z + 32)) irem(n, 3) = 2 { 0 irem(n, 3) = 0 { { (n/3 - 1/3) 2 2 , { (9/2) GAMMA(n/3 - 1/3 - RootOf(9 _Z + 27 _Z + 22)) GAMMA(n/3 + 8/3 + RootOf(9 _Z + 27 _Z + 22)) irem(n, 3) = 1 { { 0 irem(n, 3) = 2 { (n/3) 2 2 { (9/2) GAMMA(n/3 - RootOf(9 _Z + 21 _Z + 14)) GAMMA(n/3 + 7/3 + RootOf(9 _Z + 21 _Z + 14)) irem(n, 3) = 0 { } { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A123164" LREtools/SearchTable: "SearchTable successful" (n + 1) LegendreP(n + 1, 3) + (-7 n - 3) LegendreP(n, 3) (n + 1) LegendreQ(n + 1, 3) + (-7 n - 3) LegendreQ(n, 3) {--------------------------------------------------------, --------------------------------------------------------} n n "A123178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A123332" n n {2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A123333" n n {3 GAMMA(n + 4/3), 3 n!} "A123334" n n {4 GAMMA(n + 5/4), 4 n!} "A123367" n {2 , (n + 4) (n + 3) (n + 2) (n + 1) n!} "A123510" LREtools/SearchTable: "SearchTable successful" {(n + 1) (n + 2) n! LaguerreL(n, -1)} "A123511" LREtools/SearchTable: "SearchTable successful" {(n + 3) (n + 2) (n + 1) n! LaguerreL(n, -1)} "A123512" LREtools/SearchTable: "SearchTable successful" {(n + 4) (n + 3) (n + 2) (n + 1) n! LaguerreL(n, -1)} "A123525" LREtools/SearchTable: "SearchTable successful" 2 {(n + 1) n! LaguerreL(n + 1, -1)} "A123618" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 2 2 { { 4 binomial(n, n/2) (n + 1) (2 n + 1) binomial(2 n, n) { (4 n - 4) { --------------------------- n::even {----------------------------, { 2 (n + 1) , { 2 } 2 { 1/2 --------------------------------------- n::odd { (n + 2) (n + 1) (n + 2) { 2 2 2 { { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A123619" memory used=58416.0MB, alloc=1975.5MB, time=405.62 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 2 2 { { 4 binomial(n, n/2) (2 n + 1) binomial(2 n, n) { (4 n - 4) { ------------------- n::even {----------------------------, { 2 , { 2 } 2 2 { --------------------------------------- n::odd { (n + 2) (n + 1) (n + 2) { 2 2 2 { { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A123636" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) n! ((n + 2 n - 1) BesselJ(n, -2) + (n + 2) BesselJ(n - 1, -2)), (-1) n! ((n + 2 n - 1) BesselY(n, -2) + (n + 2) BesselY(n - 1, -2))} "A123637" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) n! ((n + 2 n - 1) BesselJ(n, -2) + (n + 2) BesselJ(n - 1, -2)), (-1) n! ((n + 2 n - 1) BesselY(n, -2) + (n + 2) BesselY(n - 1, -2))} "A123642" n {2 , n!} "A123680" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | | \ (n1 + 1)! | | \ | n | ) -----------| n | ) n1! binomial(2 n1, n1) (n1 + 1)!| | / (n1 + 1) n1| | / | |----- | |----- | n \n1 = 0 / \n1 = 0 / {----, ----------------------, -------------------------------------------} n! n! n! "A123681" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | | \ (n1 + 1)! | | \ | n | ) -----------| n | ) n1! binomial(2 n1, n1) (n1 + 1)!| | / (n1 + 1) n1| | / | |----- | |----- | n \n1 = 0 / \n1 = 0 / {----------, ----------------------, -------------------------------------------} (n + 1) n! (n + 1) n! (n + 1) n! "A123686" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A123687" LREtools/SearchTable: "SearchTable successful" (-n) 2 2 {4 binomial(2 n, n) (n!) LaguerreL(n, -2)} "A123922" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n 2 { 3 binomial(---, n/2) (3 n + 2) { 2 { ------------------------------- n::even { 2 {{ (n + 2) (n + 1) , { { 3 n 2 { 4 binomial(--- + 3/2, n/2 + 1/2) { 2 { --------------------------------- n::odd { (n + 2) (3 n + 1) { (-n) 3 n 2 3 n 2 { 16 16 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { ----------------------------------------------------------- n::even { 2 2 { (n + 1) (n + 2) binomial(n, n/2) { } { (-4 n + 4) 2 3 n 2 3 n 2 { 12 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) (3 n + 2) { 2 2 { ------------------------------------------------------------------------------------------------- n::odd { 2 2 2 { n (n + 1) (n + 2) binomial(n - 1, n/2 - 1/2) "A124188" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 1, 2 / 1/2\n / 1/2 \n n | 5 | |5 | (2 n + 1) binomial(2 n, n) 2 {2 , |1/2 - ----| , |---- + 1/2| , (n + 1) n!, --------------------------, n - 6 n + 3} \ 2 / \ 2 / (n + 1) (n + 2) "A124329" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A124429" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A124431" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A124435" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A124642" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n2 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n2::even||| n - 1 |n1 - 1 | |{ n2 ||| ----- |----- | |{ (n2 + 1) binomial(n2, ----) ||| n \ n1 | \ | n2 |{ 2 ||| {1, (-1) , ) (-1) | ) |-(-1) |{ |||, / | / | |{ (2 n2 - 2) ||| ----- |----- | |{ 2 (n2 + 1) ||| n1 = 0 |n2 = 0 | |{ ---------------------------------------- n2::odd ||| | | |{ n2 ||| | | |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n2 \\\ n - 1 |n1 - 1 | |{ 4 binomial(n2, ----) (n2 + 1) ||| ----- |----- | |{ 2 ||| \ n1 | \ | n2 |{ ----------------------------- n2::even||| ) (-1) | ) |-(-1) |{ n2 + 2 |||} / | / | |{ ||| ----- |----- | |{ n2 ||| n1 = 0 |n2 = 0 | |{ 2 binomial(n2 + 1, ---- + 1/2) n2::odd ||| \ \ \{ 2 /// "A124705" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124706" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124707" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124708" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124709" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124710" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124711" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124712" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124713" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124714" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124715" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124716" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124717" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124718" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124719" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124726" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124753" memory used=59063.1MB, alloc=2007.5MB, time=410.47 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 4 n { 4 binomial(---, n/3) { 3 { -------------------- irem(n, 3) = 0 { n + 1 { { 4 n { 9 binomial(--- + 8/3, n/3 + 2/3) (n + 2) {{ 3 , { ---------------------------------------- irem(n, 3) = 1 { (4 n + 5) (2 n + 1) { { 4 n { 9 binomial(--- + 4/3, n/3 + 1/3) { 3 { -------------------------------- irem(n, 3) = 2 { 4 n + 1 { /256\(n/3) 13 { 9 |---| GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) (n + 2) { \27 / 12 { ----------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(5/3 + n/3) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) (4 n + 1) { { /256\(n/3 - 1/3) { 4 |---| GAMMA(n/3 + 3/4) GAMMA(n/3 + 1/4) GAMMA(n/3 + 1/2) { \27 / , { --------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) { { /256\(n/3 + 1/3) 11 17 { 9 |---| GAMMA(n/3 + 7/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 2) (n + 3) { \27 / 12 12 { ----------------------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (4 n + 5) (2 n + 1) { /256\(n/3) 11 17 { 9 |---| GAMMA(n/3 + 7/6) GAMMA(n/3 + --) GAMMA(n/3 + --) (n + 2) (n + 3) { \27 / 12 12 { ----------------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (4 n + 5) (2 n + 1) { { /256\(n/3 - 1/3) 13 { 9 |---| GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) (n + 2) { \27 / 12 } { ----------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(5/3 + n/3) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) (4 n + 1) { { /256\(n/3 - 2/3) { 4 |---| GAMMA(n/3 + 1/4) GAMMA(n/3 + 3/4) GAMMA(n/3 + 1/2) { \27 / { --------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) "A124783" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124784" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124791" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) (hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4))} "A124799" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124803" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124804" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) , 3 , (-1) hypergeom([1/2, -n], [1], 4)} "A124817" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 4) (3 n + 2) { 2 { 3/2 -------------------------------------- n::even { (n + 3) (n + 2) (n + 1) {{ , { 3 n { 4 binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------- n::odd { (n + 2) (n + 3) { (-n) 3 n 3 n { 32 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { -------------------------------------------------------- n::even { (n + 2) (n + 3) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 12 2 (3 n - 2) (3 n + 2) (3 n + 4) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { -------------------------------------------------------------------------------------------------------- n::odd { n (n + 1) (n + 2) (n + 3) binomial(n - 1, n/2 - 1/2) "A124862" n {4 binomial(2 n, n), binomial(2 n, n)} "A125062" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1)| {n! | ) ------------|, n!} | / n1 (n1 + 1)!| |----- | \n1 = 0 / "A125107" n binomial(2 n, n) {2 , ----------------} n + 1 "A125143" LREtools/SearchTable: "SearchTable not successful" {} "A125187" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A125189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z - 1, index = 1) , RootOf(_Z - _Z - _Z - 1, index = 2) , RootOf(_Z - _Z - _Z - 1, index = 3) } "A125190" LREtools/SearchTable: "SearchTable successful" (n + 1) ((n + 2) LegendreP(n + 1, 3) + (-7 n - 6) LegendreP(n, 3)) (n + 1) ((n + 2) LegendreQ(n + 1, 3) + (-7 n - 6) LegendreQ(n, 3)) {------------------------------------------------------------------, ------------------------------------------------------------------} (n - 1) n (n - 1) n "A125267" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A125305" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A125306" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z + 2 _Z + 2 _Z + 1, index = 1) , RootOf(2 _Z + 2 _Z + 2 _Z + 1, index = 2) , RootOf(2 _Z + 2 _Z + 2 _Z + 1, index = 3) } "A125695" LREtools/SearchTable: "SearchTable successful" n n (-3) (3 LegendreP(n + 1, 1/3) - LegendreP(n, 1/3)) (-3) (3 LegendreQ(n + 1, 1/3) - LegendreQ(n, 1/3)) {---------------------------------------------------, ---------------------------------------------------} n n "A126020" n {4 (n + 4), (2 n + 1) binomial(2 n, n)} "A126042" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / /{ 0 irem(n1, 3) = 0\\ | |{ || | |{ / n1 \ || | |{ |---- - 1/3| || | |{ \ 3 / n1 n1 || | |{ 9 (27/4) GAMMA(---- + 2/3) GAMMA(---- + 1/3) || | |{ 3 3 || |n - 1 |{ -------------------------------------------------------- irem(n1, 3) = 1|| |----- |{ n1 n1 || n n | \ (-n1 - 1) |{ GAMMA(---- + 3/2) GAMMA(---- + 1) || {2 , 2 | ) 2 |{ 3 3 ||, | / |{ || |----- |{ / n1 \ || |n1 = 0 |{ |---- + 1/3| || | |{ \ 3 / n1 n1 || | |{ 2 (27/4) GAMMA(---- + 1) GAMMA(---- + 4/3) (2 n1 + 7) || | |{ 3 3 || | |{ ----------------------------------------------------------------- irem(n1, 3) = 2|| | |{ n1 n1 || | |{ GAMMA(5/3 + ----) GAMMA(---- + 13/6) (n1 + 1) || \ \{ 3 3 // / /{ n1 \\ | |{ 27 binomial(n1, ----) || | |{ 3 || |n - 1 |{ --------------------- irem(n1, 3) = 0|| |----- |{ 2 n1 + 3 || n | \ (-n1 - 1) |{ || 2 | ) 2 |{ n1 ||, | / |{ 6 binomial(n1 + 2, ---- + 2/3) || |----- |{ 3 || |n1 = 0 |{ ------------------------------ irem(n1, 3) = 1|| | |{ n1 + 1 || | |{ || \ \{ 0 irem(n1, 3) = 2// / /{ / n1 \ \\ | |{ |----| || | |{ \ 3 / n1 n1 || | |{ 2 (27/4) GAMMA(---- + 1) GAMMA(---- + 4/3) (2 n1 + 7) || | |{ 3 3 || | |{ ----------------------------------------------------------- irem(n1, 3) = 0|| | |{ n1 n1 || |n - 1 |{ GAMMA(5/3 + ----) GAMMA(---- + 13/6) (n1 + 1) || |----- |{ 3 3 || n | \ (-n1 - 1) |{ || 2 | ) 2 |{ 0 irem(n1, 3) = 1||} | / |{ || |----- |{ / n1 \ || |n1 = 0 |{ |---- - 2/3| || | |{ \ 3 / n1 n1 || | |{ 9 (27/4) GAMMA(---- + 1/3) GAMMA(---- + 2/3) || | |{ 3 3 || | |{ -------------------------------------------------------- irem(n1, 3) = 2|| | |{ n1 n1 || | |{ GAMMA(---- + 3/2) GAMMA(---- + 1) || \ \{ 3 3 // "A126068" LREtools/SearchTable: "SearchTable successful" n (-1) ((5 n + 1) hypergeom([1/2, -n], [1], 4) + (n + 1) hypergeom([1/2, -n - 1], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A126086" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A126087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ /5 n1 \ || n n | \ (-n1 - 1) |{ |---- - 5/2| || {3 , 3 | ) 3 |{ \ 2 / ||, | / |{ 2 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 2 binomial(n1, ----) || 3 | ) 3 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A126115" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 1) n1 | n n | \ 2 (-1) binomial(2 n1, n1) n1!| {2 n!, 2 n! | ) ------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A126121" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 1) n1 | n n | \ 2 (-1) (2 n1 + 1) binomial(2 n1, n1) n1!| {2 n! (4 n + 3), 2 n! (4 n + 3) | ) -----------------------------------------------------|} | / (4 n1 + 3) (4 n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A126180" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (8 n + 16 n + 37 n + 11) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 20 n - 43 n - 25) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A126185" LREtools/SearchTable: "SearchTable successful" 2 ((2 n - 1) (2 n - 3) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n + 6 n + 3) hypergeom([-1/2, -n], [1], -4)) (n - 2) {-------------------------------------------------------------------------------------------------------------------} (n + 2) n "A126187" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 ((60 n + 235 n + 249 n + 80) hypergeom([-1/2, -n - 1], [1], -4) + (-60 n - 265 n - 355 n - 160) hypergeom([-1/2, -n], [1], -4)) (n + 1) {-------------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A126189" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A126190" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (4 n - 31 n + 79 n - 36) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n + 29 n - 69 n - 36) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------------------------} n (n + 2) "A126220" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A126221" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) n1| {2 , 2 | ) --------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A126223" 2 binomial(2 n, n) (n - n + 1) {-----------------------------} (n + 1) (2 n - 1) "A126224" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-I) n! binomial(2 n, n) (3 n + 2), I n! binomial(2 n, n) (3 n + 2)} "A126322" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A126323" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 (8 n - 18 n + 82 n - 183 n + 84) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n + 14 n - 84 n + 129 n + 84) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------------------------------------------} n (n + 2) (n - 1) "A126324" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 binomial(n, n/2) { 6 4 { ------------------ n::even { -------------------------- n::even { n + 2 { n (n + 1) binomial(n, n/2) {{ , { } { 6 binomial(n - 1, n/2 - 1/2) { (2 n + 2) { ---------------------------- n::odd { 2 2 { n + 1 { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A126377" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126379" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126380" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126381" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126382" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126383" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126384" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126385" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126386" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((-2 n - 1) hypergeom([1/2, -n], [1], 4) + (n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------} n "A126568" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {(1/2) , (1/2) | ) 2 ((8 n1 + 3) hypergeom([-1/2, -n1 - 1], [1], -4) + (-8 n1 - 7) hypergeom([-1/2, -n1], [1], -4))|} | / | |----- | \n1 = 0 / "A126673" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) (n1 + 2) n1!| {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ----------------------|} | / (n1 + 3) (n1 + 1)! | |----- | \n1 = 0 / "A126674" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A126694" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(49/6) , (49/6) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (49/6) | \n1 = 0 / "A126725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 2 | |2 | {(n + 1) |- ----| n!, (n + 1) |----| n!} \ 2 / \ 2 / "A126765" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" memory used=59768.7MB, alloc=2007.5MB, time=415.40 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n (-2) (n + 1) ((2 n + 1) LegendreP(2 n + 2, I) + (2 n + 2) LegendreP(2 n, I)) {-----------------------------------------------------------------------------, (4 n + 3) n n (-2) (n + 1) ((2 n + 1) LegendreQ(2 n + 2, I) + (2 n + 2) LegendreQ(2 n, I)) -----------------------------------------------------------------------------} (4 n + 3) n "A126930" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { -2 binomial(n, n/2) n::even {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { - ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A126931" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) | | { 2 | | { 2 | | { ---------------------------------------- n1::odd | | { -------------------- n1::even| |n - 1 { n1 | |n - 1 { n1 + 2 | |----- { n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | |----- { | n n | \ { 2 | n | \ { 0 n1::odd | {(10/3) , (10/3) | ) ----------------------------------------------------------|, (10/3) | ) --------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (10/3) | |----- (10/3) | \n1 = 0 / \n1 = 0 / "A126932" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - hypergeom([1/2, -n1], [1], 4))| {(7/2) , (7/2) | ) --------------------------------------------------------------------------|} | / (n1 + 1) | |----- (7/2) | \n1 = 0 / "A126966" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1)| {2 , 2 | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A126967" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-n1 - 1) \| n n | \ | 2 binomial(2 n1, n1) n1!|| {(-2) n!, (-2) n! | ) |- ---------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A126982" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-9/4) , (-9/4) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-9/4) | \n1 = 0 / "A126983" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-1/2) , (-1/2) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-1/2) | \n1 = 0 / "A126984" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-4/3) , (-4/3) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-4/3) | \n1 = 0 / "A126985" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-64/9) , (-64/9) | ) ------------------------|} | / (n1 + 1)| |----- (n1 + 1) (-64/9) | \n1 = 0 / "A126986" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-16/5) , (-16/5) | ) ------------------------|} | / (n1 + 1)| |----- (n1 + 1) (-16/5) | \n1 = 0 / "A126987" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-25/6) , (-25/6) | ) ------------------------|} | / (n1 + 1)| |----- (n1 + 1) (-25/6) | \n1 = 0 / "A127016" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-49/8) , (-49/8) | ) ------------------------|} | / (n1 + 1)| |----- (n1 + 1) (-49/8) | \n1 = 0 / "A127017" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(-36/7) , (-36/7) | ) ------------------------|} | / (n1 + 1)| |----- (n1 + 1) (-36/7) | \n1 = 0 / "A127040" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 4) (3 n + 2) { 2 { 1/2 -------------------------------------- n::even { (n + 1) (n + 2) {{ , { 3 n { binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { 3/4 -------------------------------------------------- n::odd { (n + 2) n { (-n) 3 n 3 n { 6 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 2) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 4 2 (3 n + 4) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) "A127053" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /-81\n /-81\n | \ binomial(2 n1, n1) | {|---| , |---| | ) ----------------------|} \10 / \10 / | / /-81\(n1 + 1)| |----- (n1 + 1) |---| | \n1 = 0 \10 / / "A127065" 2 {(n - 1) , (n + 1) n!} "A127111" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {(n + 1) n!, { , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A127137" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (-1) (n/2)! binomial(n, n/2) n::even} { (n/2 - 1/2) { { (-4) (n/2 - 1/2)! n::odd { 0 n::odd "A127138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (-1/2) (n/2)! binomial(n, n/2) n::even} { (n/2 - 1/2) { { (-2) (n/2 - 1/2)! n::odd { 0 n::odd "A127144" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (-2) (n/2)! n::even} { (n/2 - 1/2) { { n (-1/2) (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { 0 n::odd "A127145" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (-1/2) (n/2)! binomial(n, n/2) (n - 1) n::even} { (n/2 - 1/2) { { (-2) (n/2 - 1/2)! (n/2 - 1/2) n::odd { 0 n::odd "A127154" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + 3 _Z + 2 _Z + 1, index = 1) , RootOf(_Z + 3 _Z + 2 _Z + 1, index = 2) , RootOf(_Z + 3 _Z + 2 _Z + 1, index = 3) } "A127189" LREtools/SearchTable: "SearchTable successful" n (-2) ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 1) hypergeom([-1/2, -n], [1], -1)) n! {---------------------------------------------------------------------------------------------------} n "A127190" LREtools/SearchTable: "SearchTable successful" n 2 (-2) (2 (n + 1) (2 n - 1) hypergeom([-1/2, -n - 1], [1], -1) + (-4 n - 4 n + 1) hypergeom([-1/2, -n], [1], -1)) n! {--------------------------------------------------------------------------------------------------------------------} n "A127231" 2 {1, (n + 1) (2 n + 1) (n!) binomial(2 n, n)} "A127275" n {4 , binomial(2 n, n)} "A127328" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - hypergeom([1/2, -n1], [1], 4))| {(-3/2) , (-3/2) | ) --------------------------------------------------------------------------|} | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / "A127358" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(5/2) , (5/2) | ) -------------------------------------------------|, (5/2) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (5/2) | |----- (5/2) | \n1 = 0 / \n1 = 0 / "A127359" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(10/3) , (10/3) | ) -------------------------------------------------|, (10/3) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (10/3) | |----- (10/3) | \n1 = 0 / \n1 = 0 / "A127360" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(17/4) , (17/4) | ) -------------------------------------------------|, (17/4) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (17/4) | |----- (17/4) | \n1 = 0 / \n1 = 0 / "A127361" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(-5/2) , (-5/2) | ) -------------------------------------------------|, (-5/2) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-5/2) | |----- (-5/2) | \n1 = 0 / \n1 = 0 / "A127362" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(-10/3) , (-10/3) | ) -------------------------------------------------|, (-10/3) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-10/3) | |----- (-10/3) | \n1 = 0 / \n1 = 0 / "A127363" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { 2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(-17/4) , (-17/4) | ) -------------------------------------------------|, (-17/4) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (-17/4) | |----- (-17/4) | \n1 = 0 / \n1 = 0 / "A127389" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A127394" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n + 1, 2 I)} "A127539" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + _Z + 1, index = 1) , RootOf(_Z + _Z + 1, index = 2) , RootOf(_Z + _Z + 1, index = 3) } "A127540" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ------------------------------------||} | / \ (n1 + 3) (n1 + 2) /| |----- | \n1 = 0 / "A127548" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n + 2) BesselI(n, 2) - 2 BesselI(n - 1, 2)), (-1) ((n + 2) BesselK(n, -2) - 2 BesselK(n - 1, -2))} "A127628" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) | {(36/5) , (36/5) | ) -----------------------|} | / (n1 + 1)| |----- (n1 + 1) (36/5) | \n1 = 0 / "A127632" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) ((2 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) + (-2 n - 2) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {----------------, -------------------------------------------------------------------------------------------------------------------------} n + 1 (n + 1) (2 n - 1) "A127846" LREtools/SearchTable: "SearchTable successful" (8 n - 1) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 3) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------} n "A127848" LREtools/SearchTable: "SearchTable successful" n n 4 (2 LegendreP(n + 1, 3/2) - 3 LegendreP(n, 3/2)) 4 (2 LegendreQ(n + 1, 3/2) - 3 LegendreQ(n, 3/2)) {--------------------------------------------------, --------------------------------------------------} n n "A127897" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A127902" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A127905" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A127927" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {(-1) ((16 n + 8) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (-9 n - 3) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(5 n + 3), binomial(2 n, n)} "A127986" n {1, 2 , n!} "A128014" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { ------------------ n::even { n binomial(n, n/2) { binomial(n, n/2) n::even {{ , { } { (2 n + 2) { binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A128015" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 8 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even {{ , { n + 2 } { (2 n - 2) { { 2 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A128057" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 (-16) { (n/2) { ------------------ n::even { (-1) binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { (n/2 - 1/2) { (n/2 + 1/2) { (-1) binomial(n - 1, n/2 - 1/2) n::odd { 2 (-16) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A128058" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 / n1 \ n - 1 / n1 \ ----- |----| 1/2 ----- |----| 1/2 \ \ 2 / 5 \ \ 2 / 5 {1, ) 5 LegendreP(n1 + 1, ----), ) 5 LegendreQ(n1 + 1, ----)} / 5 / 5 ----- ----- n1 = 0 n1 = 0 "A128079" LREtools/SearchTable: "SearchTable successful" (n - 1) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4) {----------------------------------------------------------------------------------------------------} n + 2 "A128088" LREtools/SearchTable: "SearchTable successful" ((2 n + 7) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-18 n - 45) hypergeom([1/2, -n, -n], [1, 1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------} 2 (n + 2) "A128096" memory used=60459.9MB, alloc=2007.5MB, time=420.37 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A128098" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((19 n + 92 n + 99) hypergeom([1/2, -n - 1], [1], 4) + (-15 n - 54 n - 21) hypergeom([1/2, -n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A128195" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 4 || {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A128196" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 4 || {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A128230" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2 \n / 1/2 \n | 3 | |3 | {|- ---- + 1/2| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A128386" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 1/2| || {4 , 4 | ) 2 |{ \ 2 / ||, | / |{ 48 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-2 n1 - 2) |{ 2 3 binomial(n1, ----) || 4 | ) 2 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A128387" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {6 , 6 | ) 6 |{ \ 2 / ||, | / |{ 80 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 5 binomial(n1, ----) || 6 | ) 6 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A128418" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 + 2) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A128419" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- / n2 n2 \||| n n n | \ | (n1 + 1) | \ | (-1) 2 (2 n2 + 1) (15 n2 + 23) binomial(2 n2, n2)|||| {(-1) , (-1/3) , (-1/3) | ) |-3 | ) |- -----------------------------------------------------||||} | / | | / \ (n2 + 1) (n2 + 2) /||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A128611" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 2 || n n | \ (-n1 - 1) | \ (n2 - 1) n2 binomial(2 n2, n2) (n2 - 3 n2 + 1)|| n {1, 3 , 3 | ) 3 | ) -----------------------------------------------||, 4 (6 n + 49)} | / | / (2 n2 - 5) (2 n2 - 3) (2 n2 - 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A128614" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 3) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) n! (n + 1)} "A128634" (2 n + 1) binomial(2 n, n) {1, --------------------------} (n + 1) (n + 2) "A128652" n (n - 1) n binomial(2 n, n) {4 (n + 3), --------------------------} 2 n - 1 "A128656" n (2 n + 1) binomial(2 n, n) (n + 4) {4 (n + 5), ----------------------------------} n + 2 "A128714" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) 2 2 | n n | \ 2 ((8 n1 - n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-8 n1 - 3 n1 + 1) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) ----------------------------------------------------------------------------------------------------------------------|} | / n1 (n1 + 2) | |----- | \n1 = 0 / "A128720" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A128721" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (36 n - 109 n + 91 n - 14) hypergeom([-1/2, -n - 1], [1], -4) + (-36 n + 91 n - 61 n - 14) hypergeom([-1/2, -n], [1], -4) {-----------------------------------------------------------------------------------------------------------------------------} (n - 1) n "A128723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (4 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-4 n1 - 3) hypergeom([-1/2, -n1], [1], -4)| {(-1/3) , (-1/3) | ) --------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1/3) | \n1 = 0 / "A128726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) 2 2 | n n | \ 2 ((12 n1 - 12 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 + 6 n1 + 1) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) ---------------------------------------------------------------------------------------------------------------------------|} | / n1 | |----- | \n1 = 0 / "A128729" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A128730" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) 2 | n n | \ 2 ((n1 + 1) (12 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 - 19 n1 + 1) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) ----------------------------------------------------------------------------------------------------------------------------| | / n1 | |----- | \n1 = 0 / } "A128732" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) 2 | n n | \ 2 ((24 n1 + n1 + 2) hypergeom([-1/2, -n1 - 1], [1], -4) - (3 n1 + 2) (8 n1 - 1) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) -------------------------------------------------------------------------------------------------------------------------|} | / n1 | |----- | \n1 = 0 / "A128734" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ {(1/2) , (1/2) | ) | / |----- \n1 = 0 \ (n1 + 1) 3 2 3 2 | 2 ((24 n1 - 31 n1 + 14 n1 - 6) hypergeom([-1/2, -n1 - 1], [1], -4) + (-24 n1 + 19 n1 + 6 n1 - 6) hypergeom([-1/2, -n1], [1], -4))| ---------------------------------------------------------------------------------------------------------------------------------------------|} n1 (n1 + 2) | | / "A128736" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A128737" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ {(1/2) , (1/2) | ) | / |----- \n1 = 0 \ (n1 + 1) 2 3 2 | 2 ((3 n1 - 1) (4 n1 - 13 n1 + 8) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 + 37 n1 - 7 n1 - 8) hypergeom([-1/2, -n1], [1], -4))| ----------------------------------------------------------------------------------------------------------------------------------------------|} n1 (n1 + 2) | | / "A128740" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ {(1/2) , (1/2) | ) | / |----- \n1 = 0 \ (n1 + 1) 3 2 3 2 | 2 ((24 n1 + 9 n1 + 9 n1 - 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-24 n1 - 21 n1 - 9 n1 - 1) hypergeom([-1/2, -n1], [1], -4))| -------------------------------------------------------------------------------------------------------------------------------------------|} n1 (n1 + 2) | | / "A128743" LREtools/SearchTable: "SearchTable successful" 2 2 (12 n - 12 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-12 n + 6 n + 1) hypergeom([-1/2, -n], [1], -4) {---------------------------------------------------------------------------------------------------------} n "A128746" LREtools/SearchTable: "SearchTable successful" 2 2 (20 n + 5 n + 7) hypergeom([-1/2, -n - 1], [1], -4) + (-20 n - 15 n - 5) hypergeom([-1/2, -n], [1], -4) {---------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A128748" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ (n1 + 1) {(1/2) , (1/2) | ) 2 | / |----- \n1 = 0 3 2 3 2 ((72 n1 + 247 n1 + 262 n1 + 77) hypergeom([-1/2, -n1 - 1], [1], -4) + (-72 n1 - 283 n1 - 372 n1 - 163) hypergeom([-1/2, -n1], [1], -4))/( \ | | (n1 + 2) (n1 + 3))|} | | / "A128750" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A128752" LREtools/SearchTable: "SearchTable successful" 3 2 (3 n - 1) (8 n - 9) (n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-24 n - n + 36 n + 9) hypergeom([-1/2, -n], [1], -4) {------------------------------------------------------------------------------------------------------------------------} (n - 1) n "A128811" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { (n/2 - 1/2) 3 n {{ 2 binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) (n/2 - 1/2)! (3 n + 1) (3 n - 1) , { 2 { 1/2 ------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (- n/2) 3 n 3 n { 2 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) (n/2)! { 2 2 { --------------------------------------------------------------- n::even} { (n + 1) binomial(n, n/2) { { 0 n::odd "A128882" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {1, { , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A129086" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {3 } "A129110" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) { { (-16) {1, { (n/2 - 1/2) , { -------------------------------- n::even} { 4 (-1) binomial(n - 1, n/2 - 1/2) n { (n + 3) (n + 1) binomial(n, n/2) { ---------------------------------------------- n::odd { { (n + 1) (n + 3) { 0 n::odd "A129135" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A129136" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A129147" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n n | \ (-2 I 2 ) (2 LegendreP(n1 + 1, 1/2 I 2 ) I - 2 LegendreP(n1, 1/2 I 2 ))| {(-2) , (-2) | ) ----------------------------------------------------------------------------------|, | / (n1 + 1) | |----- (n1 + 2) (-2) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n | \ (-2 I 2 ) (2 LegendreQ(n1 + 1, 1/2 I 2 ) I - 2 LegendreQ(n1, 1/2 I 2 ))| (-2) | ) ----------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-2) | \n1 = 0 / "A129148" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n n | \ (-I 7 ) (3 I 7 LegendreP(n1 + 1, 3/7 I 7 ) - 7 LegendreP(n1, 3/7 I 7 ))| {(-2) , (-2) | ) ----------------------------------------------------------------------------------|, | / (n1 + 1) | |----- (n1 + 2) (-2) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n | \ (-I 7 ) (3 I 7 LegendreQ(n1 + 1, 3/7 I 7 ) - 7 LegendreQ(n1, 3/7 I 7 ))| (-2) | ) ----------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-2) | \n1 = 0 / "A129149" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! (n + 7) (n + 6), (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------| (n + 7) (n + 6)} | / (n1 + 1)!| |----- | \n1 = 0 / "A129153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! (n + 7) (n + 8), /n - 1 \ |----- n1 | | \ (-1) | (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------| (n + 7) (n + 8)} | / (n1 + 1)!| |----- | \n1 = 0 / "A129155" memory used=61134.3MB, alloc=2007.5MB, time=425.32 memory used=61340.4MB, alloc=2007.5MB, time=428.00 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(6 _Z + 4 _Z - 2 _Z + 1, index = 1) , RootOf(6 _Z + 4 _Z - 2 _Z + 1, index = 2) , RootOf(6 _Z + 4 _Z - 2 _Z + 1, index = 3) } "A129156" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) , /n - 1 \ |----- (n1 + 1) 2 | n | \ 2 ((20 n1 + 35 n1 + 11) hypergeom([-1/2, -n1 - 1], [1], -4) - 5 (4 n1 + 5) (n1 + 1) hypergeom([-1/2, -n1], [1], -4))| (1/2) | ) -----------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 2) (n1 + 3) | |----- | \n1 = 0 / /n - 1 \ |----- (n1 + 1) 2 | n | \ 2 binomial(2 n1, n1) (n1 + 1)| (1/2) | ) --------------------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / "A129158" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" /n - 1 \ |----- (n1 + 1) 2 2 | n n | \ 2 ((12 n1 + 9 n1 + 5) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 - 15 n1 - 7) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) ---------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 2) (n1 + 3) | |----- | \n1 = 0 / /n - 1 \ |----- (n1 + 1) 2 | n | \ 2 binomial(2 n1, n1) (n1 + 1)| (1/2) | ) --------------------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / "A129160" LREtools/SearchTable: "SearchTable successful" 2 (24 n + n + 2) hypergeom([-1/2, -n - 1], [1], -4) - (3 n + 2) (8 n - 1) hypergeom([-1/2, -n], [1], -4) {-------------------------------------------------------------------------------------------------------} n "A129164" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, (2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 2) hypergeom([-1/2, -n], [1], -4)} "A129166" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(3 _Z - 4 _Z + 3 _Z - 1, index = 1) , RootOf(3 _Z - 4 _Z + 3 _Z - 1, index = 2) , RootOf(3 _Z - 4 _Z + 3 _Z - 1, index = 3) } "A129167" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) | n n | \ 2 ((5 n1 + 3) hypergeom([-1/2, -n1 - 1], [1], -4) + (-5 n1 - 5) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) --------------------------------------------------------------------------------------------------------|} | / n1 + 2 | |----- | \n1 = 0 / "A129169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (n1 + 1) 2 2 | n n | \ 2 ((12 n1 + 9 n1 + 5) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 - 15 n1 - 7) hypergeom([-1/2, -n1], [1], -4))| {(1/2) , (1/2) | ) ---------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 3) | |----- | \n1 = 0 / "A129171" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 n (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 4 n + 1) hypergeom([-1/2, -n], [1], -4) {5 , -------------------------------------------------------------------------------------------------------} n "A129173" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 n (2 n + 1) (24 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-48 n - 46 n - 1) hypergeom([-1/2, -n], [1], -4) {5 , ------------------------------------------------------------------------------------------------------------} n "A129180" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n (2 n + 1) (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) (2 n + 1) (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {(3 + 2 2 ) , (-2 2 + 3) , ---------------------------------------------------, ---------------------------------------------------} n n "A129217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- n1 | | \ (-1) | (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A129218" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- n1 | | \ (-1) | (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A129238" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! (n + 11) (n + 10), /n - 1 \ |----- n1 | | \ (-1) | (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------| (n + 11) (n + 10)} | / (n1 + 1)!| |----- | \n1 = 0 / "A129255" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! (n + 12) (n + 11), /n - 1 \ |----- n1 | | \ (-1) | (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------| (n + 12) (n + 11)} | / (n1 + 1)!| |----- | \n1 = 0 / "A129348" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! (2 n BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-2) n! (2 n BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A129400" LREtools/SearchTable: "SearchTable successful" n (-2) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------} n + 2 "A129442" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) + (-4 n - 1) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------} (n + 1) (2 n - 1) "A129458" LREtools/SearchTable: "SearchTable not successful" {} "A129507" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A129509" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A129535" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 2) (n + 1) n!} "A129637" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n n | \ | 7 (-1/4 I) (-7 LegendreQ(n1 + 1, 1/7 I 7 ) I + 7 LegendreQ(n1, 1/7 I 7 ))|| {4 , 4 | ) |-1/4 ----------------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n | \ | 7 (-1/4 I) (7 LegendreP(n1 + 1, 1/7 I 7 ) I - 7 LegendreP(n1, 1/7 I 7 ))|| 4 | ) |1/4 ---------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A129683" LREtools/SearchTable: "SearchTable successful" n {2 n! LaguerreL(n, -1)} "A129695" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -2), n! LaguerreL(n, 1)} "A129703" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {1, (-1) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A129775" memory used=62045.3MB, alloc=2007.5MB, time=432.89 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A129831" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /{ 0 n1::even\\\ |----- | |{ ||| n n | \ | n1 |{ / n1 \ ||| {(-1) , (-1) | ) |-(-1) |{ |- ---- + 1/2| |||, | / | |{ \ 2 / n1 / n1 \ ||| |----- | |{ 2 n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) |---- - 1/2|! n1::odd ||| \n1 = 0 \ \{ 2 \ 2 / /// /n - 1 / /{ / n1 \ \\\ |----- | |{ |----| ||| n | \ | n1 |{ / n1 \ \ 2 / / n1 \ ||| (-1) | ) |-(-1) |{ |---- + 1| 2 |----|! n1::even|||} | / | |{ \ 2 / \ 2 / ||| |----- | |{ ||| \n1 = 0 \ \{ 0 n1::odd /// "A129833" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1 {-----, n! LaguerreL(n + 1, -1)} n + 1 "A129840" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | | \ (2 n1 + 3) (2 n1 + 1) n1! binomial(2 n1, n1)| | \ (n1 + 1) n1! (4 n1 + 9) | {(n + 1) n!, (n + 1) n! | ) --------------------------------------------|, (n + 1) n! | ) -----------------------------|, | / 2 | | / (2 n1 + 5) (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / \n1 = 0 / /n - 1 \ |----- n1 | | \ (n1 + 1) 4 n1! (6 n1 + 13) | (n + 1) n! | ) -----------------------------|} | / (2 n1 + 5) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A129867" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {1, (n + 1) | ) n1!|, n + 1} | / | |----- | \n1 = 0 / "A129890" n n {(n + 1) 2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A129937" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even {(n + 1) n, { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A129949" {(n + 1) n!, (n + 1) (n - 1) n} "A129981" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 /{ / n1 \ \ ----- |{ | ----- |{ |- ----| | \ |{ / n1 \ | \ |{ \ 2 / n1 / n1 \ | {1, ) |{ |---- - 1/2| |, ) |{ 2 (n1 + 1) binomial(n1, ----) |----|! n1::even|} / |{ / n1 \ \ 2 / / n1 \ | / |{ 2 \ 2 / | ----- |{ |---- + 1/2| 2 |---- - 1/2|! n1::odd | ----- |{ | n1 = 0 \{ \ 2 / \ 2 / / n1 = 0 \{ 0 n1::odd / "A129996" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (n/2) { (-16) { 16 (-1) binomial(n, n/2) (n + 1) { 1/2 -------------------------------- n::even { ------------------------------------- n::even { binomial(n, n/2) (n + 1) (n + 3) { (n + 2) (n + 4) {{ , { } { (n/2 - 1/2) { (n/2 + 1/2) { 2 (-16) (n + 1) { 4 (-1) binomial(n + 1, n/2 + 1/2) { -------------------------------------------- n::odd { -------------------------------------------- n::odd { binomial(n - 1, n/2 - 1/2) (n + 2) (n + 4) n { n + 3 "A129997" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 2 1/2 (-1/2 I 2 ) (2 I 2 (2 n + 13) HermiteH(n + 1, 2 I 2 ) + (n + 46 n + 208) HermiteH(n, 2 I 2 )) {- --------------------------------------------------------------------------------------------------------} n (n - 1) (n - 2) (n - 3) "A130019" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 | n n | \ (-1/2 I 2 ) ((n1 + 15) HermiteH(n1 + 1, 2 I 2 ) - 4 I 2 (2 n1 + 15) HermiteH(n1, 2 I 2 ))| {2 , 2 | ) ----------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 - 1) (n1 - 2) n1 2 | \n1 = 0 / "A130458" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A130470" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- / n2 2 \|| | n1 2 | \ | (-1) (n2 + 3) (n2 + 1) (n2 + 2) (n2 + 4) n2!||| | (-1) (n1 + 7 n1 + 11) | ) |- -----------------------------------------------||| |n - 1 | / | 2 2 ||| |----- |----- \ ((n2 + 1) + 7 n2 + 18) (n2 + 7 n2 + 11) /|| n n | \ \n2 = 0 /| {1, (-1) , (-1) (n + 4), (n + 2) | ) -------------------------------------------------------------------------------------|, n + 2} | / (n1 + 3) (n1 + 2) | |----- | \n1 = 0 / "A130471" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n n \ n1 | \ | (-1) (n2 + 1) (n2 + 2) (n2 + 3) (n2 + 4) n2!|| {1, (-1) , (-1) (n + 9/2), ) (-1) (n1 + 5) | ) |- ----------------------------------------------||} / | / \ (n2 + 5) (n2 + 6) /| ----- |----- | n1 = 0 \n2 = 0 / "A130494" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) n1! (n1 + 3 n1 + 3)} / ----- n1 = 0 "A130591" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A130619" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A130636" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A130637" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A130655" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) (n - 1) {------------------------, (n + 1) (n + 2) (3 (2 n + 1) (3 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) - 5 (4 n + 1) (n + 1) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) ----------------------------------------------------------------------------------------------------------------------------------------------} (n + 1) (n + 2) (2 n - 1) "A130679" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 1) (4 n1 + 7) (4 n1 + 11) n1!|| {(n + 1) (-1) n!, (n + 2) (n + 1) n!, (n + 1) (-1) n! | ) |- ------------------------------------------||, | / \ (n1 + 2) (n1 + 1)! /| |----- | \n1 = 0 / /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A130783" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (2 n + 2) n::even { binomial(n, n/2) {2 (n + 1), { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A130905" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |- ---- + 1/2| | | { 2 |----|! n1::even| |n - 1 { \ 2 / n1 / n1 \ | |n - 1 { \ 2 / | |----- { 2 n1 binomial(n1 - 1, ---- - 1/2) |---- - 1/2|! n1::odd | |----- { | | \ { 2 \ 2 / | | \ { 0 n1::odd | {n! | ) -------------------------------------------------------------------------------|, n! | ) ---------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A130907" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 | | \ (-1/2 I 2 ) HermiteH(n1 + 1, 1/2 I 2 )| {n! | ) --------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A130915" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A130976" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 4 5 binomial(2 n1, n1)| {25 , 25 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A130977" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 5 6 binomial(2 n1, n1)| {36 , 36 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A130978" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 6 7 binomial(2 n1, n1)| {49 , 49 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A130979" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-6 n1 - 6) | n n | \ 7 2 binomial(2 n1, n1)| {64 , 64 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A130980" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 8 3 binomial(2 n1, n1)| {81 , 81 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A131428" binomial(2 n, n) {1, ----------------} n + 1 "A131430" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 \ binomial(2 n1, n1) (3 n1 + 11 n1 + 4) {1, 4 n + 3, ) --------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A131441" LREtools/SearchTable: "SearchTable successful" (-n) {2 (2 n + 1) (hypergeom([-n - 1], [1/2], 1/2) - hypergeom([-n], [1/2], 1/2)) binomial(2 n, n) n!} "A131521" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 9 10 binomial(2 n1, n1)| {100 , 100 | ) ------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A131763" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 7) - 7 LegendreP(n, 7) LegendreQ(n + 1, 7) - 7 LegendreQ(n, 7) {---------------------------------------, ---------------------------------------} n n "A131765" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 9) - 9 LegendreP(n, 9) LegendreQ(n + 1, 9) - 9 LegendreQ(n, 9) {---------------------------------------, ---------------------------------------} n n "A131846" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 11) - 11 LegendreP(n, 11) LegendreQ(n + 1, 11) - 11 LegendreQ(n, 11) {------------------------------------------, ------------------------------------------} n n "A131869" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 15) - 15 LegendreP(n, 15) LegendreQ(n + 1, 15) - 15 LegendreQ(n, 15) {------------------------------------------, ------------------------------------------} n n "A131926" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 13) - 13 LegendreP(n, 13) LegendreQ(n + 1, 13) - 13 LegendreQ(n, 13) {------------------------------------------, ------------------------------------------} n n "A131927" LREtools/SearchTable: "SearchTable successful" (17 n + 17) LegendreP(n + 1, 17) + (-577 n - 289) LegendreP(n, 17) (17 n + 17) LegendreQ(n + 1, 17) + (-577 n - 289) LegendreQ(n, 17) {------------------------------------------------------------------, ------------------------------------------------------------------} (n - 1) n (n - 1) n "A131965" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A132164" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A132262" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n | 3 5 | | 3 5 | {|7/2 - ------| , |7/2 + ------| , \ 2 / \ 2 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 3 5 | ||| / 1/2\n |----- | |----- |7/2 + ------| (3 LegendreP(n2 + 1, 3) - LegendreP(n2, 3))||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) -------------------------------------------------------------------|||, \ 2 / | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 3 5 | ||| / 1/2\n |----- | |----- |7/2 + ------| (3 LegendreQ(n2 + 1, 3) - LegendreQ(n2, 3))||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) -------------------------------------------------------------------|||} \ 2 / | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A132306" LREtools/SearchTable: "SearchTable successful" {(4 n (4 n + 7) (n - 5) (4 n + 5) hypergeom([-n - 3, -2 n - 5], [-2 n - 7/2], 1/4) 4 3 2 / 2 + (n + 431 n + 1481 n + 1669 n + 594) hypergeom([-n - 2, -2 n - 3], [-2 n - 3/2], 1/4)) binomial(4 n, 2 n) (4 n + 1) (4 n + 3) / ((n + 1) / (n + 2) (2 n + 1) (2 n + 3) (7 n + 11))} "A132310" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 (2 n1 + 1) binomial(2 n1, n1)| {3 , 3 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A132364" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A132371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1! n1} / ----- n1 = 0 "A132373" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {7 , 7 | ) 7 |{ \ 2 / ||, | / |{ 96 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 6 binomial(n1, ----) || 7 | ) 7 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A132374" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-3 n1 - 3) |{ |---- - 1/2| || {8 , 8 | ) 2 |{ \ 2 / ||, | / |{ 112 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-3 n1 - 3) |{ 2 7 binomial(n1, ----) || 8 | ) 2 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A132375" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ /7 n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 7/2| || {9 , 9 | ) 3 |{ \ 2 / ||, | / |{ 2 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ /3 n1\ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-2 n1 - 2) |{ 2 2 binomial(n1, ----) || 9 | ) 3 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A132461" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A132595" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=62737.6MB, alloc=2007.5MB, time=437.89 /n - 1 \ |----- / (-n1 - 1) n1 \| n n | \ |2 3 8 ((30 n1 - 2) LegendreP(n1 + 1, 3/2) + (-10 n1 + 3) LegendreP(n1, 3/2))|| {(3/2) , (3/2) | ) |---------------------------------------------------------------------------------------||, | / \ n1 /| |----- | \n1 = 0 / /n - 1 \ |----- / (-n1 - 1) n1 \| n | \ |2 3 8 ((30 n1 - 2) LegendreQ(n1 + 1, 3/2) + (-10 n1 + 3) LegendreQ(n1, 3/2))|| (3/2) | ) |---------------------------------------------------------------------------------------||} | / \ n1 /| |----- | \n1 = 0 / "A132647" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 7 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A132788" (2 n + 1) binomial(2 n, n) {--------------------------, n + 1} (n + 1) (n + 2) "A132790" n (2 n + 1) binomial(2 n, n) {2 , --------------------------, n + 1} (n + 1) (n + 2) "A132863" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 4 binomial(2 n1, n1)| {(-9) , (-9) | ) ----------------------|} | / (n1 + 1) | |----- (n1 + 1) (-9) | \n1 = 0 / "A132864" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 5 binomial(2 n1, n1)| {(-16) , (-16) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-16) | \n1 = 0 / "A132865" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 6 binomial(2 n1, n1)| {(-25) , (-25) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-25) | \n1 = 0 / "A132866" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 7 binomial(2 n1, n1)| {(-36) , (-36) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-36) | \n1 = 0 / "A132869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 9 binomial(2 n1, n1)| {(-64) , (-64) | ) ----------------------|} | / (n1 + 1)| |----- (n1 + 1) (-64) | \n1 = 0 / "A132889" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 16 { 2 { -------------------------- n::even { 4 binomial(n, n/2) (n + 1) { 2 2 { --------------------------- n::even (2 n + 1) binomial(2 n, n) n { (n + 1) binomial(n, n/2) { n + 2 {----------------------------, { , { } (n + 1) (n + 2) { (4 n + 4) { 2 2 { 2 { 16 binomial(n - 1, n/2 - 1/2) n { ------------------------------------------- n::odd { --------------------------------- n::odd { 2 { 2 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { (n + 1) "A132894" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4))} "A132897" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 10 binomial(2 n1, n1)| {(-81) , (-81) | ) -----------------------|} | / (n1 + 1) | |----- (n1 + 1) (-81) | \n1 = 0 / "A132900" LREtools/SearchTable: "SearchTable successful" n (-3) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------} n + 2 "A133053" LREtools/SolveLRE: "Reduced the order of" (n+4)*(2*n+5)*(n+5)^2*E^3-(n+4)*(2*n+7)*(7*n^2+42*n+59)*E^2-3*(2*n+5)*(n+2)*(7*n^2+42*n+59)*E+27*(2*n+7 )*(n+2)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" 2 (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------} 2 (n + 2) "A133054" memory used=63108.7MB, alloc=2007.5MB, time=441.35 memory used=63430.4MB, alloc=2007.5MB, time=444.28 LREtools/SolveLRE: "Reduced the order of" (2*n+5)*(n+4)*(7*n^2+42*n+59)*(n+5)^2*(n+6)^3*E^4-2*(2*n+9)*(2*n+5)*(n+4)*(5*n^2+35*n+54)*(7*n^2+56*n+ 108)*(n+5)^2*E^3-6*(2*n+7)*(n+4)*(n+3)*(7*n^2+42*n+59)*(5*n^2+35*n+54)*(7*n^2+56*n+108)*E^2+54*(2*n+9)*(2*n+5)*(n+3)*(7*n^2+42*n+59)*(5*n^2+35*n+54)* (n+2)^2*E+729*(n+3)*(2*n+9)*(7*n^2+56*n+108)*(n+2)^2*(n+1)^3 "to two: Symmetric cube" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" n 3 (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {- --------------------------------------------------------------------------} 3 (n + 2) "A133106" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { 2 binomial(n, n/2) (8 n + 56 n + 95) (n + 1) (n + 3) n { ----------------------------------------------------- n::even {(n + 5) (n + 4) (n + 3) 2 , { n + 2 , { { 16 binomial(n - 1, n/2 - 1/2) n (n + 2) (n + 4) (2 n + 7) { --------------------------------------------------------- n::odd { n + 1 { n { 2 4 (n + 2) (n + 4) (2 n + 7) { ------------------------------ n::even { (n + 1) binomial(n, n/2) { } { (2 n + 2) 2 { 2 (n + 3) (8 n + 56 n + 95) { 1/4 ------------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) "A133107" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {2 , 2 n} "A133158" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (-n1 - 1) n1 | {(3/2) , (3/2) | ) (2 3 4 ((4 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -2) + (-4 n1 - 3) hypergeom([-1/2, -n1], [1], -2)))|} | / | |----- | \n1 = 0 / "A133221" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 2 (n/2)! { (- n/2) { --------------- n::even { 2 binomial(n, n/2) (n/2)! n::even { n {{ , { } { (- n/2 + 1/2) { (n/2 + 1/2) { 2 binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 2 2 (n/2 + 1/2)! { --------------------------- n::odd { n + 1 "A133305" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 9) - 9 LegendreP(n, 9) LegendreQ(n + 1, 9) - 9 LegendreQ(n, 9) {---------------------------------------, ---------------------------------------} n n "A133306" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 11) - 11 LegendreP(n, 11) LegendreQ(n + 1, 11) - 11 LegendreQ(n, 11) {------------------------------------------, ------------------------------------------} n n "A133307" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 13) - 13 LegendreP(n, 13) LegendreQ(n + 1, 13) - 13 LegendreQ(n, 13) {------------------------------------------, ------------------------------------------} n n "A133308" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 15) - 15 LegendreP(n, 15) LegendreQ(n + 1, 15) - 15 LegendreQ(n, 15) {------------------------------------------, ------------------------------------------} n n "A133309" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 17) - 17 LegendreP(n, 17) LegendreQ(n + 1, 17) - 17 LegendreQ(n, 17) {------------------------------------------, ------------------------------------------} n n "A133443" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(10/3) , (10/3) | ) ---------------------------------------------------|, (10/3) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (10/3) | |----- (10/3) | \n1 = 0 / \n1 = 0 / "A133444" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | / { n1 \ | { (2 n1 - 2) | | { -2 binomial(n1, ----) n1::even| | { 2 2 | | { 2 | | { - ------------------------------- n1::odd | | { | |n - 1 { n1 | |n - 1 { n1 | |----- { n1 binomial(n1 - 1, ---- - 1/2) | |----- { binomial(n1 + 1, ---- + 1/2) n1::odd | n n | \ { 2 | n | \ { 2 | {(17/4) , (17/4) | ) ---------------------------------------------------|, (17/4) | ) ----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (17/4) | |----- (17/4) | \n1 = 0 / \n1 = 0 / "A133471" LREtools/SearchTable: "SearchTable not successful" {} "A133602" n binomial(2 n, n) (3 n - 1) (-1) binomial(2 n, n) {--------------------------, 1/2 ----------------------} (n + 1) (2 n - 1) n - 1/2 "A133603" n binomial(2 n, n) (3 n - 1) (-1) binomial(2 n, n) {--------------------------, 1/2 ----------------------} (n + 1) (2 n - 1) n - 1/2 "A133656" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A133751" n {2 , (n + 2) (n + 1) n!} "A133798" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {1, (n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A133915" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n n | \ (2 n1 + 1) binomial(2 n1, n1)| {4 , (-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A134184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A134316" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- | n \ n1 | \ n2 | {1, (-1) , ) (-1) | ) (-(-1) (n2 + 3) (n2 + 2) (n2 + 1) n2!)|} / | / | ----- |----- | n1 = 0 \n2 = 0 / "A134389" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n - 1) { ------------------ n::even { 2 n binomial(n, n/2) n::even n { n binomial(n, n/2) { {2 , { , { (2 n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) } { (2 n - 2) { 1/2 -------------------------------------------- n::odd { 4 2 { n { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A134425" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- (-n1 - 1) | |----- (-n1 - 1) | n n | \ 6 (3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3))| n | \ 6 (3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3))| {6 , 6 | ) ------------------------------------------------------|, 6 | ) ------------------------------------------------------|} | / n1 + 2 | | / n1 + 2 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A134432" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {(n + 1) n! (n - 1), (n + 1) n! (n - 1) | ) ---------------------|} | / (n1 + 1)! n1 (n1 - 1)| |----- | \n1 = 0 / "A134515" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \ \\ |----- | |----- | |----- n2 | || n n | \ n1 | n | \ | n1 | \ (-1) | || {(-1) (n + 1) (n + 2), (-1) (n + 1) (n + 2) | ) (-(-1) n1!)|, (-1) (n + 1) (n + 2) | ) |-(-1) | ) ---------| n1!||} | / | | / | | / (n2 + 1)!| || |----- | |----- | |----- | || \n1 = 0 / \n1 = 0 \ \n2 = 0 / // "A134565" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 2) { 2 { - ---------------------------- n::even { (n + 1) (n + 2) {{ , { 3 n { binomial(--- + 3/2, n/2 + 1/2) { 2 { ------------------------------ n::odd { n + 2 { (-n) 3 n 3 n { 2 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 1) (n + 2) binomial(n, n/2) { } { (-2 n + 2) 3 n 3 n { 2 2 (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { - --------------------------------------------------------------------------------------------- n::odd { n (n + 1) (n + 2) binomial(n - 1, n/2 - 1/2) "A134635" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) n1| {2 , 2 | ) --------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A134646" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A134648" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A134757" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n - 2) hypergeom([1/2, -n], [1], 4))} "A134758" {n, binomial(2 n, n)} "A134759" {n + 1, binomial(2 n, n)} "A134760" {1, binomial(2 n, n)} "A134762" {1, binomial(2 n, n)} "A134833" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- n2 || | | \ (-2) || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || n n | \ n1! | n | \ \n2 = 0 /| {(-2) , (-2) | ) ------------|, (-2) | ) ----------------------|} | / (n1 + 1)| | / (n1 + 1) | |----- (-2) | |----- (-2) | \n1 = 0 / \n1 = 0 / "A134920" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / n1 \ \ / / n1 \ \ | |----| 1/2 | | |----| 1/2 | |n - 1 \ 2 / 5 | |n - 1 \ 2 / 5 | |----- 5 LegendreP(n1 + 1, ----)| |----- 5 LegendreQ(n1 + 1, ----)| | \ 5 | | \ 5 | {1, n, n | ) -------------------------------|, n | ) -------------------------------|} | / (n1 + 1) n1 | | / (n1 + 1) n1 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A135052" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=64081.2MB, alloc=2007.5MB, time=448.96 LREtools/SolveLRE: "Reduced the order of" (n+7)*E^4+(-4*n-22)*E^3+(8*n+20)*E-4*n-4 "to two: Half integer product u(n/2) * u(n/2+1/2)" E^2+(-4 *n-5)*E+(n+1)*(2*n+1) LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n - 1 ----- \ 1/2 n1 /{ / n1 1/2 n1 1/2 {1, ) (2 - 2 ) |{ |hypergeom([3/4, - ---- - 1], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) / \{ \ 2 2 ----- n1 = 0 1/2 n1 1/2 n1 1/2 + (3 + 2 2 ) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) 2 2 1/2 n1 1/2 n1 1/2 \ 1/2 + (-2 - 2 ) hypergeom([3/4, - ---- - 1], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 )| (-2028 + 1434 2 ) , 2 2 / n1::even / 1/2 n1 1/2 n1 1/2 6 |-2 (-3 + 2 2 ) (2 n1 + 3) hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) \ 2 2 1/2 n1 1/2 n1 1/2 + (2 - 2) n1 hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1], [3/2], -2 - 2 2 ) 2 2 1/2 n1 1/2 n1 1/2 + 3 (2 - 2) (n1 + 1) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) 2 2 n1 1/2 n1 1/2 \ 1/2 \ + 2 n1 hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1], [3/2], -2 - 2 2 )| (12 2 - 17)/(n1 + 3) , n1::odd|, 2 2 / / n - 1 ----- \ 1/2 n1 /{ / ) (2 - 2 ) |{ 6 | / \{ \ ----- n1 = 0 1/2 n1 1/2 n1 1/2 -2 (-3 + 2 2 ) (2 n1 + 3) hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) 2 2 1/2 n1 1/2 n1 1/2 + (2 - 2) n1 hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1], [3/2], -2 - 2 2 ) 2 2 1/2 n1 1/2 n1 1/2 + 3 (2 - 2) (n1 + 1) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) 2 2 n1 1/2 n1 1/2 \ 1/2 + 2 n1 hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1], [3/2], -2 - 2 2 )| (12 2 - 17)/(n1 + 3) , n1::even 2 2 / / n1 1/2 n1 1/2 |hypergeom([3/4, - ---- - 1], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- - 1/2], [3/2], -2 - 2 2 ) \ 2 2 1/2 n1 1/2 n1 1/2 + (3 + 2 2 ) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 ) hypergeom([3/4, - ----], [3/2], -2 - 2 2 ) 2 2 1/2 n1 1/2 n1 1/2 \ 1/2 + (-2 - 2 ) hypergeom([3/4, - ---- - 1], [3/2], -2 - 2 2 ) hypergeom([3/4, - ---- + 1/2], [3/2], -2 - 2 2 )| (-2028 + 1434 2 ) , n1::odd 2 2 / \ |} / "A135218" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-5) (2 n1 - 1) | {n! (n + 6) (n + 13 n + 27), n! (n + 6) (n + 13 n + 27) | ) -----------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! (n1 + 7) ((n1 + 1) + 13 n1 + 40) (n1 + 6) (n1 + 13 n1 + 27)| \n1 = 0 / "A135307" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A135310" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 4 _Z + 5 _Z + 1 "A135334" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(1/2 - 1/2 I) , (1/2 + 1/2 I) , /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n | \ | \ (1/2 + 1/2 I) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) ---------------------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A135335" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 3 _Z + 4 _Z - 2 _Z + 1 "A135336" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ | ----- | | \ (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)| | ) ----------------------------------------| |n - 1 / (n2 + 3) (n2 + 2) (n2 + 1) | |----- ----- | n n | \ n2 = 0 | {1, (-1/2) , (-1/2) | ) -----------------------------------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A135337" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z + 2 _Z + 1, index = 1) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 2) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 3) } "A135339" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (5 n1 - 2) | {(-1/2) , (-1/2) | ) ----------------------------------|} | / (n1 + 1)| |----- (n1 + 2) (2 n1 - 1) (-1/2) | \n1 = 0 / "A135390" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) {----------------------------------------------------------------------} n + 1 "A135394" LREtools/SearchTable: "SearchTable successful" ((-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4) + (7 n + 11) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4)) binomial(2 n, n) (2 n + 1) {------------------------------------------------------------------------------------------------------------------------------------} 2 (n + 2) "A135395" LREtools/SearchTable: "SearchTable successful" ((-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4) + (7 n + 11) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4)) binomial(2 n, n) (2 n + 1) (2 n + 3) {----------------------------------------------------------------------------------------------------------------------------------------------} 2 (n + 2) (n + 3) "A135401" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/4) { (-1) HermiteH(n/2, -I) n::even { {{ (n/2 + 1/2) (n/2 + 3/2) (n/2 + 1/2) , { (-I) ((-1) HermiteH(n/2 + 3/2, -I) I + 2 (-1) HermiteH(n/2 + 1/2, -I)) { - ------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { (-I) HermiteH(n/2, I) n::even { { (n/2 + 1/2) , { (-I) (HermiteH(n/2 + 3/2, I) I + 2 HermiteH(n/2 + 1/2, I)) { - --------------------------------------------------------------------- n::odd { n + 1 { (n/2) { (-I) (HermiteH(n/2 + 3/2, I) I + 2 HermiteH(n/2 + 1/2, I)) { - --------------------------------------------------------------- n::even { n + 1 , { { (n/2 - 1/2) { (-I) HermiteH(n/2, I) n::odd { (n/2) (n/2 + 3/2) (n/2 + 1/2) { (-I) ((-1) HermiteH(n/2 + 3/2, -I) I + 2 (-1) HermiteH(n/2 + 1/2, -I)) { - ------------------------------------------------------------------------------------------------- n::even { n + 1 } { { (n/2 - 1/2) (n/2) { (-I) (-1) HermiteH(n/2, -I) n::odd "A135413" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A135457" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) binomial(2 n1, n1) n1! || {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |------------------------------------||} | / \binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A135489" LREtools/SearchTable: "SearchTable successful" memory used=64728.1MB, alloc=2007.5MB, time=453.54 n {(-1) ((16 n + 8) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (-9 n - 3) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(5 n + 3)} "A135512" 2 n binomial(2 n, n) (n + n + 1) {4 , -----------------------------} (n + 1) (n + 2) "A135582" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n 3 2 n 3 2 n 3 2 n {(-1) , 2 , RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A135593" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(n + 2) (n + 1) (-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) + 2 HermiteH(n, 1/2 I 2 ) I)} "A135799" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \ \\ |----- | |----- | |----- n2 | || n n | \ n1 | n | \ | n1 | \ (-1) | || {(-1) (n + 1), (-1) (n + 1) | ) (-(-1) n1!)|, (-1) (n + 1) | ) |-(-1) | ) ---------| n1!||} | / | | / | | / (n2 + 1)!| || |----- | |----- | |----- | || \n1 = 0 / \n1 = 0 \ \n2 = 0 / // "A135801" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) (n + 3) (n + 2) (n + 1), (-1) (n + 3) (n + 2) (n + 1) | ) (-(-1) n1!)|, | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 3) (n + 2) (n + 1) | ) |-(-1) | ) ---------| n1!||} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135802" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) (n + 1) (n + 4) (n + 3) (n + 2), (-1) (n + 1) | ) (-(-1) n1!)| (n + 4) (n + 3) (n + 2), | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 1) | ) |-(-1) | ) ---------| n1!|| (n + 4) (n + 3) (n + 2)} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135803" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1), (-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) | ) (-(-1) n1!)|, | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) | ) |-(-1) | ) ---------| n1!||} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135804" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) (n + 3) (n + 2) (n + 1) (n + 6) (n + 5) (n + 4), (-1) (n + 3) (n + 2) (n + 1) | ) (-(-1) n1!)| (n + 6) (n + 5) (n + 4), | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 3) (n + 2) (n + 1) | ) |-(-1) | ) ---------| n1!|| (n + 6) (n + 5) (n + 4)} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135805" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) (n + 1) (n + 4) (n + 3) (n + 2) (n + 5) (n + 6) (n + 7), (-1) (n + 1) | ) (-(-1) n1!)| (n + 4) (n + 3) (n + 2) (n + 5) (n + 6) (n + 7), | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 1) | ) |-(-1) | ) ---------| n1!|| (n + 4) (n + 3) (n + 2) (n + 5) (n + 6) (n + 7)} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135806" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) (n + 6) (n + 7) (n + 8), /n - 1 \ |----- | n | \ n1 | (-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) | ) (-(-1) n1!)| (n + 6) (n + 7) (n + 8), | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) | ) |-(-1) | ) ---------| n1!|| (n + 6) (n + 7) (n + 8)} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135807" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(-1) (n + 3) (n + 2) (n + 1) (n + 6) (n + 5) (n + 4) (n + 7) (n + 8) (n + 9), /n - 1 \ |----- | n | \ n1 | (-1) (n + 3) (n + 2) (n + 1) | ) (-(-1) n1!)| (n + 6) (n + 5) (n + 4) (n + 7) (n + 8) (n + 9), | / | |----- | \n1 = 0 / /n - 1 / /n1 - 1 \ \\ |----- | |----- n2 | || n | \ | n1 | \ (-1) | || (-1) (n + 3) (n + 2) (n + 1) | ) |-(-1) | ) ---------| n1!|| (n + 6) (n + 5) (n + 4) (n + 7) (n + 8) (n + 9)} | / | | / (n2 + 1)!| || |----- | |----- | || \n1 = 0 \ \n2 = 0 / // "A135809" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A135810" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A135811" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A135812" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A135813" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A136029" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A136046" LREtools/SearchTable: "SearchTable successful" ((n - 1) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) {-----------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 1) "A136092" LREtools/SearchTable: "SearchTable successful" 7 6 5 4 3 2 {((4 n + 113 n + 1357 n + 9014 n + 42934 n + 137915 n + 233481 n + 148014) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 6 5 4 3 2 / - 9 (n + 1) (4 n + 105 n + 1144 n + 6648 n + 22648 n + 43233 n + 34938) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) / ((n + 1) / 3 3 2 2 (n + 2) (n + 3) (n + 4) (n + 5) (n + 6))} "A136128" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) n1!| {(n + 2) (n + 1) (1/2) n!, (n + 2) (n + 1) (1/2) n! | ) ----------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A136281" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A136284" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A136304" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 3 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 3 _Z + 2 _Z - 1, index = 3) } "A136328" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { 8 binomial(n, n/2) binomial(2 n, n) (2 n + 1) 3 2 { ----------------------------------------------- n::even (2 n + 1) binomial(2 n, n) { n + 1 {----------------------------, { , 2 { 2 (n + 1) { 64 binomial(n - 1, n/2 - 1/2) binomial(2 n - 2, n - 1) n (2 n - 1) (2 n + 1) (2 n + 3) { --------------------------------------------------------------------------------------- n::odd { 3 { (n + 1) { n { 8 16 (2 n + 1) (2 n + 3) binomial(2 n, n) { ------------------------------------------ n::even { 3 2 { (n + 1) binomial(n, n/2) { } { (4 n + 4) { 2 (2 n + 1) binomial(2 n + 2, n + 1) { --------------------------------------------- n::odd { 2 2 { (n + 1) binomial(n + 1, n/2 + 1/2) "A136574" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, n, ) (n1 + 1) n1!} / ----- n1 = 0 "A136576" LREtools/SearchTable: "SearchTable successful" n n (2 I) (-LegendreP(n, I) + LegendreP(n + 1, I) I) (2 I) (-LegendreQ(n, I) + LegendreQ(n + 1, I) I) {-------------------------------------------------, -------------------------------------------------} n + 2 n + 2 "A136580" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- | n \ n1 | \ n2 | {1, (-1) , ) (-1) | ) (-(-1) (n2 + 2) (n2 + 1) n2!)|} / | / | ----- |----- | n1 = 0 \n2 = 0 / "A136591" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) n!, (- 1/2 + 1/2 I 3 ) n!} "A136658" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A137265" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A137400" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / | / 1/2\n / 1/2\n / 1/2\n | | 3 2 | | 3 2 | | 3 2 | | {|2 - ------| , |2 + ------| , |2 - ------| | \ 2 / \ 2 / \ 2 / | | \ / / 1/2\(-n2 - 1) \ n - 1 |n1 - 1 | 3 2 | 2 | ----- / 1/2\n1 / 1/2\(-n1 - 1) |----- |2 + ------| (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 13 n2 + 34)| \ | 3 2 | | 3 2 | | \ \ 2 / | ) |2 + ------| |2 - ------| | ) --------------------------------------------------------------------------------------------| / \ 2 / \ 2 / | / (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) | ----- |----- | n1 = 0 \n2 = 0 / \ | | | |} | | / "A137482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=65375.7MB, alloc=2039.5MB, time=458.24 / / n1 \ \ | |- ---- + 1/2| / 1/2 1/2 \| |n - 1 \ 2 / n1 | 2 1/2 2 || |----- 2 (-1) |HermiteH(n1 + 1, ----) - 2 (n1 + 1) HermiteH(n1, ----)|| | \ \ 2 2 /| {n! | ) ----------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A137591" LREtools/SearchTable: "SearchTable successful" (n/2 + 1/2) 1/2 1/2 1/2 2 (n + 1) (-2 LegendreP(n + 1, 2 ) + 2 LegendreP(n, 2 )) n! {- -----------------------------------------------------------------------------, n (n/2 + 1/2) 1/2 1/2 1/2 2 (n + 1) (-2 LegendreQ(n + 1, 2 ) + 2 LegendreQ(n, 2 )) n! - -----------------------------------------------------------------------------} n "A137635" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A137636" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A137638" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1/2) } "A137644" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A137697" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n (2 ) binomial(2 n, n) (-2 ) binomial(2 n, n) {------------------------, -------------------------} n + 1 n + 1 "A137720" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 binomial(2 n1, n1)| {3 , 3 | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A137729" n 2 n! (1/2) (n!) binomial(2 n, n) {----, -----------------------------} n n "A137775" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-3) | {n! (n + 9 n + 17), n! (n + 9 n + 17) | ) ---------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 9 n1 + 26) (n1 + 9 n1 + 17)| \n1 = 0 / "A137886" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 4 3 2 3 2 {(-1) n! (n + 3) (n + 2) (n + 1) ((n + 7 n + 17 n + 19 n + 9) BesselI(n, 2) + (-n - 7 n - 16 n - 13) BesselI(n - 1, 2)), n 4 3 2 3 2 (-1) n! (n + 3) (n + 2) (n + 1) ((n + 7 n + 17 n + 19 n + 9) BesselK(n, -2) + (-n - 7 n - 16 n - 13) BesselK(n - 1, -2))} "A137953" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A137954" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A137959" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A138016" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { / n1 \ / n1 \ \ | { |---- + 1| |----|! n1::even| | { \ 2 / \ 2 / | | { | |n - 1 { / n1 \ / n1 \ | |----- { |---- + 3/2| |---- + 1/2|! n1::odd | | \ { \ 2 / \ 2 / | {1, (n + 1) | ) --------------------------------------------|, | / (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / / { (-n1) n1 / n1 \ \ | { 2 (n1 + 1) (n1 + 3) binomial(n1, ----) |----|! n1::even| | { 2 \ 2 / | | { | |n - 1 { (-n1 + 1) n1 / n1 \ | |----- { 2 n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) |---- - 1/2|! n1::odd | | \ { 2 \ 2 / | (n + 1) | ) -----------------------------------------------------------------------------------|, n + 1} | / (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / "A138156" n (2 n + 1) binomial(2 n, n) (3 n + 4) {4 , ------------------------------------} (n + 1) (n + 2) "A138164" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / | | |n - 1 | / 1/2\n / 1/2\n / 1/2\n |----- | | 5 | | 5 | | 5 | | \ | {|3/2 - ----| , |3/2 + ----| , |3/2 - ----| | ) | \ 2 / \ 2 / \ 2 / | / | |----- | |n1 = 0 | | | \ \ / /{ 3 n2 n2 \\\ | |{ 4 binomial(----, ----) n2 ||| |n1 - 1 |{ 2 2 ||| |----- / 1/2\(-n2 - 1) |{ ------------------------- n2::even||| 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ (n2 + 1) (n2 + 2) ||| 2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ ||| | / \ 2 / |{ 3 n2 n2 ||| |----- |{ binomial(---- - 3/2, ---- - 1/2) (3 n2 + 1) (3 n2 - 1) ||| |n2 = 0 |{ 2 2 ||| | |{ 3/2 ------------------------------------------------------ n2::odd ||| \ \{ (n2 + 2) (n2 + 1) n2 /// / / | | \ | | | | | | |n - 1 | | / 1/2\n |----- | | | 5 | | \ | 1/2 n1 1/2 (-n1 - 1) |, |3/2 - ----| | ) |2 (3 + 5 ) (3 - 5 ) | \ 2 / | / | | |----- | | |n1 = 0 | | | | / | | | | \ \ / /{ (-n2) 3 n2 3 n2 n2 \\\\ | |{ 6 4 (3 n2 + 1) binomial(3 n2, ----) binomial(----, ----) |||| | |{ 2 2 2 |||| | |{ ------------------------------------------------------------- n2::even|||| |n1 - 1 |{ n2 |||| |----- / 1/2\(-n2 - 1) |{ (n2 + 1) (n2 + 2) binomial(n2, ----) |||| | \ | 5 | |{ 2 |||| | ) |3/2 + ----| |{ ||||} | / \ 2 / |{ (-2 n2 - 2) 3 n2 3 n2 n2 |||| |----- |{ 16 2 n2 binomial(3 n2 + 3, ---- + 3/2) binomial(---- + 3/2, ---- + 1/2) |||| |n2 = 0 |{ 2 2 2 |||| | |{ ---------------------------------------------------------------------------------- n2::odd |||| | |{ n2 |||| | |{ (n2 + 2) (3 n2 + 2) binomial(n2 + 1, ---- + 1/2) |||| \ \{ 2 //// "A138175" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 3 n1 n1 \ | { 3 binomial(----, ----) | | { 2 2 | | { ---------------------- n1::even| | { n1 + 1 | | { | | { 3 n1 n1 | | { 8 binomial(---- + 3/2, ---- + 1/2) | |n - 1 { 2 2 | |----- { ---------------------------------- n1::odd | n n | \ { 3 n1 + 1 | {(8/3) , (8/3) | ) ----------------------------------------------------|, | / (n1 + 1) | |----- (8/3) | \n1 = 0 / / { (-n1) 3 n1 3 n1 n1 \ | { 16 4 binomial(3 n1, ----) binomial(----, ----) | | { 2 2 2 | | { --------------------------------------------------- n1::even| | { n1 | | { binomial(n1, ----) (n1 + 1) | | { 2 | | { | | { (-2 n1 + 2) 3 n1 3 n1 n1 | | { 6 2 (3 n1 - 2) binomial(3 n1 - 3, ---- - 3/2) binomial(---- - 3/2, ---- - 1/2) | | { 2 2 2 | | { ----------------------------------------------------------------------------------------- n1::odd | |n - 1 { n1 | |----- { n1 (n1 + 1) binomial(n1 - 1, ---- - 1/2) | n | \ { 2 | (8/3) | ) -----------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (8/3) | \n1 = 0 / "A138240" LREtools/SearchTable: "SearchTable successful" n 4 ((n + 1) hypergeom([-1/2, -n - 1], [1], -2) - n hypergeom([-1/2, -n], [1], -2)) {----------------------------------------------------------------------------------} n "A138350" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 { 16 { 16 binomial(n, n/2) (n + 1) { 1/2 ---------------------------------- n::even { - ---------------------------- n::even { 2 2 { 2 { (n + 1) (n + 3) binomial(n, n/2) {{ (n + 2) , { } { { (4 n - 4) { 2 { 2 2 (n + 1) { 4 binomial(n + 1, n/2 + 1/2) { - --------------------------------------- n::odd { ----------------------------- n::odd { 2 2 2 { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A138351" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A138354" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A138356" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A138413" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (2 n1 + 2) | n n | \ 2 binomial(4 n1, 2 n1) (7 n1 + 1)| {(1/4) , (1/4) | ) -------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A138414" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (2 n1 + 2) | n n | \ 2 (4 n1 + 1) binomial(4 n1, 2 n1) (14 n1 + 9)| {(1/4) , (1/4) | ) -------------------------------------------------------|} | / (n1 + 1) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / "A138415" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {1, (1/2) , (1/2) | ) 2 (hypergeom([-1/2, -n1 - 1], [1], -4) - hypergeom([-1/2, -n1], [1], -4))|} | / | |----- | \n1 = 0 / "A138461" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) + 3 hypergeom([1/2, -n1], [1], 4))| {(-1) , (-3/2) , (-3/2) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / "A138462" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (n + 1) LegendreP(2 n + 2, 3) + (-17 n - 13) LegendreP(2 n, 3) (n + 1) LegendreQ(2 n + 2, 3) + (-17 n - 13) LegendreQ(2 n, 3) {--------------------------------------------------------------, --------------------------------------------------------------} (4 n + 3) n (4 n + 3) n "A138523" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) (n1 + 1) (2 n1 + 3) (2 n1 + 1) (n1!) binomial(2 n1, n1)} / ----- n1 = 0 "A138524" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) (n1!) binomial(2 n1, n1)} / ----- n1 = 0 "A138525" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) (n1 + 1) (2 n1 + 1) (n1!) binomial(2 n1, n1)} / ----- n1 = 0 "A138564" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 3 3 3 {1, ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A139149" {1, (n + 2) (n + 1) n!} "A139150" {1, (n + 3) (n + 2) (n + 1) n!} "A139151" {1, (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139152" {1, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139153" {1, (n + 3) (n + 2) (n + 1) n!} "A139154" {1, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139155" {1, (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139156" {1, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139157" {1, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139172" {1, (n + 2) (n + 1) n!} "A139173" {1, (n + 3) (n + 2) (n + 1) n!} "A139174" {1, (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139175" {1, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139176" {1, (n + 3) (n + 2) (n + 1) n!} "A139177" {1, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139183" {1, (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139184" {1, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139185" {1, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A139262" n {4 , binomial(2 n, n) (n + 1)} "A139376" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n | 5 | |5 | 3 2 n 3 2 n 3 2 n {|1/2 - ----| , |---- + 1/2| , RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } \ 2 / \ 2 / "A139379" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- / n2 \|| n n n | \ n1 (-n1 - 1) | \ | (-1) (2 n2 + 1) binomial(2 n2, n2)||| {(-1) , 2 , 2 | ) (-1) 2 | ) |- ------------------------------------|||} | / | / \ (n2 + 1) (n2 + 2) /|| |----- |----- || \n1 = 0 \n2 = 0 // "A139464" {(n + 1) n!, 2 n + 1} "A139468" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), (- 1/2 + 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), binomial(2 n, n)} "A139469" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), (- 1/2 + 1/2 I 3 ) hypergeom([1/2, -n], [1], 4), binomial(2 n, n)} "A139678" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A140526" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 {n! (n + 6 n + 13 n + 16 n + 9), /n - 1 \ |----- n1 2 2 | 4 3 2 | \ (-1) (n1 + 3 n1 + 3) | n! (n + 6 n + 13 n + 16 n + 9) | ) --------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 6 (n1 + 1) + 13 (n1 + 1) + 16 n1 + 25) (n1 + 6 n1 + 13 n1 + 16 n1 + 9)| \n1 = 0 / "A140662" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((13 n + 55 n + 54) hypergeom([1/2, -n - 1], [1], 4) - 3 (5 n + 14) (n + 1) hypergeom([1/2, -n], [1], 4)) {1, ----------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A140710" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (-n1 - 1) 2 | {2 , 2 | ) 2 (n1 + 1) n1!|} | / | |----- | \n1 = 0 / "A140980" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 \ 2 ((n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -2) - n1 hypergeom([-1/2, -n1], [1], -2)) {1, ) ---------------------------------------------------------------------------------------} / n1 ----- n1 = 0 "A141057" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A141146" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (n - 2) hypergeom([-n, -n, -n], [1, 1], -1) {-----------------------------------------------------------------------------------------------------} n + 2 "A141147" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + hypergeom([-n, -n, -n], [1, 1], -1)} "A141223" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(9/2) , (9/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (9/2) | \n1 = 0 / "A141253" memory used=66110.8MB, alloc=2039.5MB, time=463.35 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) n1 binomial(2 n1, n1)| {(n + 1) | ) --------------------------------|, n + 1} | / (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / "A141342" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 \\ |----- |----- n2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2) (LegendreP(n2 + 1, 2) - 2 LegendreP(n2, 2))|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) --------------------------------------------------||, | / | / 1/2 (n2 + 1) || |----- |----- n2 (2 + 5 ) || \n1 = 0 \n2 = 0 // /n - 1 /n1 - 1 \\ |----- |----- n2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2) (LegendreQ(n2 + 1, 2) - 2 LegendreQ(n2, 2))|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) --------------------------------------------------||} | / | / 1/2 (n2 + 1) || |----- |----- n2 (2 + 5 ) || \n1 = 0 \n2 = 0 // "A141344" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 13 | | 13 | {|4/3 - -----| , |4/3 + -----| , \ 3 / \ 3 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 n2 | 13 | ||| / 1/2\n |----- | |----- (-1) |4/3 + -----| binomial(2 n2, n2)||| | 13 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 3 / ||| |4/3 - -----| | ) |3 (4 + 13 ) (4 - 13 ) | ) ------------------------------------------------|||} \ 3 / | / | | / n2 + 1 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A141351" binomial(2 n, n) {1, ----------------} n + 1 "A141353" n binomial(2 n, n) {2 , ----------------} n + 1 "A141364" binomial(2 n, n) {1, ----------------} n + 1 "A141827" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n!) n, (n!) n | ) ------------------------|} | / 2 | |----- ((n1 + 1)!) (n1 + 1) n1| \n1 = 0 / "A141828" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 3 3 | \ 1 | {(n!) n, (n!) n | ) ------------------------|} | / 3 | |----- ((n1 + 1)!) (n1 + 1) n1| \n1 = 0 / "A141834" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 2 n1 + 5 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A142703" LREtools/SearchTable: "SearchTable successful" n {(n + 1) (-2) n! LaguerreL(n + 1, -n - 3/2, -1/2)} "A142704" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A142722" LREtools/SearchTable: "SearchTable successful" {BesselI(n + 1/2, 1), BesselK(n + 1/2, -1)} "A142970" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 1) (1/2) n! binomial(2 n, n) (n + 1), /n - 1 \ |----- / n1 \| n | \ | 2 (-1) (2 n1 + 1) binomial(2 n1, n1) n1! || (2 n + 1) (1/2) n! binomial(2 n, n) (n + 1) | ) |-----------------------------------------------------------------||} | / \(n1 + 1) (n1 + 2) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A142979" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 1) n! (2 n + 3), (n + 1) n! (2 n + 3) | ) ----------------------------------------|} | / (n1 + 2) (n1 + 1)! (2 n1 + 5) (2 n1 + 3)| |----- | \n1 = 0 / "A142980" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) (-1) n1! | {(n + 1) n! (2 n + 6 n + 5), (n + 1) n! (2 n + 6 n + 5) | ) ---------------------------------------------------------------|} | / 2 2 | |----- (n1 + 2) (n1 + 1)! (2 (n1 + 1) + 6 n1 + 11) (2 n1 + 6 n1 + 5)| \n1 = 0 / "A142981" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) n! (2 n + 3) (2 n + 6 n + 7), /n - 1 \ |----- n1 | 2 | \ (n1 + 1) (-1) n1! | (n + 1) n! (2 n + 3) (2 n + 6 n + 7) | ) -------------------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 2) (n1 + 1)! (2 n1 + 5) (2 (n1 + 1) + 6 n1 + 13) (2 n1 + 3) (2 n1 + 6 n1 + 7)| \n1 = 0 / "A142982" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {(n + 1) n! (2 n + 12 n + 34 n + 48 n + 27), (n + 1) n! (2 n + 12 n + 34 n + 48 n + 27) /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | | ) ------------------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 2) (n1 + 1)! (2 (n1 + 1) + 12 (n1 + 1) + 34 (n1 + 1) + 48 n1 + 75) (2 n1 + 12 n1 + 34 n1 + 48 n1 + 27)| \n1 = 0 / "A142983" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A142984" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) (-1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) -----------------------------|} | / 2 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1)!| \n1 = 0 / "A142985" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 2) (n + 1) n! (2 n + 8 n + 9), /n - 1 \ |----- n1 | 2 | \ (n1 + 1) (-1) n1! | (n + 2) (n + 1) n! (2 n + 8 n + 9) | ) ------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1)! (2 (n1 + 1) + 8 n1 + 17) (2 n1 + 8 n1 + 9)| \n1 = 0 / "A142986" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 2 {(n + 2) (n + 1) n! (n + 4 n + 6), /n - 1 \ |----- n1 | 2 2 | \ (n1 + 1) (-1) n1! | (n + 2) (n + 1) n! (n + 4 n + 6) | ) ----------------------------------------------------------------------|} | / 2 2 2 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1)! ((n1 + 1) + 4 n1 + 10) (n1 + 4 n1 + 6)| \n1 = 0 / "A142987" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {(n + 2) (n + 1) n! (2 n + 16 n + 58 n + 104 n + 75), (n + 2) (n + 1) n! (2 n + 16 n + 58 n + 104 n + 75) /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | | ) ------------------------------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1)! (2 (n1 + 1) + 16 (n1 + 1) + 58 (n1 + 1) + 104 n1 + 179) (2 n1 + 16 n1 + 58 n1 + 104 n1 + 75)| \n1 = 0 / "A142988" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A142989" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 3) (n + 2) (n + 1) n! (2 n + 5), (n + 3) (n + 2) (n + 1) n! (2 n + 5) | ) ----------------------------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)! (2 n1 + 7) (2 n1 + 5)| |----- | \n1 = 0 / "A142990" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 3) (n + 2) (n + 1) n! (n + 5 n + 7), /n - 1 \ |----- n1 | 2 | \ (n1 + 1) (-1) n1! | (n + 3) (n + 2) (n + 1) n! (n + 5 n + 7) | ) -----------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)! ((n1 + 1) + 5 n1 + 12) (n1 + 5 n1 + 7)| \n1 = 0 / "A142991" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 2 {(n + 3) (n + 2) (n + 1) n! (2 n + 5) (n + 5 n + 9), (n + 3) (n + 2) (n + 1) n! (2 n + 5) (n + 5 n + 9) /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | | ) ---------------------------------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)! (2 n1 + 7) ((n1 + 1) + 5 n1 + 14) (2 n1 + 5) (n1 + 5 n1 + 9)| \n1 = 0 / "A142995" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 2 | \ (n1!) | {(n!) (2 n + 1), (n!) (2 n + 1) | ) ----------------------------------|} | / 2 | |----- ((n1 + 1)!) (2 n1 + 3) (2 n1 + 1)| \n1 = 0 / "A142999" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2| 2 2 | \ (-1) (n1!) | {(n!) , (n!) | ) -------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A143003" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 | 3 2 3 2 | \ (n1!) | {(n!) (2 n + 2 n + 1), (n!) (2 n + 2 n + 1) | ) --------------------------------------------------------|} | / 3 2 2 | |----- ((n1 + 1)!) (2 (n1 + 1) + 2 n1 + 3) (2 n1 + 2 n1 + 1)| \n1 = 0 / "A143004" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 4 3 2 3 4 3 2 {(n!) (3 n + 6 n + 9 n + 6 n + 2), (n!) (3 n + 6 n + 9 n + 6 n + 2) /n - 1 \ |----- 3 | | \ (n1!) | | ) ----------------------------------------------------------------------------------------------------|} | / 3 4 3 2 4 3 2 | |----- ((n1 + 1)!) (3 (n1 + 1) + 6 (n1 + 1) + 9 (n1 + 1) + 6 n1 + 8) (3 n1 + 6 n1 + 9 n1 + 6 n1 + 2)| \n1 = 0 / "A143005" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 3 6 5 4 3 2 3 6 5 4 3 2 | \ 3 / {(n!) (10 n + 30 n + 85 n + 120 n + 121 n + 66 n + 18), (n!) (10 n + 30 n + 85 n + 120 n + 121 n + 66 n + 18) | ) (n1!) / ( | / / |----- \n1 = 0 3 6 5 4 3 2 ((n1 + 1)!) (10 (n1 + 1) + 30 (n1 + 1) + 85 (n1 + 1) + 120 (n1 + 1) + 121 (n1 + 1) + 66 n1 + 84) \ | 6 5 4 3 2 | (10 n1 + 30 n1 + 85 n1 + 120 n1 + 121 n1 + 66 n1 + 18))|} | | / "A143006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 8 7 6 5 4 3 2 3 {(n!) (35 n + 140 n + 630 n + 1400 n + 2595 n + 3020 n + 2500 n + 1200 n + 288), (n!) /n - 1 |----- 8 7 6 5 4 3 2 | \ 3 / 3 (35 n + 140 n + 630 n + 1400 n + 2595 n + 3020 n + 2500 n + 1200 n + 288) | ) (n1!) / (((n1 + 1)!) | / / |----- \n1 = 0 8 7 6 5 4 3 2 (35 (n1 + 1) + 140 (n1 + 1) + 630 (n1 + 1) + 1400 (n1 + 1) + 2595 (n1 + 1) + 3020 (n1 + 1) + 2500 (n1 + 1) + 1200 n1 + 1488) \ | 8 7 6 5 4 3 2 | (35 n1 + 140 n1 + 630 n1 + 1400 n1 + 2595 n1 + 3020 n1 + 2500 n1 + 1200 n1 + 288))|} | | / "A143013" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A143020" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 4 (3 n + 2) (8 n + 9) binomial(---, n/2) { 2 { - ------------------------------------------- n::even { (n + 1) (n + 2) (n + 3) {{ , { (2 n + 2) 3 n { 2 (9 n + 11) binomial(--- + 3/2, n/2 + 1/2) { 2 { ---------------------------------------------------- n::odd { (n + 2) (n + 3) { 3 n 3 n { 4 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) (9 n + 11) { 2 2 { ------------------------------------------------------------ n::even { binomial(n, n/2) (n + 3) (n + 2) (n + 1) { } { 3 n 3 n { 16 binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) (3 n - 2) (3 n + 2) (8 n + 9) { 2 2 { - -------------------------------------------------------------------------------------------- n::odd { binomial(n - 1, n/2 - 1/2) (n + 3) (n + 2) (n + 1) n "A143021" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 (5 n + 2) binomial(---, n/2) { 2 { 1/2 ------------------------------- n::even { n + 1 {{ , { (2 n - 2) 3 n { 3 2 (3 n - 1) (3 n + 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - --------------------------------------------------------------- n::odd { n (n + 1) { 3 n 3 n { 6 (3 n + 1) binomial(---, n/2) binomial(3 n, ---) { 2 2 { - ------------------------------------------------- n::even { (n + 1) binomial(n, n/2) { } { 3 n 3 n { (5 n + 2) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { --------------------------------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) (3 n + 2) "A143023" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 (3 n + 2) binomial(---, n/2) { 2 { 1/2 ------------------------------- n::even { (n + 1) (n + 2) {{ , { (2 n - 2) 3 n { 3 2 (3 n - 1) (3 n + 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - --------------------------------------------------------------- n::odd { n (n + 1) (n + 2) { 3 n 3 n { 12 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { - -------------------------------------------------- n::even { binomial(n, n/2) (n + 1) (n + 2) { } { 3 n 3 n { 2 binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ------------------------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) (n + 2) "A143045" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A143165" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {2 n!, (2 n + 1) (1/2) n! binomial(2 n, n), / / /n1 - 1 \ \\ | | |----- (-2 n2 - 1) n2 | || | | n1 | \ 2 (-1) (2 n2 + 1) binomial(2 n2, n2) n2!| || | |2 4 | ) -----------------------------------------------------| n1!|| |n - 1 | | / (n2 + 1)! | || |----- | |----- | || n | \ | \n2 = 0 / || (2 n + 1) (1/2) n! binomial(2 n, n) | ) |------------------------------------------------------------------------||} | / \ (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! /| |----- | \n1 = 0 / "A143167" n {4 GAMMA(n + 7/4), (2 n + 1) n! binomial(2 n, n)} "A143217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2| | \ (n1 + 1) (n1!) | {n! | ) ----------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A143330" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A143339" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" n n {(-I) , I } "A143360" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 4 binomial(n, n/2) (3 n + 4) (n + 1) { 3/2 ------------------------ n::even { ------------------------------------ n::even { (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n - 2) { 6 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 (n + 1) (3 n + 4) { ------------------------------------ n::odd { -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A143363" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A143412" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/4), (-1) BesselK(n + 1/2, -1/4)} "A143413" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n - 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((2 n - 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A143414" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n - 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((2 n - 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A143415" LREtools/SearchTable: "SearchTable successful" n n (-1) ((2 n - 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)) (-1) ((2 n - 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2)) {---------------------------------------------------------------, -----------------------------------------------------------------} (n + 1) n (n + 1) n "A143464" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 3 2 | | 3 2 | {|2 - ------| , |2 + ------| , \ 2 / \ 2 / / / / 1/2\(-n2 - 1) \\ |n - 1 |n1 - 1 | 3 2 | || / 1/2\n |----- / 1/2\n1 / 1/2\(-n1 - 1) |----- |2 + ------| binomial(2 n2, n2) (5 n2 + 4)|| | 3 2 | | \ | 3 2 | | 3 2 | | \ \ 2 / || |2 - ------| | ) |2 + ------| |2 - ------| | ) ---------------------------------------------------||} \ 2 / | / \ 2 / \ 2 / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A143546" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /3125\(n/2) { 5 |----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 1/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) { \256 / { --------------------------------------------------------------------------------- n::even { GAMMA(n/2 + 1) GAMMA(n/2 + 3/4) GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/4) {{ , { /3125\(n/2 + 1/2) 11 13 { 24 |----| GAMMA(n/2 + 7/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + --) GAMMA(n/2 + --) (2 n + 3) (n + 1) { \256 / 10 10 { ---------------------------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 3/2) GAMMA(n/2 + 5/4) GAMMA(n/2 + 7/4) (5 n + 1) (5 n + 3) { /3125\(n/2) 11 13 { 24 |----| GAMMA(n/2 + 7/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + --) GAMMA(n/2 + --) (2 n + 3) (n + 1) { \256 / 10 10 { ---------------------------------------------------------------------------------------------------- n::even { GAMMA(n/2 + 1) GAMMA(n/2 + 3/2) GAMMA(n/2 + 5/4) GAMMA(n/2 + 7/4) (5 n + 1) (5 n + 3) { } { /3125\(n/2 - 1/2) { 5 |----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) GAMMA(n/2 + 1/5) { \256 / { --------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 5/4) GAMMA(n/2 + 1/2) GAMMA(n/2 + 3/4) "A143556" memory used=66831.9MB, alloc=2039.5MB, time=468.33 memory used=67276.5MB, alloc=2039.5MB, time=472.20 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A143560" memory used=67841.9MB, alloc=2041.0MB, time=476.89 memory used=68335.7MB, alloc=2039.5MB, time=480.84 memory used=68875.3MB, alloc=2039.5MB, time=484.40 LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A143583" LREtools/SearchTable: "SearchTable successful" n {16 hypergeom([1/2, 1/2, -n], [1, 1], 1)} "A143646" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 2 13 | | 2 13 | {|7/3 - -------| , |7/3 + -------| , \ 3 / \ 3 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 2 13 | ||| / 1/2\n |----- | |----- |7/3 + -------| binomial(2 n2, n2) (5 n2 + 4)||| | 2 13 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 3 / ||| |7/3 - -------| | ) |-3 (-1) (-7 + 2 13 ) (7 + 2 13 ) | ) ------------------------------------------------------|||} \ 3 / | / | | / (n2 + 1) (n2 + 2) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A143749" LREtools/SearchTable: "SearchTable successful" n 9 ((99 n + 99) LegendreP(n + 1, 11/9) + (-161 n - 121) LegendreP(n, 11/9)) {---------------------------------------------------------------------------, n (n - 1) n 9 ((99 n + 99) LegendreQ(n + 1, 11/9) + (-161 n - 121) LegendreQ(n, 11/9)) ---------------------------------------------------------------------------} n (n - 1) "A143918" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 2) (n1 + 1)| n! | ) -----------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, -----------------------------} n n "A143926" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { //9 n \ \ { 192 ||--- + 6| hypergeom([- n/4, - n/4 - 1/2], [n/2 + 2], 4) + (-4 n - 6) hypergeom([- n/4, - n/4 + 1/2], [n/2 + 1], 4)| binomial(n, n/2) {{ \\ 2 / / { - ----------------------------------------------------------------------------------------------------------------------------------------- , { n (13 n + 18) n::even - 32 (3/2 (3 n + 7) (3 n + 5) hypergeom([- n/4 - 1/4, - n/4 - 3/4], [n/2 + 5/2], 4) 2 , + (-61/4 n - 61 n - 231/4) hypergeom([- n/4 - 1/4, - n/4 + 1/4], [n/2 + 3/2], 4)) binomial(n + 1, n/2 + 1/2)/(n (n + 1) (13 n + 31)) , n::odd { { { n { - 8 4 (3/2 (3 n + 7) (3 n + 5) hypergeom([- n/4 - 1/4, - n/4 - 3/4], [n/2 + 5/2], 4) { { 2 / 2 + (-61/4 n - 61 n - 231/4) hypergeom([- n/4 - 1/4, - n/4 + 1/4], [n/2 + 3/2], 4)) / (n (n + 1) (13 n + 31) binomial(n, n/2)) , n::even / (2 n - 2) //9 n \ \ 48 2 ||--- + 6| hypergeom([- n/4, - n/4 - 1/2], [n/2 + 2], 4) + (-4 n - 6) hypergeom([- n/4, - n/4 + 1/2], [n/2 + 1], 4)| \\ 2 / / } - ---------------------------------------------------------------------------------------------------------------------------------- , n::odd 2 n (13 n + 18) binomial(n - 1, n/2 - 1/2) "A143927" LREtools/SearchTable: "SearchTable successful" ((11 n + 9) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + (9 n + 9) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) (2 n + 1) {-------------------------------------------------------------------------------------------------------------------------------------------} (n + 1) (3 n + 4) (13 n + 9) "A143954" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) binomial(2 n1, n1)| {1, (3 n + 2) | ) -----------------------------|, 3 n + 2} | / (3 n1 + 5) (3 n1 + 2) | |----- | \n1 = 0 / "A143955" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 (n1 - 1) binomial(2 n1, n1) {1, ) ------------------------------} / (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A143990" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! BesselI(n - 1/2, 1), (-1) n! BesselK(n - 1/2, -1)} "A144085" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144086" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144087" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144097" LREtools/SearchTable: "SearchTable successful" {hypergeom([-3 n, -n + 1], [2], 2)} "A144141" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 2)} "A144142" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 3)} "A144143" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 4)} "A144165" LREtools/SearchTable: "SearchTable successful" (n + 1) ((27 n + 44) LegendreP(n + 1, 5) + (-3 n - 4) LegendreP(n, 5)) (n + 1) ((27 n + 44) LegendreQ(n + 1, 5) + (-3 n - 4) LegendreQ(n, 5)) {----------------------------------------------------------------------, ----------------------------------------------------------------------} (n + 2) (n + 3) (n + 2) (n + 3) "A144166" LREtools/SearchTable: "SearchTable successful" 2 2 (n + 1) ((147 n + 567 n + 526) LegendreP(n + 1, 5) + (-15 n - 51 n - 38) LegendreP(n, 5)) {-------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) 2 2 (n + 1) ((147 n + 567 n + 526) LegendreQ(n + 1, 5) + (-15 n - 51 n - 38) LegendreQ(n, 5)) -------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A144297" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144301" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n - 1/2, 1), (-1) BesselK(n - 1/2, -1)} "A144345" n n {3 GAMMA(n + 4/3), 3 GAMMA(n + 5/3)} "A144346" n n {3 GAMMA(n + 7/3), 3 GAMMA(n + 8/3)} "A144347" n n {5 GAMMA(n + 8/5), 5 GAMMA(n + 9/5)} "A144349" n n {6 GAMMA(n + 5/3), 6 GAMMA(n + 11/6)} "A144416" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144422" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144495" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ n1 + 3 | {n! (n + 3 n + 1), n! (n + 3 n + 1) | ) -------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 3 n1 + 4) (n1 + 3 n1 + 1)| \n1 = 0 / "A144496" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {n! (n + 10 n + 33 n + 44 n + 21), n! (n + 10 n + 33 n + 44 n + 21) /n - 1 \ |----- 2 | | \ n1 + 8 n1 + 14 | | ) -----------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 10 (n1 + 1) + 33 (n1 + 1) + 44 n1 + 65) (n1 + 10 n1 + 33 n1 + 44 n1 + 21)| \n1 = 0 / "A144497" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ {n! (n + 21 n + 172 n + 705 n + 1522 n + 1623 n + 653), n! (n + 21 n + 172 n + 705 n + 1522 n + 1623 n + 653) | ) ( | / |----- \n1 = 0 3 2 / 6 5 4 3 2 n1 + 15 n1 + 68 n1 + 91) / ((n1 + 1)! ((n1 + 1) + 21 (n1 + 1) + 172 (n1 + 1) + 705 (n1 + 1) + 1522 (n1 + 1) + 1623 n1 + 2276) / \ | 6 5 4 3 2 | (n1 + 21 n1 + 172 n1 + 705 n1 + 1522 n1 + 1623 n1 + 653))|} | | / "A144498" LREtools/SearchTable: "SearchTable successful" n n {(-1) (2 n BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) (2 n BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A144499" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((4 n + 2 n + 1) BesselI(n + 1/2, 1) - 2 n BesselI(n - 1/2, 1)), (-1) ((4 n + 2 n + 1) BesselK(n + 1/2, -1) - 2 n BesselK(n - 1/2, -1))} "A144500" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) (4 n (2 n + 3 n + 2) BesselI(n + 1/2, 1) - (2 n + 1) BesselI(n - 1/2, 1)), n 2 2 (-1) (4 n (2 n + 3 n + 2) BesselK(n + 1/2, -1) - (2 n + 1) BesselK(n - 1/2, -1))} "A144501" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144503" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" memory used=69551.4MB, alloc=2295.5MB, time=489.21 LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 1), (-1) BesselK(n, -1)} "A144506" LREtools/SearchTable: "SearchTable successful" n 3 2 2 {(-1) ((4 n - 6 n + 7 n - 1) BesselI(n - 1/2, 1) + (2 n - 2 n + 2) BesselI(n + 1/2, 1)), n 3 2 2 (-1) ((4 n - 6 n + 7 n - 1) BesselK(n - 1/2, -1) + (2 n - 2 n + 2) BesselK(n + 1/2, -1))} "A144508" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A144509" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A144511" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 9)|| (2 n + 1) binomial(2 n, n) {(-1) , (-1) | ) |- -----------------------------------------------------------------------------||, --------------------------, 2 n + 3} | / | 2 || n + 1 |----- \ (2 n1 + 3) (n1 + 1) /| \n1 = 0 / "A144513" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) (-n BesselI(n - 1/2, 1) + (2 n + n + 1) BesselI(n + 1/2, 1)), (-1) (-n BesselK(n - 1/2, -1) + (2 n + n + 1) BesselK(n + 1/2, -1))} "A144514" LREtools/SearchTable: "SearchTable successful" n 2 3 2 {(-1) ((-2 n - 2 n - 2) BesselI(n - 1/2, 1) + (4 n + 6 n + 7 n + 1) BesselI(n + 1/2, 1)), n 2 3 2 (-1) ((-2 n - 2 n - 2) BesselK(n - 1/2, -1) + (4 n + 6 n + 7 n + 1) BesselK(n + 1/2, -1))} "A144632" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 4 3 2 | \ 3 2 {(16 n + 64 n + 88 n + 48 n + 11) | ) ((12 n1 + 43 n1 + 47 n1 + 13) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) | / |----- \n1 = 0 2 / + (n1 + 1) hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1)) (2 n1 + 3) (n1 + 1) / ( / \ | 4 3 2 4 3 2 | 2 (16 (n1 + 1) + 64 (n1 + 1) + 88 (n1 + 1) + 48 n1 + 59) (16 n1 + 64 n1 + 88 n1 + 48 n1 + 11))|, 8 n + 16 n + 7, | | / 4 3 2 16 n + 64 n + 88 n + 48 n + 11} "A144635" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 5 (2 n1 + 1) binomial(2 n1, n1)| {5 , 5 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A144636" LREtools/SearchTable: "SearchTable successful" n {(-6/5 + 3/5 I) ((15 n + 20) hypergeom([5/6, - 2/3 - n], [5/3], 2/5 - 4/5 I) - (12 + 6 I) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], 2/5 - 4/5 I)) GAMMA(n - 1/3) GAMMA(n + 1/3)/GAMMA(n + 2/3)} "A144637" LREtools/SearchTable: "SearchTable successful" n {(-6/5 + 3/5 I) ( 2 5 (3 n + 4) (9 n - 4) hypergeom([5/6, - 2/3 - n], [5/3], 2/5 - 4/5 I) - (2 + I) (63 n + 6 n - 19) hypergeom([5/6, 1/3 - n], [5/3], 2/5 - 4/5 I)) GAMMA(n - 2/3) GAMMA(n - 1/3)/GAMMA(n + 2/3)} "A144638" LREtools/SearchTable: "SearchTable successful" n {(2 + I) (-6/5 + 3/5 I) ((30 n + 40) hypergeom([5/6, - 2/3 - n], [5/3], 2/5 - 4/5 I) - (2 + I) (9 n + 7) hypergeom([5/6, 1/3 - n], [5/3], 2/5 - 4/5 I)) GAMMA(n - 2/3)} "A144639" LREtools/SearchTable: "SearchTable successful" n 2 {(-6/5 + 3/5 I) (5 (3 n + 4) (3 n - 2) hypergeom([5/6, - 2/3 - n], [5/3], 2/5 - 4/5 I) 3 2 + (2 + I) (9 n - 63 n + 19 n + 17) hypergeom([5/6, 1/3 - n], [5/3], 2/5 - 4/5 I)) GAMMA(n - 4/3) GAMMA(n - 2/3)/GAMMA(n + 2/3)} "A144647" LREtools/SearchTable: "SearchTable successful" n n {(-1) (-2 BesselI(n - 1/2, 1) + (2 n - 1) BesselI(n + 1/2, 1)), (-1) (-2 BesselK(n - 1/2, -1) + (2 n - 1) BesselK(n + 1/2, -1))} "A144657" (2 n + 1) binomial(2 n, n) {--------------------------, n + 1} n + 1 "A144658" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 9)|| (2 n + 1) binomial(2 n, n) {(-1) , (-1) | ) |- -----------------------------------------------------------------------------||, --------------------------, 2 n + 3} | / | 2 || n + 1 |----- \ (2 n1 + 3) (n1 + 1) /| \n1 = 0 / "A144659" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) (4 BesselI(n - 1/2, 1) + (n - 3 n + 4) BesselI(n + 1/2, 1)), (-1) (4 BesselK(n - 1/2, -1) + (n - 3 n + 4) BesselK(n + 1/2, -1))} "A144660" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 9)|| {1, (-1) , (-1) | ) |- -----------------------------------------------------------------------------||} | / | 2 || |----- \ (2 n1 + 3) (n1 + 1) /| \n1 = 0 / "A144661" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 /n1 - 1 |----- | |----- |----- n n | \ (2 n1 + 1) binomial(2 n1, n1)| n | \ | \ {1, (-1/2) , (-1/2) | ) -----------------------------|, (-1/2) | ) (2 n1 + 1) binomial(2 n1, n1) | ) (4 n2 + 3) (2 n2 + 1) (4 n2 + 1) | / (n1 + 1) | | / | / |----- (n1 + 1) (-1/2) | |----- |----- \n1 = 0 / \n1 = 0 \n2 = 0 2 5 4 3 2 / 2 3 binomial(2 n2, n2) binomial(4 n2, 2 n2) (10773 n2 + 87840 n2 + 283395 n2 + 452444 n2 + 357612 n2 + 112000) / ((n2 + 2) (n2 + 1) / \ \ | | | / (n1 + 1) | (3 n2 + 7) (3 n2 + 4) (3 n2 + 5) binomial(2 n2 + 2, n2 + 1))| / ((n1 + 1) (-1/2) )|} | / | | | / / "A144685" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n + 3) { 1/2 ------------------------ n::even n { binomial(n, n/2) (4 n + 4) n::even { (n + 1) binomial(n, n/2) {2 (n + 6), { , { } { (2 n + 3) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A144700" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A144768" 9 {n , n!} "A144904" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A144905" LREtools/SearchTable: "SearchTable successful" n n (-1) n! BesselI(n + 1/2, 1) (-1) n! BesselK(n + 1/2, -1) {----------------------------, -----------------------------} n n "A144952" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 (2 n1 + 3) | |{ 1/2 --------------------------- n1::even| n - 1 /{ n1 \ n - 1 |{ n1 | ----- |{ binomial(n1, ----) (4 n1 + 4) n1::even| ----- |{ (n1 + 1) binomial(n1, ----) | n \ |{ 2 | \ |{ 2 | {1, n, 2 (n + 2), ) |{ |, ) |{ |} / |{ n1 | / |{ (2 n1 - 2) | ----- |{ (2 n1 + 3) binomial(n1 + 1, ---- + 1/2) n1::odd | ----- |{ 2 2 (n1 + 1) | n1 = 0 \{ 2 / n1 = 0 |{ ------------------------------- n1::odd | |{ n1 | |{ n1 binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A145138" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 6 _Z + 5 _Z - 1 "A145219" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 - 1) (n1 + 2)| {(n + 1) n!, (n + 1) n! | ) ------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A145220" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 - 1) (n1 + 2)| {(n + 1) (n + 2) n!, (n + 1) (n + 2) n! | ) ------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A145221" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (-1) | {n! | ) ------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A145222" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A145223" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) n1 (-1) | {(n + 1) (n + 2) n!, n! | ) ------------------| (n + 1) (n + 2)} | / (n1 + 1)! | |----- | \n1 = 0 / "A145495" 2 n n binomial(3 n, n) binomial(6 n, 3 n) n (-1) binomial(3 n, n) binomial(6 n, 3 n) {--------------------------------------, -------------------------------------------} 6 n - 1 6 n - 1 "A145562" n {4 GAMMA(n + 9/4), (2 n + 3) (2 n + 1) n! binomial(2 n, n)} "A145844" memory used=70262.6MB, alloc=2327.5MB, time=494.05 LREtools/SearchTable: "SearchTable successful" {((2 n + 1) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1) + (-8 n - 8) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)) binomial(2 n, n)/((n + 1) (n + 2))} "A145845" LREtools/SearchTable: "SearchTable successful" {((2 n + 3) (2 n + 1) (n + 4) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1) 2 / 3 - 16 (n + 3) (n + 1) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)) binomial(2 n, n) / ((n + 1) (n + 2) )} / "A145847" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A145867" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A145886" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| 2 2 | \ (n1 + 1) (-1) | {(n + 1) n!, (n + 1) n! | ) ---------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A145887" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| 2 2 | \ (n1 + 3) n1 (-1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A147615" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A147855" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ (4 n1 + 1) binomial(4 n1, n1) (415 n1 + 509 n1 + 150)| {(-1/8) , (-1/8) | ) ------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (3 n1 + 2) (3 n1 + 1) (-1/8) | \n1 = 0 / "A148092" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A148162" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A148554" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 ((-9 n - 3) LegendreP(n, 1/15 I 15 ) + 15 (3 n + 5) LegendreP(n + 1, 1/15 I 15 ) I) {---------------------------------------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 ((-9 n - 3) LegendreQ(n, 1/15 I 15 ) + 15 (3 n + 5) LegendreQ(n + 1, 1/15 I 15 ) I) ---------------------------------------------------------------------------------------------------------------} n + 2 "A148703" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A149187" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A150500" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A151019" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {5 , 5 | ) 5 |{ \ 2 / ||, | / |{ 96 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 6 binomial(n1, ----) || 5 | ) 5 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A151087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" (4 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n + 2 "A151090" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-2/5 I) (-LegendreP(n1, I) + LegendreP(n1 + 1, I) I)|| {5 , 5 | ) |1/5 -------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ | (-2/5 I) (-LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I)|| 5 | ) |1/5 -------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A151093" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A151162" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 1/2| || {4 , 4 | ) 2 |{ \ 2 / ||, | / |{ 48 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-2 n1 - 2) |{ 2 3 binomial(n1, ----) || 4 | ) 2 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A151170" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A151241" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {5 , 5 | ) 5 |{ \ 2 / ||, | / |{ 96 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 6 binomial(n1, ----) || 5 | ) 5 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A151251" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-2/5 I) (-LegendreP(n1, I) + LegendreP(n1 + 1, I) I)|| {5 , 5 | ) |1/5 -------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ | (-2/5 I) (-LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I)|| 5 | ) |1/5 -------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A151253" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n n | \ | 11 (-1/5 I) (11 LegendreP(n1 + 1, 1/11 I 11 ) I - 11 LegendreP(n1, 1/11 I 11 ))|| {5 , 5 | ) |1/5 ----------------------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n | \ | 11 (-1/5 I) (11 LegendreQ(n1 + 1, 1/11 I 11 ) I - 11 LegendreQ(n1, 1/11 I 11 ))|| 5 | ) |1/5 ----------------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A151254" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 /{ 0 n1::even\\ |----- |{ || n n | \ (-n1 - 1) |{ (3 n1 - 3) || {5 , 5 | ) 5 |{ 2 ||, | / |{ ---------------------------------------- n1::odd || |----- |{ n1 || |n1 = 0 |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // /n - 1 /{ n1 n1 \\ |----- |{ 2 2 binomial(n1, ----) || n | \ (-n1 - 1) |{ 2 || 5 | ) 5 |{ ------------------------ n1::even||} | / |{ n1 + 2 || |----- |{ || \n1 = 0 \{ 0 n1::odd // "A151265" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {3 } "A151281" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ /5 n1 \ || n n | \ (-n1 - 1) |{ |---- - 5/2| || {3 , 3 | ) 3 |{ \ 2 / ||, | / |{ 2 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 2 binomial(n1, ----) || 3 | ) 3 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A151282" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n n | \ | 7 (-1/4 I) (7 LegendreP(n1 + 1, 1/7 I 7 ) I - 7 LegendreP(n1, 1/7 I 7 ))|| {4 , 4 | ) |1/4 ---------------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n | \ | 7 (-1/4 I) (7 LegendreQ(n1 + 1, 1/7 I 7 ) I - 7 LegendreQ(n1, 1/7 I 7 ))|| 4 | ) |1/4 ---------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A151292" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n n | \ | 11 (-1/5 I) (11 LegendreP(n1 + 1, 1/11 I 11 ) I - 11 LegendreP(n1, 1/11 I 11 ))|| {5 , 5 | ) |1/5 ----------------------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n | \ | 11 (-1/5 I) (11 LegendreQ(n1 + 1, 1/11 I 11 ) I - 11 LegendreQ(n1, 1/11 I 11 ))|| 5 | ) |1/5 ----------------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A151312" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" LREtools/SolveLRE: "Solutions may be linearly dependent" { 48 ((n/2 + 1) hypergeom([1/2, -n - 2], [1], 4) - 1/2 n hypergeom([1/2, -n], [1], 4)) binomial(n, n/2) { ----------------------------------------------------------------------------------------------------- n::even { 2 n + 3 {{ , { 4 ((n + 2) hypergeom([1/2, -n - 3], [1], 4) + (-n - 4) hypergeom([1/2, -n - 1], [1], 4)) binomial(n + 1, n/2 + 1/2) (n + 3) { --------------------------------------------------------------------------------------------------------------------------- n::odd { (2 n + 5) (n + 1) { n { 4 4 (n + 3) ((n + 2) hypergeom([1/2, -n - 3], [1], 4) + (-n - 4) hypergeom([1/2, -n - 1], [1], 4)) { --------------------------------------------------------------------------------------------------- n::even { 2 { (n + 1) (2 n + 5) binomial(n, n/2) } { { (2 n - 2) { 48 2 ((n/2 + 1) hypergeom([1/2, -n - 2], [1], 4) - 1/2 n hypergeom([1/2, -n], [1], 4)) { ----------------------------------------------------------------------------------------------- n::odd { n (2 n + 3) binomial(n - 1, n/2 - 1/2) "A151318" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 15 ) 15 (15 LegendreP(n + 1, 1/15 I 15 ) I - 3 LegendreP(n, 1/15 I 15 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 15 ) 15 (15 LegendreQ(n + 1, 1/15 I 15 ) I - 3 LegendreQ(n, 1/15 I 15 ))} "A151323" LREtools/SearchTable: "SearchTable successful" n {(-2) (hypergeom([1/4, -n - 1], [1], 4) - hypergeom([1/4, -n], [1], 4))} "A151341" LREtools/SearchTable: "SearchTable successful" n (-1) binomial(2 n, n) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------} (n + 1) (n + 2) "A151362" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" binomial(2 n, n) (hypergeom([1/2, -2 n - 2], [1], 4) + 3 hypergeom([1/2, -2 n], [1], 4)) {----------------------------------------------------------------------------------------} (4 n + 3) (n + 1) "A151366" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A151368" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A151372" memory used=70967.0MB, alloc=2327.5MB, time=499.05 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A151375" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (n + 5) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------} (n + 3) (n + 2) "A151379" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { 4 binomial(2 n, n/2) binomial(---, n/2) { 2 { --------------------------------------- n::even { n + 1 {{ , { 3 n { 16 binomial(2 n - 2, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) (2 n - 1) { 2 { ------------------------------------------------------------------------ n::odd { 2 { (n + 1) { n { 4 4 binomial(2 n, n) { ------------------------- n::even { 2 { (n + 1) binomial(n, n/2) } { { (2 n + 2) { 2 binomial(2 n + 2, n + 1) { -------------------------------------------- n::odd { (n + 1) (2 n + 1) binomial(n + 1, n/2 + 1/2) "A151380" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n1 \ (-1) ((n1 + 1) hypergeom([1/2, -n1 - 1], [1], 4) + (n1 + 5) hypergeom([1/2, -n1], [1], 4)) {1, ) --------------------------------------------------------------------------------------------} / (n1 + 3) (n1 + 2) ----- n1 = 0 "A151388" n n 16 GAMMA(n + 1/2) GAMMA(n + 5/6) 16 GAMMA(n + 1/2) GAMMA(n + 7/6) {---------------------------------, ---------------------------------} GAMMA(n + 2) GAMMA(n + 5/3) GAMMA(n + 2) GAMMA(n + 4/3) "A151410" LREtools/SearchTable: "SearchTable successful" n (-1) binomial(2 n, n) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n + 1 "A151464" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A151483" LREtools/SearchTable: "SearchTable successful" n (-2) (n hypergeom([1/2, -n - 1], [1], 4) + (3 n + 12) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------} (n + 3) (n + 2) "A151881" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| /n - 1 \ |n - 1 | / (n2 + 2) (n2 + 1)!|| |----- | |----- |----- || | \ (n1 + 1) n1! | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ----------------------------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A151882" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- | | \ (n1 + 1) (-1) n1!| | \ (n1 + 1) n1! (n1 + 4)| {(n + 1) n!, (n + 1) n! | ) -------------------|, (n + 1) n! | ) ---------------------|} | / 2 | | / 2 | |----- (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)! | \n1 = 0 / \n1 = 0 / "A151883" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" /n - 1 \ /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | |----- | | \ (n1 + 1) (-I) n1!| | \ (n1 + 1) I n1! | | \ (n1 + 1) n1! (n1 + 4)| {(n + 1) n!, (n + 1) n! | ) -------------------|, (n + 1) n! | ) -------------------|, (n + 1) n! | ) ---------------------|, | / 2 | | / 2 | | / 2 | |----- (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)! | \n1 = 0 / \n1 = 0 / \n1 = 0 / /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! (n1 + 4)| (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! | \n1 = 0 / "A151884" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" /n - 1 \ /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | |----- | | \ (n1 + 1) (-I) n1!| | \ (n1 + 1) I n1! | | \ (n1 + 1) n1! (n1 + 4)| {(n + 1) n!, (n + 1) n! | ) -------------------|, (n + 1) n! | ) -------------------|, (n + 1) n! | ) ---------------------|, | / 2 | | / 2 | | / 2 | |----- (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)!| |----- (n1 + 2) (n1 + 1)! | \n1 = 0 / \n1 = 0 / \n1 = 0 / /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! n1| (n + 1) n! | ) ----------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! | \n1 = 0 / "A152059" LREtools/SearchTable: "SearchTable successful" {n! binomial(2 n, n) hypergeom([-n], [n], -1)} "A152171" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A152172" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A152173" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A152225" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A152254" LREtools/SearchTable: "SearchTable successful" n n {2 LegendreP(n, 3), 2 LegendreQ(n, 3)} "A152297" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A152548" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {{ , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A152600" LREtools/SearchTable: "SearchTable successful" n n 2 ((4 n + 4) LegendreP(n + 1, 4) + (-31 n - 16) LegendreP(n, 4)) 2 ((4 n + 4) LegendreQ(n + 1, 4) + (-31 n - 16) LegendreQ(n, 4)) {-----------------------------------------------------------------, -----------------------------------------------------------------} (n - 1) n (n - 1) n "A152601" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 4) - 4 LegendreP(n, 4)) 2 (LegendreQ(n + 1, 4) - 4 LegendreQ(n, 4)) {--------------------------------------------, --------------------------------------------} n n "A152661" n 2 {(-1) n!, n! (2 n + 4 n + 1)} "A152663" n (n + 1) (-1) n! (n + 2) (n + 1) n! {----------------, ------------------} n + 3 n + 3 "A152668" n 2 {(n + 1) (-1) n!, (n + 1) n! (2 n + 12 n + 13)} "A152681" LREtools/SearchTable: "SearchTable successful" n n (-3) (3 LegendreP(n + 1, 1/3) - LegendreP(n, 1/3)) (-3) (3 LegendreQ(n + 1, 1/3) - LegendreQ(n, 1/3)) {---------------------------------------------------, ---------------------------------------------------} n n "A152689" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1! | 2 {(n - 3 n + 1) | ) ---------------------------------------|, n - 1, n - 3 n + 1} | / 2 2 | |----- ((n1 + 1) - 3 n1 - 2) (n1 - 3 n1 + 1)| \n1 = 0 / "A152873" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { (-n) 2 2 2 { 1/8 ((n/2)!) (n + 4) (n + 2) n::even { 2 4 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) (n + 5) n::even {{ , { } { 2 { (-2 n + 2) 2 2 2 2 { 1/4 ((n/2 + 1/2)!) (n + 5) (n + 3) n::odd { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 4) n::odd "A152875" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { (-n) 2 2 2 { 1/4 ((n/2)!) (n + 2) n::even { 2 4 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) n::even {{ , { } { 2 { (-2 n + 2) 2 2 2 2 { (n/2 + 3/2) ((n/2 + 1/2)!) n::odd { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd "A152887" 2 n {n! (2 n + 6 n + 3), (-1) n! (2 n + 3)} "A153008" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n binomial(2 n, n) (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {----------------, -------------------------------------------------------------------------} n + 1 n + 2 "A153229" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 | {(-1) , (-1) | ) (-(-1) n1!)|} | / | |----- | \n1 = 0 / "A153232" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A153293" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A153334" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even n { n binomial(n, n/2) n::even { binomial(n, n/2) {2 (n + 2), { , { } { binomial(n + 1, n/2 + 1/2) (n/2 + 1/2) n::odd { (2 n - 2) { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A153335" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (2 n + 2) n::even { binomial(n, n/2) {2 (n + 2), { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A153336" n {4 (2 n + 3), (2 n + 1) binomial(2 n, n)} "A153337" n {n binomial(2 n, n), 4 (n + 1)} "A153338" n {4 (n + 1), (2 n + 1) binomial(2 n, n)} "A153760" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 7 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {1} "A154028" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (-n) {(n + 2) n, (-1) (n + 2) n, { , { 2 binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2)! n::odd { { 0 n::odd "A154029" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (n/2) {(n + 4) n, (-1) (n + 4) n, { , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A154030" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { n {(n + 4) n, (-1) (n + 4) n, { , { 2 (n/2)! n::even} { n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { { 0 n::odd "A154225" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (-n) 2 2 {(n + 2) (n + 1) n, (n + 2) (n + 1) n (-1) , { , { 4 binomial(n, n/2) ((n/2)!) n::even} { 2 { { ((n/2 - 1/2)!) n::odd { 0 n::odd "A154226" memory used=71667.5MB, alloc=2327.5MB, time=504.17 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 2 2 n 2 2 { 0 n::even { (-n) 3 3 {(n + 2) n , (-1) (n + 2) n , { , { 8 binomial(n, n/2) ((n/2)!) n::even} { 3 { { ((n/2 - 1/2)!) n::odd { 0 n::odd "A154232" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even {(n - 4) (n + 4) n, (-1) (n - 4) (n + 4) n, { , { 2 2 { GAMMA(n/2 - 1/2 - RootOf(_Z + _Z - 1)) GAMMA(n/2 + 1/2 + RootOf(_Z + _Z - 1)) n::odd { 2 2 { GAMMA(n/2 - 1/2 RootOf(_Z - 5)) GAMMA(n/2 + 1/2 RootOf(_Z - 5)) n::even} { { 0 n::odd "A154623" LREtools/SearchTable: "SearchTable successful" n 2 5 ((n + 1) (8 n - 15) hypergeom([-3/2, -n - 1], [1], -4/5) + (-8 n - 5 n - 3) hypergeom([-3/2, -n], [1], -4/5)) {-----------------------------------------------------------------------------------------------------------------} n "A154825" LREtools/SearchTable: "SearchTable successful" n n 3 (3 LegendreP(n + 1, 1/3) - LegendreP(n, 1/3)) 3 (3 LegendreQ(n + 1, 1/3) - LegendreQ(n, 1/3)) {------------------------------------------------, ------------------------------------------------} n n "A155051" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | /{ n1 \ |{ ------------------------------------ n1::even| |{ 2 binomial(n1, ----) | n - 1 |{ n1 | n - 1 |{ 2 | ----- |{ (n1 + 1) (n1 + 3) binomial(n1, ----) | ----- |{ -------------------- n1::even| \ |{ 2 | \ |{ n1 + 2 | {1, ) |{ |, ) |{ |} / |{ (2 n1 - 2) | / |{ n1 | ----- |{ 2 | ----- |{ 2 binomial(n1 + 1, ---- + 1/2) | n1 = 0 |{ ---------------------------------------- n1::odd | n1 = 0 |{ 2 | |{ n1 | |{ ------------------------------ n1::odd | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | \{ n1 + 3 / \{ 2 / "A155069" LREtools/SearchTable: "SearchTable successful" (3 n + 3) LegendreP(n + 1, 3) + (-17 n - 9) LegendreP(n, 3) (3 n + 3) LegendreQ(n + 1, 3) + (-17 n - 9) LegendreQ(n, 3) {-----------------------------------------------------------, -----------------------------------------------------------} (n - 1) n (n - 1) n "A155084" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 4 3 | | 4 3 | {|2 - ------| , |2 + ------| , \ 3 / \ 3 / / / / 1/2\(-n2 - 1) \\ |n - 1 |n1 - 1 | 4 3 | || / 1/2\n |----- / 1/2\n1 / 1/2\(-n1 - 1) |----- |2 + ------| (2 n2 + 1) binomial(2 n2, n2)|| | 4 3 | | \ | 4 3 | | 4 3 | | \ \ 3 / || |2 - ------| | ) |2 + ------| |2 - ------| | ) ---------------------------------------------------||} \ 3 / | / \ 3 / \ 3 / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A155159" {1, n! n} "A155160" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { (n/2) {{ /3 n \ , { (n + 3) (n + 1) 2 (n/2)! binomial(n, n/2) n::even} { |--- - 3/2| { { \ 2 / { 0 n::odd { 1/4 2 (n + 1) (n + 3) (n/2 - 1/2)! n::odd "A155519" n 2 (-1) n! n! (2 n + 4 n + 1) {--------, -------------------} n n "A155521" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 + 1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A155587" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) {1, ) ------------------} / n1 + 1 ----- n1 = 0 "A155857" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n - 1) BesselI(n, 2) - BesselI(n - 1, 2)), (-1) ((n - 1) BesselK(n, -2) - BesselK(n - 1, -2))} "A155862" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n / 1/2\n | 3 17 | | 3 17 | | 3 17 | {|13/4 - -------| , |13/4 + -------| , |13/4 - -------| \ 4 / \ 4 / \ 4 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 3 17 | ||| |----- | |----- |13/4 + -------| (3 LegendreP(n2 + 1, 3) - LegendreP(n2, 3))||| | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 4 / ||| | ) |4 (13 + 3 17 ) (13 - 3 17 ) | ) ---------------------------------------------------------------------|||, | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// / 1/2\n | 3 17 | |13/4 - -------| \ 4 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 3 17 | ||| |----- | |----- |13/4 + -------| (3 LegendreQ(n2 + 1, 3) - LegendreQ(n2, 3))||| | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 4 / ||| | ) |4 (13 + 3 17 ) (13 - 3 17 ) | ) ---------------------------------------------------------------------|||} | / | | / n2 + 2 ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A155867" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A156016" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-I 3 ) 3 (3 LegendreP(n + 1, 3 I) I + 3 LegendreP(n, 3 I)) {-----------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-I 3 ) 3 (3 LegendreQ(n + 1, 3 I) I + 3 LegendreQ(n, 3 I)) -----------------------------------------------------------------------------} n "A156017" LREtools/SearchTable: "SearchTable successful" n n 2 (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) 2 (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {--------------------------------------------, --------------------------------------------} n n "A156106" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n 1/2 n {(9/4 - 3/4 I 15 ) , (9/4 + 3/4 I 15 ) , (9/4 - 3/4 I 15 ) /n - 1 /n1 - 1 \\ |----- |----- n2 1/2 (-n2 - 1) || | \ 1/2 n1 1/2 (-n1 - 1) | \ 2 (9/4 + 3/4 I 15 ) (3 n2 + 1) binomial(3 n2, n2)|| | ) (9/4 + 3/4 I 15 ) (9/4 - 3/4 I 15 ) | ) --------------------------------------------------------------||} | / | / (2 n2 + 1) (2 n2 + 3) || |----- |----- || \n1 = 0 \n2 = 0 // "A156195" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {6 , 6 | ) 6 |{ \ 2 / ||, | / |{ 80 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 5 binomial(n1, ----) || 6 | ) 6 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A156361" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 1/2| || {7 , 7 | ) 7 |{ \ 2 / ||, | / |{ 96 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-n1 - 1) |{ 2 6 binomial(n1, ----) || 7 | ) 7 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A156362" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-3 n1 - 3) |{ |---- - 1/2| || {8 , 8 | ) 2 |{ \ 2 / ||, | / |{ 112 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ (n1 + 2) n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-3 n1 - 3) |{ 2 7 binomial(n1, ----) || 8 | ) 2 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A156554" LREtools/SearchTable: "SearchTable successful" 2 3 2 {((2 n + 1) (n + 1) hypergeom([-2 n - 2, 2 n + 3, -n - 1], [1, 1], 1) + (-58 n - 93 n - 47 n - 7) hypergeom([-2 n, -n, 2 n + 1], [1, 1], 1)) / 2 2 (2 n + 1) / ((48 n + 66 n + 23) n )} / "A156566" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ /7 n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 7/2| || {9 , 9 | ) 3 |{ \ 2 / ||, | / |{ 2 || |----- |{ ---------------------------------------- n1::odd || |n1 = 0 |{ n1 || | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ /3 n1\ \\ |n - 1 |{ |----| || |----- |{ \ 2 / n1 || n | \ (-2 n1 - 2) |{ 2 2 binomial(n1, ----) || 9 | ) 3 |{ 2 ||} | / |{ ---------------------------- n1::even|| |----- |{ n1 + 2 || |n1 = 0 |{ || \ \{ 0 n1::odd // "A156577" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 /{ 0 n1::even\\ |----- |{ || n n | \ (-n1 - 1) |{ (n1 - 1) || {10 , 10 | ) 10 |{ 12 ||, | / |{ ---------------------------------------- n1::odd || |----- |{ n1 || |n1 = 0 |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // /n - 1 /{ n1 n1 \\ |----- |{ 2 3 binomial(n1, ----) || n | \ (-n1 - 1) |{ 2 || 10 | ) 10 |{ ------------------------ n1::even||} | / |{ n1 + 2 || |----- |{ || \n1 = 0 \{ 0 n1::odd // "A156626" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -1)} "A156886" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, 3 n + 1], [1], -1)} "A156887" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, 4 n + 1], [1], -1)} "A156894" LREtools/SearchTable: "SearchTable successful" 2 4 (n + 1) (2 n + 1) hypergeom([2 n + 3, -n - 1], [1], -1) + (-95 n - 87 n - 16) hypergeom([-n, 2 n + 1], [1], -1) {------------------------------------------------------------------------------------------------------------------} (17 n + 11) n "A156906" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ 0 n1::even\\\ |n - 1 | |{ ||| |----- | |{ / n1 \ ||| n n | \ | n1 |{ |---- - 1/2| ||| {(-1) , (-1) | ) |-(-1) |{ \ 2 / n1 |||, | / | |{ 2 (-1) binomial(n1 - 1, ---- - 1/2) ||| |----- | |{ 2 ||| |n1 = 0 | |{ ----------------------------------------------- n1::odd ||| \ \ \{ n1 + 1 /// / / /{ / n1 \ \\\ |n - 1 | |{ |----| ||| |----- | |{ \ 2 / ||| n | \ | n1 |{ 2 (-16) ||| (-1) | ) |-(-1) |{ ------------------------------ n1::even|||} | / | |{ n1 ||| |----- | |{ (n1 + 1) n1 binomial(n1, ----) ||| |n1 = 0 | |{ 2 ||| | | |{ ||| \ \ \{ 0 n1::odd /// "A157002" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A157003" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A157004" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A157021" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A157100" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A157125" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A157127" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A157143" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A157328" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 4 binomial(2 n1, n1)| {(-2) , (-2) | ) ----------------------|} | / (n1 + 1) | |----- (n1 + 1) (-2) | \n1 = 0 / "A157418" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (2 I) ((n1 - 4) LegendreP(n1, I) + 3 I (n1 + 4) LegendreP(n1 + 1, I)) I| {(-1) (n + 5), (-1) (n + 5) | ) -------------------------------------------------------------------------|, | / (n1 + 2) (n1 + 5) (n1 + 6) | |----- | \n1 = 0 / /n - 1 \ |----- n1 | n | \ (2 I) ((n1 - 4) LegendreQ(n1, I) + 3 I (n1 + 4) LegendreQ(n1 + 1, I)) I| (-1) (n + 5) | ) -------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 5) (n1 + 6) | |----- | \n1 = 0 / "A157674" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { 0 n2::even\\ | | { || | | { / n2 \ || | | { |---- - 1/2| || | | { \ 2 / n2 || | | { 2 (-1) binomial(n2 - 1, ---- - 1/2) || |n - 1 |n1 - 1 { 2 || |----- |----- { ----------------------------------------------- n2::odd || 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { n2 + 1 || {(-I 3 ) , (3 I) , (-I 3 ) | ) 1/3 I (-1) 3 | ) -----------------------------------------------------------------||, | / | / 1/2 (n2 + 1) || |----- |----- (3 I) || \n1 = 0 \n2 = 0 // / / { / n2 \ \\ | | { |----| || | | { \ 2 / || | | { 2 (-16) || | | { ------------------------------ n2::even|| | | { n2 || | | { (n2 + 1) n2 binomial(n2, ----) || |n - 1 |n1 - 1 { 2 || |----- |----- { || 1/2 n | \ n1 1/2 | \ { 0 n2::odd || (-I 3 ) | ) 1/3 I (-1) 3 | ) ------------------------------------------------||} | / | / 1/2 (n2 + 1) || |----- |----- (3 I) || \n1 = 0 \n2 = 0 // "A158005" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=72382.2MB, alloc=2327.5MB, time=509.21 LREtools/SearchTable: "SearchTable successful" {(n + 4) (n + 3) (n + 2) (n + 1) n!, ( 9 8 7 6 5 4 3 2 (2 n + 63 n - 32352 n - 649242 n - 5386926 n - 24163785 n - 63339428 n - 97012332 n - 80399232 n - 27851040) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 7 6 5 4 3 2 2 / - 9 (2 n + 57 n - 3004 n - 57279 n - 390676 n - 1290276 n - 2090304 n - 1334880) (n + 1) hypergeom([1/2, -n, -n], [1, 1], 4)) / ( / 2 2 2 2 2 (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) )} "A158196" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (2 n1 + 7) | {(16/3) , (16/3) | ) --------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (16/3) | \n1 = 0 / "A158197" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (n1 + 3) | {(25/4) , (25/4) | ) --------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (25/4) | \n1 = 0 / "A158243" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\n1 1/2 \ /n - 1 \ |n - 1 | 2 | 2 | |----- n1 | |----- |- ----| HermiteH(n1 + 1, ----)| | \ (-1) | | \ \ 2 / 2 | {n! | ) ---------|, n! | ) ---------------------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A158422" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 3 n 3 n 3 n {RootOf(_Z - _Z - 1, index = 1) , RootOf(_Z - _Z - 1, index = 2) , RootOf(_Z - _Z - 1, index = 3) } "A158432" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=72839.7MB, alloc=2327.5MB, time=512.29 memory used=73072.2MB, alloc=2327.5MB, time=514.63 memory used=73266.0MB, alloc=2327.5MB, time=516.97 memory used=73436.5MB, alloc=2327.5MB, time=519.29 memory used=73639.4MB, alloc=2327.5MB, time=521.66 memory used=73867.3MB, alloc=2327.5MB, time=524.05 memory used=74118.9MB, alloc=2327.5MB, time=526.43 memory used=74303.8MB, alloc=2327.5MB, time=528.71 memory used=74447.0MB, alloc=2327.5MB, time=530.97 memory used=74590.5MB, alloc=2327.5MB, time=533.10 memory used=74739.0MB, alloc=2327.5MB, time=535.28 memory used=74894.8MB, alloc=2327.5MB, time=537.52 memory used=75050.7MB, alloc=2327.5MB, time=539.70 memory used=75211.6MB, alloc=2327.5MB, time=541.94 memory used=75375.8MB, alloc=2327.5MB, time=544.25 memory used=75544.4MB, alloc=2327.5MB, time=546.45 memory used=75712.9MB, alloc=2327.5MB, time=548.50 memory used=75882.6MB, alloc=2327.5MB, time=550.62 memory used=76050.6MB, alloc=2327.5MB, time=552.99 memory used=76235.4MB, alloc=2327.5MB, time=555.40 memory used=76402.3MB, alloc=2327.5MB, time=557.69 memory used=76572.0MB, alloc=2327.5MB, time=559.98 memory used=76751.6MB, alloc=2327.5MB, time=562.41 memory used=76930.9MB, alloc=2327.5MB, time=564.74 memory used=77111.6MB, alloc=2327.5MB, time=567.14 memory used=77291.6MB, alloc=2327.5MB, time=569.54 LREtools/SearchTable: "SearchTable successful" 16 15 14 13 12 11 10 {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (-16 (64 n + 5184 n + 194960 n + 4126032 n + 49866636 n + 262007020 n - 1710807365 n 9 8 7 6 5 4 3 - 46704775257 n - 469782406010 n - 2994707639814 n - 13380896688885 n - 43158035910961 n - 100504890792632 n - 165302946045804 n 2 4 19 - 182478710482416 n - 121441911769728 n - 36851679627264) (n + 1) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1) + (2 n + 1) (128 n 18 17 16 15 14 13 12 11 + 10944 n + 437536 n + 7725872 n - 5247432 n - 3087176844 n - 72158180794 n - 964237070895 n - 8864392369586 n 10 9 8 7 6 5 - 59894449658959 n - 306932079216774 n - 1211542115687745 n - 3704403528912046 n - 8754476615030073 n - 15832007025232856 n 4 3 2 - 21485411510253100 n - 21156662667656544 n - 14259214791402048 n - 5878908797870592 n - 1117496072282112) / 3 3 3 3 3 hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1)) binomial(2 n, n) / ((n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) / 3 3 (n + 7) (n + 8) (n + 9))} "A158495" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1)| {2 , 2 | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A158499" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1)| {2 , 2 | ) -----------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A158513" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 5)} "A158516" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 6)} "A158530" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 7)} "A158531" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 8)} "A158532" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 9)} "A158534" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 10)} "A158535" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 11)} "A158538" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 12)} "A158542" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 13)} "A158545" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 14)} "A158580" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 15)} "A158617" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 16)} "A158696" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 17)} "A158700" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 18)} "A158702" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 19)} "A158703" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 20)} "A158727" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 21)} "A158751" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 22)} "A158752" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 23)} "A158783" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 24)} "A158802" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A158811" LREtools/SearchTable: "SearchTable successful" n {3 HermiteH(n, 1/3)} "A158826" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A158903" LREtools/SearchTable: "SearchTable successful" n {3 HermiteH(n, 2/3)} "A158954" LREtools/SearchTable: "SearchTable successful" n {2 HermiteH(n, 1/4)} "A158958" LREtools/SearchTable: "SearchTable successful" n {2 HermiteH(n, 3/4)} "A158960" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 1/5)} "A158961" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 2/5)} "A158965" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 3/5)} "A158967" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 4/5)} "A158968" LREtools/SearchTable: "SearchTable successful" n {3 HermiteH(n, 1/6)} "A158969" LREtools/SearchTable: "SearchTable successful" n {3 HermiteH(n, 5/6)} "A158980" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 1/7)} "A158981" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 2/7)} "A158987" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 3/7)} "A158991" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 4/7)} "A159005" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 5/7)} "A159013" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 6/7)} "A159014" LREtools/SearchTable: "SearchTable successful" n {4 HermiteH(n, 1/8)} "A159017" LREtools/SearchTable: "SearchTable successful" n {4 HermiteH(n, 3/8)} "A159019" LREtools/SearchTable: "SearchTable successful" n {4 HermiteH(n, 5/8)} "A159028" LREtools/SearchTable: "SearchTable successful" n {4 HermiteH(n, 7/8)} "A159030" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 1/9)} "A159139" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A159175" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A159197" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 2/9)} "A159232" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 4/9)} "A159240" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 5/9)} "A159242" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 7/9)} "A159245" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 8/9)} "A159247" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 1/10)} "A159249" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 3/10)} "A159252" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 7/10)} "A159279" LREtools/SearchTable: "SearchTable successful" n {5 HermiteH(n, 9/10)} "A159280" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 1/11)} "A159281" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 2/11)} "A159307" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 3/11)} "A159324" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 3) n1| {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A159326" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 4/11)} "A159327" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 5/11)} "A159449" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 6/11)} "A159450" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 7/11)} "A159454" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 8/11)} "A159460" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 9/11)} "A159470" LREtools/SearchTable: "SearchTable successful" n 10 {11 HermiteH(n, --)} 11 "A159472" LREtools/SearchTable: "SearchTable successful" n {6 HermiteH(n, 1/12)} "A159480" LREtools/SearchTable: "SearchTable successful" n {6 HermiteH(n, 5/12)} "A159485" LREtools/SearchTable: "SearchTable successful" n {6 HermiteH(n, 7/12)} "A159486" LREtools/SearchTable: "SearchTable successful" n 11 {6 HermiteH(n, --)} 12 "A159488" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 1/13)} "A159492" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 2/13)} "A159494" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 3/13)} "A159496" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 4/13)} "A159497" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 5/13)} "A159498" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 6/13)} "A159500" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 7/13)} "A159501" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 8/13)} "A159502" memory used=78026.3MB, alloc=2327.5MB, time=574.40 LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 9/13)} "A159504" LREtools/SearchTable: "SearchTable successful" n 10 {13 HermiteH(n, --)} 13 "A159505" LREtools/SearchTable: "SearchTable successful" n 11 {13 HermiteH(n, --)} 13 "A159506" LREtools/SearchTable: "SearchTable successful" n 12 {13 HermiteH(n, --)} 13 "A159507" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 1/14)} "A159508" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 3/14)} "A159509" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 5/14)} "A159510" LREtools/SearchTable: "SearchTable successful" n {7 HermiteH(n, 9/14)} "A159511" LREtools/SearchTable: "SearchTable successful" n 11 {7 HermiteH(n, --)} 14 "A159512" LREtools/SearchTable: "SearchTable successful" n 13 {7 HermiteH(n, --)} 14 "A159513" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 1/15)} "A159514" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 2/15)} "A159515" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 4/15)} "A159516" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 7/15)} "A159517" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 8/15)} "A159518" LREtools/SearchTable: "SearchTable successful" n 11 {15 HermiteH(n, --)} 15 "A159519" LREtools/SearchTable: "SearchTable successful" n 13 {15 HermiteH(n, --)} 15 "A159520" LREtools/SearchTable: "SearchTable successful" n 14 {15 HermiteH(n, --)} 15 "A159521" LREtools/SearchTable: "SearchTable successful" n {8 HermiteH(n, 1/16)} "A159522" LREtools/SearchTable: "SearchTable successful" n {8 HermiteH(n, 3/16)} "A159523" LREtools/SearchTable: "SearchTable successful" n {8 HermiteH(n, 5/16)} "A159524" LREtools/SearchTable: "SearchTable successful" n {8 HermiteH(n, 7/16)} "A159525" LREtools/SearchTable: "SearchTable successful" n {8 HermiteH(n, 9/16)} "A159526" LREtools/SearchTable: "SearchTable successful" n 11 {8 HermiteH(n, --)} 16 "A159527" LREtools/SearchTable: "SearchTable successful" n 13 {8 HermiteH(n, --)} 16 "A159528" LREtools/SearchTable: "SearchTable successful" n 15 {8 HermiteH(n, --)} 16 "A159529" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 1/17)} "A159530" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 2/17)} "A159531" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 3/17)} "A159532" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 4/17)} "A159533" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 5/17)} "A159534" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 6/17)} "A159535" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 7/17)} "A159536" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 8/17)} "A159537" LREtools/SearchTable: "SearchTable successful" n {17 HermiteH(n, 9/17)} "A159538" LREtools/SearchTable: "SearchTable successful" n 10 {17 HermiteH(n, --)} 17 "A159539" LREtools/SearchTable: "SearchTable successful" n 11 {17 HermiteH(n, --)} 17 "A159540" LREtools/SearchTable: "SearchTable successful" n 12 {17 HermiteH(n, --)} 17 "A159541" LREtools/SearchTable: "SearchTable successful" n 13 {17 HermiteH(n, --)} 17 "A159542" LREtools/SearchTable: "SearchTable successful" n 14 {17 HermiteH(n, --)} 17 "A159543" LREtools/SearchTable: "SearchTable successful" n 15 {17 HermiteH(n, --)} 17 "A159544" LREtools/SearchTable: "SearchTable successful" n 16 {17 HermiteH(n, --)} 17 "A159545" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 1/18)} "A159546" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 5/18)} "A159552" LREtools/SearchTable: "SearchTable successful" n {9 HermiteH(n, 7/18)} "A159561" LREtools/SearchTable: "SearchTable successful" n 11 {9 HermiteH(n, --)} 18 "A159562" LREtools/SearchTable: "SearchTable successful" n 13 {9 HermiteH(n, --)} 18 "A159563" LREtools/SearchTable: "SearchTable successful" n 17 {9 HermiteH(n, --)} 18 "A159564" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 1/19)} "A159618" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 2/19)} "A159620" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 3/19)} "A159621" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 4/19)} "A159622" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 5/19)} "A159644" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 6/19)} "A159645" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 7/19)} "A159646" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 8/19)} "A159647" LREtools/SearchTable: "SearchTable successful" n {19 HermiteH(n, 9/19)} "A159648" LREtools/SearchTable: "SearchTable successful" n 10 {19 HermiteH(n, --)} 19 "A159649" LREtools/SearchTable: "SearchTable successful" n 11 {19 HermiteH(n, --)} 19 "A159650" LREtools/SearchTable: "SearchTable successful" n 12 {19 HermiteH(n, --)} 19 "A159651" LREtools/SearchTable: "SearchTable successful" n 13 {19 HermiteH(n, --)} 19 "A159652" LREtools/SearchTable: "SearchTable successful" n 14 {19 HermiteH(n, --)} 19 "A159653" LREtools/SearchTable: "SearchTable successful" n 15 {19 HermiteH(n, --)} 19 "A159654" LREtools/SearchTable: "SearchTable successful" n 16 {19 HermiteH(n, --)} 19 "A159655" LREtools/SearchTable: "SearchTable successful" n 17 {19 HermiteH(n, --)} 19 "A159656" LREtools/SearchTable: "SearchTable successful" n 18 {19 HermiteH(n, --)} 19 "A159657" LREtools/SearchTable: "SearchTable successful" n {10 HermiteH(n, 1/20)} "A159658" LREtools/SearchTable: "SearchTable successful" n {10 HermiteH(n, 3/20)} "A159659" LREtools/SearchTable: "SearchTable successful" n {10 HermiteH(n, 7/20)} "A159660" LREtools/SearchTable: "SearchTable successful" n {10 HermiteH(n, 9/20)} "A159663" LREtools/SearchTable: "SearchTable successful" n 11 {10 HermiteH(n, --)} 20 "A159670" LREtools/SearchTable: "SearchTable successful" n 13 {10 HermiteH(n, --)} 20 "A159676" LREtools/SearchTable: "SearchTable successful" n 17 {10 HermiteH(n, --)} 20 "A159682" LREtools/SearchTable: "SearchTable successful" n 19 {10 HermiteH(n, --)} 20 "A159705" LREtools/SearchTable: "SearchTable successful" n {21 HermiteH(n, 1/21)} "A159706" LREtools/SearchTable: "SearchTable successful" n {21 HermiteH(n, 2/21)} "A159707" LREtools/SearchTable: "SearchTable successful" n {21 HermiteH(n, 4/21)} "A159709" LREtools/SearchTable: "SearchTable successful" n {21 HermiteH(n, 5/21)} "A159745" LREtools/SearchTable: "SearchTable successful" n {21 HermiteH(n, 8/21)} "A159753" LREtools/SearchTable: "SearchTable successful" n 10 {21 HermiteH(n, --)} 21 "A159761" LREtools/SearchTable: "SearchTable successful" n 11 {21 HermiteH(n, --)} 21 "A159762" LREtools/SearchTable: "SearchTable successful" n 13 {21 HermiteH(n, --)} 21 "A159763" LREtools/SearchTable: "SearchTable successful" n 16 {21 HermiteH(n, --)} 21 "A159769" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) } "A159770" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A159771" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A159772" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A159776" LREtools/SearchTable: "SearchTable successful" n 17 {21 HermiteH(n, --)} 21 "A159784" LREtools/SearchTable: "SearchTable successful" n 19 {21 HermiteH(n, --)} 21 "A159806" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 1/22)} "A159807" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 3/22)} "A159808" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 5/22)} "A159826" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 7/22)} "A159831" LREtools/SearchTable: "SearchTable successful" n {11 HermiteH(n, 9/22)} "A159832" LREtools/SearchTable: "SearchTable successful" n 13 {11 HermiteH(n, --)} 22 "A159840" memory used=78809.0MB, alloc=2359.5MB, time=579.69 LREtools/SearchTable: "SearchTable successful" n 15 {11 HermiteH(n, --)} 22 "A159850" LREtools/SearchTable: "SearchTable successful" n 17 {11 HermiteH(n, --)} 22 "A159851" LREtools/SearchTable: "SearchTable successful" n 19 {11 HermiteH(n, --)} 22 "A159857" LREtools/SearchTable: "SearchTable successful" n 21 {11 HermiteH(n, --)} 22 "A159858" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 1/23)} "A159859" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 2/23)} "A159865" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 3/23)} "A159868" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 4/23)} "A159869" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 5/23)} "A159870" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 6/23)} "A159871" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 7/23)} "A159872" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 8/23)} "A159873" LREtools/SearchTable: "SearchTable successful" n {23 HermiteH(n, 9/23)} "A159874" LREtools/SearchTable: "SearchTable successful" n 10 {23 HermiteH(n, --)} 23 "A159875" LREtools/SearchTable: "SearchTable successful" n 11 {23 HermiteH(n, --)} 23 "A159877" LREtools/SearchTable: "SearchTable successful" n 12 {23 HermiteH(n, --)} 23 "A159882" LREtools/SearchTable: "SearchTable successful" n 13 {23 HermiteH(n, --)} 23 "A159883" LREtools/SearchTable: "SearchTable successful" n 14 {23 HermiteH(n, --)} 23 "A159884" LREtools/SearchTable: "SearchTable successful" n 15 {23 HermiteH(n, --)} 23 "A159889" LREtools/SearchTable: "SearchTable successful" n 16 {23 HermiteH(n, --)} 23 "A159904" LREtools/SearchTable: "SearchTable successful" n 17 {23 HermiteH(n, --)} 23 "A159921" LREtools/SearchTable: "SearchTable successful" n 18 {23 HermiteH(n, --)} 23 "A159925" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 3 2 n 3 2 {(-1) ((n + 2 n - n - 1) BesselJ(n, -2) + (n + 2) n BesselJ(n - 1, -2)), (-1) ((n + 2 n - n - 1) BesselY(n, -2) + (n + 2) n BesselY(n - 1, -2)) } "A159926" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((n + 1) (n + 2 n - 2) BesselJ(n, -2) + (n + 3 n + 1) BesselJ(n - 1, -2)), n 2 2 (-1) ((n + 1) (n + 2 n - 2) BesselY(n, -2) + (n + 3 n + 1) BesselY(n - 1, -2))} "A159928" LREtools/SearchTable: "SearchTable successful" n 4 3 2 3 2 {(-1) ((n + 5 n + 5 n - 3 n - 3) BesselJ(n, -2) + (n + 5 n + 6 n + 1) BesselJ(n - 1, -2)), n 4 3 2 3 2 (-1) ((n + 5 n + 5 n - 3 n - 3) BesselY(n, -2) + (n + 5 n + 6 n + 1) BesselY(n - 1, -2))} "A159943" LREtools/SearchTable: "SearchTable successful" n 19 {23 HermiteH(n, --)} 23 "A159946" LREtools/SearchTable: "SearchTable successful" n 20 {23 HermiteH(n, --)} 23 "A159947" LREtools/SearchTable: "SearchTable successful" n 21 {23 HermiteH(n, --)} 23 "A159948" LREtools/SearchTable: "SearchTable successful" n 22 {23 HermiteH(n, --)} 23 "A159949" LREtools/SearchTable: "SearchTable successful" n {12 HermiteH(n, 1/24)} "A159954" LREtools/SearchTable: "SearchTable successful" n {12 HermiteH(n, 5/24)} "A159960" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" 2 {(n + 1) (2 n + 1) (n!) binomial(2 n, n)} "A159967" LREtools/SearchTable: "SearchTable successful" n {12 HermiteH(n, 7/24)} "A159968" LREtools/SearchTable: "SearchTable successful" n 11 {12 HermiteH(n, --)} 24 "A159969" LREtools/SearchTable: "SearchTable successful" n 13 {12 HermiteH(n, --)} 24 "A159972" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / binomial(2 n1, n1)\| {(-1) , (-1) | ) |- ------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A159996" LREtools/SearchTable: "SearchTable successful" n 17 {12 HermiteH(n, --)} 24 "A159997" LREtools/SearchTable: "SearchTable successful" n 19 {12 HermiteH(n, --)} 24 "A159998" LREtools/SearchTable: "SearchTable successful" n 23 {12 HermiteH(n, --)} 24 "A160003" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 1/25)} "A160004" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 2/25)} "A160005" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 3/25)} "A160008" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 4/25)} "A160010" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 6/25)} "A160011" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 7/25)} "A160012" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 8/25)} "A160013" LREtools/SearchTable: "SearchTable successful" n {25 HermiteH(n, 9/25)} "A160037" LREtools/SearchTable: "SearchTable successful" n 11 {25 HermiteH(n, --)} 25 "A160038" LREtools/SearchTable: "SearchTable successful" n 12 {25 HermiteH(n, --)} 25 "A160059" LREtools/SearchTable: "SearchTable successful" n 13 {25 HermiteH(n, --)} 25 "A160060" LREtools/SearchTable: "SearchTable successful" n 14 {25 HermiteH(n, --)} 25 "A160061" LREtools/SearchTable: "SearchTable successful" n 16 {25 HermiteH(n, --)} 25 "A160062" LREtools/SearchTable: "SearchTable successful" n 17 {25 HermiteH(n, --)} 25 "A160063" LREtools/SearchTable: "SearchTable successful" n 18 {25 HermiteH(n, --)} 25 "A160064" LREtools/SearchTable: "SearchTable successful" n 19 {25 HermiteH(n, --)} 25 "A160065" LREtools/SearchTable: "SearchTable successful" n 21 {25 HermiteH(n, --)} 25 "A160066" LREtools/SearchTable: "SearchTable successful" n 22 {25 HermiteH(n, --)} 25 "A160067" LREtools/SearchTable: "SearchTable successful" n 23 {25 HermiteH(n, --)} 25 "A160068" LREtools/SearchTable: "SearchTable successful" n 24 {25 HermiteH(n, --)} 25 "A160069" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 1/26)} "A160070" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 3/26)} "A160071" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 5/26)} "A160072" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 7/26)} "A160073" LREtools/SearchTable: "SearchTable successful" n {13 HermiteH(n, 9/26)} "A160074" LREtools/SearchTable: "SearchTable successful" n 11 {13 HermiteH(n, --)} 26 "A160075" LREtools/SearchTable: "SearchTable successful" n 15 {13 HermiteH(n, --)} 26 "A160076" LREtools/SearchTable: "SearchTable successful" n 17 {13 HermiteH(n, --)} 26 "A160077" LREtools/SearchTable: "SearchTable successful" n 19 {13 HermiteH(n, --)} 26 "A160082" LREtools/SearchTable: "SearchTable successful" n 21 {13 HermiteH(n, --)} 26 "A160083" LREtools/SearchTable: "SearchTable successful" n 23 {13 HermiteH(n, --)} 26 "A160084" LREtools/SearchTable: "SearchTable successful" n 25 {13 HermiteH(n, --)} 26 "A160087" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 1/27)} "A160088" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 2/27)} "A160103" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 4/27)} "A160104" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 5/27)} "A160107" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 7/27)} "A160131" LREtools/SearchTable: "SearchTable successful" n {27 HermiteH(n, 8/27)} "A160132" LREtools/SearchTable: "SearchTable successful" n 10 {27 HermiteH(n, --)} 27 "A160139" LREtools/SearchTable: "SearchTable successful" n 11 {27 HermiteH(n, --)} 27 "A160140" LREtools/SearchTable: "SearchTable successful" n 13 {27 HermiteH(n, --)} 27 "A160141" LREtools/SearchTable: "SearchTable successful" n 14 {27 HermiteH(n, --)} 27 "A160142" LREtools/SearchTable: "SearchTable successful" n 16 {27 HermiteH(n, --)} 27 "A160146" LREtools/SearchTable: "SearchTable successful" n 17 {27 HermiteH(n, --)} 27 "A160147" LREtools/SearchTable: "SearchTable successful" n 19 {27 HermiteH(n, --)} 27 "A160148" LREtools/SearchTable: "SearchTable successful" n 20 {27 HermiteH(n, --)} 27 "A160150" LREtools/SearchTable: "SearchTable successful" n 22 {27 HermiteH(n, --)} 27 "A160151" LREtools/SearchTable: "SearchTable successful" n 23 {27 HermiteH(n, --)} 27 "A160152" LREtools/SearchTable: "SearchTable successful" n 25 {27 HermiteH(n, --)} 27 "A160153" LREtools/SearchTable: "SearchTable successful" n 26 {27 HermiteH(n, --)} 27 "A160184" LREtools/SearchTable: "SearchTable successful" n {14 HermiteH(n, 1/28)} "A160192" LREtools/SearchTable: "SearchTable successful" n {14 HermiteH(n, 3/28)} "A160193" LREtools/SearchTable: "SearchTable successful" n {14 HermiteH(n, 5/28)} "A160194" LREtools/SearchTable: "SearchTable successful" n {14 HermiteH(n, 9/28)} "A160195" LREtools/SearchTable: "SearchTable successful" n 11 {14 HermiteH(n, --)} 28 "A160196" LREtools/SearchTable: "SearchTable successful" n 13 {14 HermiteH(n, --)} 28 "A160197" LREtools/SearchTable: "SearchTable successful" n 15 {14 HermiteH(n, --)} 28 "A160219" LREtools/SearchTable: "SearchTable successful" n 17 {14 HermiteH(n, --)} 28 "A160220" LREtools/SearchTable: "SearchTable successful" n 19 {14 HermiteH(n, --)} 28 "A160221" LREtools/SearchTable: "SearchTable successful" n 23 {14 HermiteH(n, --)} 28 "A160222" LREtools/SearchTable: "SearchTable successful" n 25 {14 HermiteH(n, --)} 28 "A160223" LREtools/SearchTable: "SearchTable successful" n 27 {14 HermiteH(n, --)} 28 "A160224" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 1/29)} "A160225" memory used=79625.9MB, alloc=2359.5MB, time=585.15 LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 2/29)} "A160226" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 3/29)} "A160231" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 4/29)} "A160236" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 5/29)} "A160237" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 6/29)} "A160246" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 7/29)} "A160251" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 8/29)} "A160252" LREtools/SearchTable: "SearchTable successful" n {29 HermiteH(n, 9/29)} "A160253" LREtools/SearchTable: "SearchTable successful" n 10 {29 HermiteH(n, --)} 29 "A160254" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) } 6 5 4 3 2 %1 := _Z - 3 _Z + 3 _Z - 3 _Z + 4 _Z - 3 _Z + 2 "A160259" LREtools/SearchTable: "SearchTable successful" n 11 {29 HermiteH(n, --)} 29 "A160260" LREtools/SearchTable: "SearchTable successful" n 12 {29 HermiteH(n, --)} 29 "A160261" LREtools/SearchTable: "SearchTable successful" n 13 {29 HermiteH(n, --)} 29 "A160263" LREtools/SearchTable: "SearchTable successful" n 14 {29 HermiteH(n, --)} 29 "A160269" LREtools/SearchTable: "SearchTable successful" n 15 {29 HermiteH(n, --)} 29 "A160270" LREtools/SearchTable: "SearchTable successful" n 16 {29 HermiteH(n, --)} 29 "A160279" LREtools/SearchTable: "SearchTable successful" n 17 {29 HermiteH(n, --)} 29 "A160280" LREtools/SearchTable: "SearchTable successful" n 18 {29 HermiteH(n, --)} 29 "A160281" LREtools/SearchTable: "SearchTable successful" n 19 {29 HermiteH(n, --)} 29 "A160282" LREtools/SearchTable: "SearchTable successful" n 20 {29 HermiteH(n, --)} 29 "A160283" LREtools/SearchTable: "SearchTable successful" n 21 {29 HermiteH(n, --)} 29 "A160284" LREtools/SearchTable: "SearchTable successful" n 22 {29 HermiteH(n, --)} 29 "A160285" LREtools/SearchTable: "SearchTable successful" n 23 {29 HermiteH(n, --)} 29 "A160286" LREtools/SearchTable: "SearchTable successful" n 24 {29 HermiteH(n, --)} 29 "A160287" LREtools/SearchTable: "SearchTable successful" n 25 {29 HermiteH(n, --)} 29 "A160288" LREtools/SearchTable: "SearchTable successful" n 26 {29 HermiteH(n, --)} 29 "A160289" LREtools/SearchTable: "SearchTable successful" n 27 {29 HermiteH(n, --)} 29 "A160290" LREtools/SearchTable: "SearchTable successful" n 28 {29 HermiteH(n, --)} 29 "A160291" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 1/30)} "A160292" LREtools/SearchTable: "SearchTable successful" n {15 HermiteH(n, 7/30)} "A160293" LREtools/SearchTable: "SearchTable successful" n 11 {15 HermiteH(n, --)} 30 "A160294" LREtools/SearchTable: "SearchTable successful" n 13 {15 HermiteH(n, --)} 30 "A160295" LREtools/SearchTable: "SearchTable successful" n 17 {15 HermiteH(n, --)} 30 "A160296" LREtools/SearchTable: "SearchTable successful" n 19 {15 HermiteH(n, --)} 30 "A160297" LREtools/SearchTable: "SearchTable successful" n 23 {15 HermiteH(n, --)} 30 "A160298" LREtools/SearchTable: "SearchTable successful" n 29 {15 HermiteH(n, --)} 30 "A160299" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 1/31)} "A160300" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 2/31)} "A160301" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 3/31)} "A160302" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 4/31)} "A160303" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 5/31)} "A160304" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 6/31)} "A160305" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 7/31)} "A160306" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 8/31)} "A160307" LREtools/SearchTable: "SearchTable successful" n {31 HermiteH(n, 9/31)} "A160308" LREtools/SearchTable: "SearchTable successful" n 10 {31 HermiteH(n, --)} 31 "A160309" LREtools/SearchTable: "SearchTable successful" n 11 {31 HermiteH(n, --)} 31 "A160310" LREtools/SearchTable: "SearchTable successful" n 12 {31 HermiteH(n, --)} 31 "A160311" LREtools/SearchTable: "SearchTable successful" n 13 {31 HermiteH(n, --)} 31 "A160312" LREtools/SearchTable: "SearchTable successful" n 14 {31 HermiteH(n, --)} 31 "A160313" LREtools/SearchTable: "SearchTable successful" n 15 {31 HermiteH(n, --)} 31 "A160314" LREtools/SearchTable: "SearchTable successful" n 16 {31 HermiteH(n, --)} 31 "A160315" LREtools/SearchTable: "SearchTable successful" n 17 {31 HermiteH(n, --)} 31 "A160316" LREtools/SearchTable: "SearchTable successful" n 18 {31 HermiteH(n, --)} 31 "A160317" LREtools/SearchTable: "SearchTable successful" n 19 {31 HermiteH(n, --)} 31 "A160328" LREtools/SearchTable: "SearchTable successful" n 20 {31 HermiteH(n, --)} 31 "A160329" LREtools/SearchTable: "SearchTable successful" n 21 {31 HermiteH(n, --)} 31 "A160330" LREtools/SearchTable: "SearchTable successful" n 22 {31 HermiteH(n, --)} 31 "A160334" LREtools/SearchTable: "SearchTable successful" n 23 {31 HermiteH(n, --)} 31 "A160335" LREtools/SearchTable: "SearchTable successful" n 24 {31 HermiteH(n, --)} 31 "A160336" LREtools/SearchTable: "SearchTable successful" n 25 {31 HermiteH(n, --)} 31 "A160344" LREtools/SearchTable: "SearchTable successful" n 26 {31 HermiteH(n, --)} 31 "A160345" LREtools/SearchTable: "SearchTable successful" n 27 {31 HermiteH(n, --)} 31 "A160346" LREtools/SearchTable: "SearchTable successful" n 28 {31 HermiteH(n, --)} 31 "A160347" LREtools/SearchTable: "SearchTable successful" n 29 {31 HermiteH(n, --)} 31 "A160349" LREtools/SearchTable: "SearchTable successful" n 30 {31 HermiteH(n, --)} 31 "A160361" LREtools/SearchTable: "SearchTable successful" n {16 HermiteH(n, 1/32)} "A160362" LREtools/SearchTable: "SearchTable successful" n {16 HermiteH(n, 3/32)} "A160363" LREtools/SearchTable: "SearchTable successful" n {16 HermiteH(n, 5/32)} "A160374" LREtools/SearchTable: "SearchTable successful" n {16 HermiteH(n, 7/32)} "A160376" LREtools/SearchTable: "SearchTable successful" n {16 HermiteH(n, 9/32)} "A160391" LREtools/SearchTable: "SearchTable successful" n 11 {16 HermiteH(n, --)} 32 "A160396" LREtools/SearchTable: "SearchTable successful" n 13 {16 HermiteH(n, --)} 32 "A160397" LREtools/SearchTable: "SearchTable successful" n 15 {16 HermiteH(n, --)} 32 "A160398" LREtools/SearchTable: "SearchTable successful" n 17 {16 HermiteH(n, --)} 32 "A160431" LREtools/SearchTable: "SearchTable successful" n 19 {16 HermiteH(n, --)} 32 "A160435" LREtools/SearchTable: "SearchTable successful" n 21 {16 HermiteH(n, --)} 32 "A160436" LREtools/SearchTable: "SearchTable successful" n 23 {16 HermiteH(n, --)} 32 "A160437" LREtools/SearchTable: "SearchTable successful" n 25 {16 HermiteH(n, --)} 32 "A160441" LREtools/SearchTable: "SearchTable successful" n 27 {16 HermiteH(n, --)} 32 "A160442" LREtools/SearchTable: "SearchTable successful" n 29 {16 HermiteH(n, --)} 32 "A160443" LREtools/SearchTable: "SearchTable successful" n 31 {16 HermiteH(n, --)} 32 "A160445" LREtools/SearchTable: "SearchTable successful" n 20 {21 HermiteH(n, --)} 21 "A160565" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A160568" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A160702" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A160823" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A160852" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A160906" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 (3 n1 + 1) binomial(3 n1, n1) (5 n1 + 6)| {8 , 8 | ) -----------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A160999" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A161125" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) (n - 1) HermiteH(n, 1/2 I 2 )} "A161128" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- | \ | \ n2! | {1, ) n1! n1 | ) ---------|, n!} / | / (n2 + 1)!| ----- |----- | n1 = 0 \n2 = 0 / "A161130" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=80431.3MB, alloc=2359.5MB, time=590.52 /n - 1 \ |----- n1 | | \ (-1) n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A161132" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1) (n/2)! BesselI(n/2 + 1/2, 1/2) n::even { {{ (n/2 + 1/2) , { (-1) (n/2 + 1/2)! (BesselI(n/2, 1/2) - BesselI(n/2 + 1, 1/2)) { - ------------------------------------------------------------------------ n::odd { n + 1 { (n/2) { (-1) (n/2)! BesselK(n/2 + 1/2, -1/2) n::even { { (n/2 + 1/2) , { (-1) (n/2 + 1/2)! (BesselK(n/2 + 1, -1/2) - BesselK(n/2, -1/2)) { -------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { (-1/4) binomial(n, n/2) (n/2)! (-BesselI(n/2, 1/2) + BesselI(n/2 + 1, 1/2)) n::even { , { (n/2 - 1/2) { (-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n BesselI(n/2 + 1/2, 1/2) n::odd { (n/2) { (-1/4) binomial(n, n/2) (n/2)! (BesselK(n/2 + 1, -1/2) - BesselK(n/2, -1/2)) n::even { } { (n/2 - 1/2) { (-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n BesselK(n/2 + 1/2, -1/2) n::odd "A161370" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 - 1)| {2 n!, 2 n! | ) ---------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A161474" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { {{ 0 irem(n, 3) = 1, { (n/3 - 1/3) , { { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 1 { (n/3 - 2/3) { { (n/3 + 1/3) 3 (n/3 - 2/3)! irem(n, 3) = 2 { 0 irem(n, 3) = 2 { (n/3) { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 0 { } { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A161634" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A161738" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (- n/2) { 2 binomial(2 n, n) (n/2)! n::even {{ , { (- n/2 + 1/2) { 2 (2 n - 1) binomial(2 n - 2, n - 1) (n/2 - 1/2)! n::odd { / 3 n\ { |- ---| { \ 2 / 3 n { 1/2 2 n binomial(2 n, n/2) binomial(---, n/2) (n/2)! n::even { 2 { { / 3 n \ } { |- --- - 3/2| { \ 2 / 3 n { 2 (n + 1) binomial(2 n + 2, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) (n/2 + 1/2)! { 2 { 1/2 ----------------------------------------------------------------------------------------------- n::odd { 2 n + 1 "A161936" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ (-1) 2 (n1 + 3)| {(n + 1) 2 n!, (n + 1) 2 n! | ) --------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A161937" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n n | \ (-1) 2 | {(n + 1) 2 n!, (n + 1) 2 n! | ) -----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A162162" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162326" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-n + 3) hypergeom([-1/2, -n], [1], -8) {------------------------------------------------------------------------------------} n "A162475" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162476" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162477" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162478" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162479" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162480" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162481" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162482" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A162533" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A162543" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A162548" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A162748" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) (HermiteH(n + 1, 2 I) - 4 I HermiteH(n, 2 I))} "A162969" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n/2)! n::even { 2 (n + 3) binomial(n, n/2) (n/2)! n::even { {{ , { (n/2 + 1/2)! (n + 3) } { (-n + 1) { -------------------- n::odd { 2 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A162970" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (n + 1) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I)} "A162972" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 1) n2!)|| /n - 1 \ |n - 1 | / || |----- n1 | |----- |----- || | \ (-1) | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) --------------------------------------|, | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 2) (n2 + 1) n2!)|| |n - 1 | / || |----- |----- || | \ \n2 = 0 /| (n + 1) n! | ) -----------------------------------------------|, | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / / /n1 - 1 / /n2 - 1 \\\\ | |----- | |----- n3 |||| | n1 | \ | n2 | \ (-1) |||| | (-1) | ) |-(-1) (n2 + 2) (n2 + 1) n2! | ) ------------------|||| |n - 1 | / | | / (n3 + 2) (n3 + 1)!|||| |----- |----- | |----- |||| | \ \n2 = 0 \ \n3 = 0 ///| (n + 1) n! | ) ---------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A162973" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 2) (n2 + 1) n2!)|| /n - 1 \ |n - 1 | / || |----- n1 | |----- |----- || | \ (-1) | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) -----------------------------------------------|, | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / /n1 - 1 / /n2 - 1 \\\\ | |----- | |----- n3 |||| | n1 | \ | n2 | \ (-1) |||| | (-1) | ) |-(-1) (n2 + 2) (n2 + 1) n2! | ) ------------------|||| |n - 1 | / | | / (n3 + 2) (n3 + 1)!|||| |----- |----- | |----- |||| | \ \n2 = 0 \ \n3 = 0 ///| (n + 1) n! | ) ---------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A162985" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A163493" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A163650" LREtools/SearchTable: "SearchTable successful" {(n + 1) hypergeom([1/2, -n - 1], [1], 4) + (2 n + 4) hypergeom([1/2, -n], [1], 4)} "A163765" LREtools/SearchTable: "SearchTable successful" n (-1) ((4 n + 7) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 5) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------} n + 2 "A163773" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n + 1) binomial(n, n/2) n::even {hypergeom([1/2, -n - 1], [1], 4) + (n + 1) hypergeom([1/2, -n], [1], 4), { , { binomial(n + 1, n/2 + 1/2) n::odd { n { 4 { 1/2 ------------------------ n::even { (n + 1) binomial(n, n/2) { } { (2 n - 2) { 2 (n + 1) { 1/2 ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A163774" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (2 n + 1) binomial(2 n, n) {(-1) hypergeom([1/2, -n - 1], [1], 4), --------------------------} n + 1 "A163775" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (2 n + 3) (2 n + 1) binomial(2 n, n) {(-1) ((2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 2) hypergeom([1/2, -n], [1], 4)), ------------------------------------} n + 1 "A163824" n n {2 , 3 , binomial(2 n, n)} "A163843" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { (n + 1) binomial(n, n/2) n::even {(-1) (hypergeom([1/2, -n - 1], [1], 4) + (-n - 1) hypergeom([1/2, -n], [1], 4)), { , { binomial(n + 1, n/2 + 1/2) n::odd { n { 4 { 1/2 ------------------------ n::even { (n + 1) binomial(n, n/2) { } { (2 n - 2) { 2 (n + 1) { 1/2 ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A163844" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) {--------------------------, (2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 2) hypergeom([-1/2, -n], [1], -4)} n + 1 "A163865" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (2 n - 2) hypergeom([1/2, -n], [1], 4))} "A163869" LREtools/SearchTable: "SearchTable successful" 2 {8 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-16 n - 32 n - 17) hypergeom([-1/2, -n], [1], -4)} "A163872" LREtools/SearchTable: "SearchTable successful" n {(-1) ((2 n + 2) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 1) hypergeom([1/2, -n], [1], 4))} "A164137" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 6 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A164139" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {n + 1} "A164140" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A164141" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A164142" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A164144" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164145" memory used=81127.9MB, alloc=2391.5MB, time=595.80 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A164146" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A164148" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A164153" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 6 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A164155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164156" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A164157" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 7 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1, 15 n - 14} "A164158" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 1} "A164159" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 7 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 1/2 n 1/2 n {1, (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A164160" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164162" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 8 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {1} "A164163" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A164164" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164165" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n {1, (-1) } "A164166" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164168" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A164169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164170" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {n + 2} "A164171" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A164173" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A164175" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164176" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164177" memory used=81887.3MB, alloc=2423.5MB, time=601.18 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" 1/2 n 1/2 n {1, (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A164192" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {1} "A164193" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164195" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A164197" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 9 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {1, 3 n - 4} "A164198" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 7 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1, 15 n + 31} "A164200" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A164205" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164206" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {1} "A164208" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {1} "A164211" memory used=82675.1MB, alloc=2423.5MB, time=606.30 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A164213" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164214" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A164216" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A164218" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {1} "A164220" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {1} "A164222" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A164229" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A164233" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A164244" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 8 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {1, (1/2) , n + 13/9} "A164246" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A164248" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A164258" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {n + 1} "A164259" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A164586" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(-1) } "A164651" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2)|| {1, (2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) --------------------------------------||} | / | / n2 + 1 || |----- |----- || \n1 = 0 \n2 = 0 // "A164990" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { n binomial(n, n/2) n::even { ------------------ n::even n { { n binomial(n, n/2) {2 (n + 3), { binomial(n + 1, n/2 + 1/2) (n - 1) (n + 1) , { } { 1/2 ------------------------------------------ n::odd { (2 n - 2) { n { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A164991" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ------------------ n::even n { n binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n + 2) { 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A165201" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) n2|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -----------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A165203" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (5 n2 + 9 n2 + 2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ---------------------------------------------------------||} | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A165233" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-I) n!, I n!} "A165407" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / 1/2\n / 1/2 \n / 1/2\n | 5 | |5 | | 5 | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| \ 2 / \ 2 / \ 2 / / / / /{ 0 irem(n2, 3) = 0\\\\ | | | |{ |||| | | | |{ 0 irem(n2, 3) = 1|||| |n - 1 | |n1 - 1 |{ |||| |----- | |----- / 1/2 \(-n2 - 1) |{ /2 n2 \ |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ |---- - 4/3| |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ \ 3 / n2 ||||, | / | | / \ 2 / |{ 2 GAMMA(---- + 1/2) |||| |----- | |----- |{ 3 |||| |n1 = 0 | |n2 = 0 |{ ------------------------------- irem(n2, 3) = 2|||| | | | |{ n2 |||| | | | |{ GAMMA(---- + 2) |||| \ \ \ \{ 3 //// / 1/2\n | 5 | |1/2 - ----| \ 2 / / / / /{ 0 irem(n2, 3) = 0\\\\ | | | |{ |||| | | | |{ /2 n2 \ |||| |n - 1 | |n1 - 1 |{ |---- - 2/3| |||| |----- | |----- / 1/2 \(-n2 - 1) |{ \ 3 / n2 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ 2 GAMMA(---- + 1/2) |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 3 ||||, | / | | / \ 2 / |{ ------------------------------- irem(n2, 3) = 1|||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ GAMMA(---- + 2) |||| | | | |{ 3 |||| | | | |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// / 1/2\n | 5 | |1/2 - ----| \ 2 / / / / /{ 2 n2 n2 \\\\ |n - 1 | |n1 - 1 |{ 3 binomial(----, ----) |||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ---------------------- irem(n2, 3) = 0|||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ n2 + 3 ||||} | / | | / \ 2 / |{ |||| |----- | |----- |{ 0 irem(n2, 3) = 1|||| |n1 = 0 | |n2 = 0 |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// "A165409" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" 1/2 n 1/2 n {(1 - 2 ) , (1 + 2 ) , / / /{ 0 irem(n2, 3) = 0\\\ | | |{ ||| |n - 1 |n1 - 1 |{ 0 irem(n2, 3) = 1||| |----- |----- |{ ||| 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ n2 n2 ||| (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) |{ 2 GAMMA(---- + 1/2) |||, | / | / |{ 3 ||| |----- |----- |{ 1/4 --------------------- irem(n2, 3) = 2||| |n1 = 0 |n2 = 0 |{ n2 ||| | | |{ GAMMA(---- + 2) ||| \ \ \{ 3 /// / / /{ 0 irem(n2, 3) = 0\\\ | | |{ ||| |n - 1 |n1 - 1 |{ (n2 - 1) n2 ||| |----- |----- |{ 2 GAMMA(---- + 1/2) ||| 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ 3 ||| (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) |{ --------------------------- irem(n2, 3) = 1|||, | / | / |{ n2 ||| |----- |----- |{ GAMMA(---- + 2) ||| |n1 = 0 |n2 = 0 |{ 3 ||| | | |{ ||| \ \ \{ 0 irem(n2, 3) = 2/// / / /{ / n2 \ \\\ | | |{ |----| ||| |n - 1 |n1 - 1 |{ \ 3 / 2 n2 n2 ||| |----- |----- |{ 3 2 binomial(----, ----) ||| 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ 3 3 ||| (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) |{ ------------------------------ irem(n2, 3) = 0|||} | / | / |{ n2 + 3 ||| |----- |----- |{ ||| |n1 = 0 |n2 = 0 |{ 0 irem(n2, 3) = 1||| | | |{ ||| \ \ \{ 0 irem(n2, 3) = 2/// "A165431" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A165433" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A165532" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\n / 1/2\n / 1/2\ / 1/2\n / 1/2\ / 1/2\n |----- | | 5 | | 5 | | 5 | | 8 5 | | 5 | | 8 5 | | 5 | 1/2 | \ | {|3/2 - ----| , |3/2 + ----| , |3/2 - ----| |n + 2 - ------|, |3/2 + ----| |n + 2 + ------|, |3/2 - ----| (5 n + 10 - 8 5 ) | ) |2 \ 2 / \ 2 / \ 2 / \ 5 / \ 2 / \ 5 / \ 2 / | / | |----- | \n1 = 0 \ 1/2 n1 1/2 (-n1 - 1) 2 (3 + 5 ) (3 - 5 ) (5 n1 + 25 n1 - 58) / / 1/2\(-n2 - 1) / 1/2\ \ |n1 - 1 | 5 | | 8 5 | 4 3 2 | |----- |3/2 + ----| |n2 + 3 - ------| (16 n2 + 49 n2 - 193 n2 + 90 n2 + 32) binomial(2 n2, n2)| // 1/2\ | \ \ 2 / \ 5 / | / || 8 5 | | ) ---------------------------------------------------------------------------------------------------| / ||n1 + 2 - ------| | / 2 2 | / \\ 5 / |----- (n2 + 1) (n2 + 2) (5 (n2 + 1) + 25 n2 - 33) (5 n2 + 25 n2 - 58) | \n2 = 0 / \\ || \|| 1/2 ||| (5 n1 + 15 - 8 5 )|||} /|| || // "A165533" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n RootOf(%1, index = 7) } 7 6 5 4 3 2 %1 := _Z - 11 _Z + 48 _Z - 110 _Z + 141 _Z - 98 _Z + 32 _Z - 4 "A165534" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 3 2 | n | \ (2 n1 + 1) binomial(2 n1, n1) (2 n1 - 25 n1 - 109 n1 - 64 n1 + 36)| {4 , (6 n + 13) | ) ---------------------------------------------------------------------|, 6 n - 7, 6 n + 13} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (6 n1 + 19) (6 n1 + 13) | |----- | \n1 = 0 / "A165535" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 5 | | 5 | 3 2 n 3 2 n {|3/2 - ----| , |3/2 + ----| , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , \ 2 / \ 2 / 3 2 n RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A165536" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 / /n1 - 1 / / 1/2\n / 1/2\n / 1/2\n |----- | |----- | n | 5 | | 5 | n | 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 1/2 (-n2 - 1) {2 , |3/2 - ----| , |3/2 + ----| , (1/2) (n - 32/9), |3/2 - ----| | ) |2 (3 + 5 ) (3 - 5 ) | ) |2 (3 + 5 ) \ 2 / \ 2 / \ 2 / | / | | / | |----- | |----- | \n1 = 0 \ \n2 = 0 \ /n2 - 1 |----- | \ (n3 + 1) 3 2 (9 n2 + 4) | ) 2 ((n3 + 1) (476 n3 + 1107 n3 + 827 n3 + 124) hypergeom([-1/2, -n3 - 1], [1], -4) | / |----- \n3 = 0 \\\\\ ||||| 4 3 2 ||||| + (-476 n3 - 1821 n3 - 2634 n3 - 1621 n3 - 308) hypergeom([-1/2, -n3], [1], -4))/((n3 + 2) (n3 + 3) (n3 + 4) (9 n3 + 13) (18 n3 + 8))|||||} ||||| ||||| ///// "A165537" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 10 _Z + 24 _Z - 20 _Z + 4 "A165539" memory used=83457.5MB, alloc=2423.5MB, time=611.80 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A165540" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2\n | 5 | | 5 | (2 n + 3) (2 n + 1) binomial(2 n, n) (n - 4) {|3/2 - ----| , |3/2 + ----| , --------------------------------------------} \ 2 / \ 2 / (n + 4) (n + 3) (n + 2) "A165543" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n n n | \ | \ (1/2 + 1/2 I) (2 n2 + 1) binomial(2 n2, n2) n2|| {(1/2 - 1/2 I) , (1/2 + 1/2 I) , (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) -------------------------------------------------------||, | / | / (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // /n - 1 /n1 - 1 |----- |----- n | \ | \ (-n2 - 1) (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) (1/2 + 1/2 I) ( | / | / |----- |----- \n1 = 0 \n2 = 0 2 (2 n2 + 1) (43 n2 + 59 n2 + 6) hypergeom([n2 + 1, -n2 - 1], [-n2 - 1/2], 1/4) \\ || 2 || - (4 n2 + 1) (23 n2 + 45 n2 + 18) hypergeom([n2, -n2], [-n2 + 1/2], 1/4)) binomial(2 n2, n2)/((n2 + 1) (n2 + 2) (n2 + 3) (2 n2 - 1))||} || || // "A165546" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A165792" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A165793" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A165813" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A165814" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A165961" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A165968" LREtools/SearchTable: "SearchTable successful" n (-2) n! (n + 1) LaguerreL(n + 1, -n - 3/2, -1/2) {-------------------------------------------------} n "A165976" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 binomial(2 n, n) {-------------------} n + 1 "A166076" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 / /n1 - 1 \\\ |----- | |----- 1/2 n2 1/2 1/2 1/2 ||| n n n | \ | n1 (-n1) | \ (-I 7 ) (5 I 7 LegendreP(n2 + 1, 5/7 I 7 ) - 7 LegendreP(n2, 5/7 I 7 ))||| {(-1) , (-2/3) , (-1) | ) |-2 3 | ) ----------------------------------------------------------------------------------|||, | / | | / (n2 + 1) ||| |----- | |----- (n2 + 2) (-2/3) ||| \n1 = 0 \ \n2 = 0 /// /n - 1 / /n1 - 1 \\\ |----- | |----- 1/2 n2 1/2 1/2 1/2 ||| n | \ | n1 (-n1) | \ (-I 7 ) (5 I 7 LegendreQ(n2 + 1, 5/7 I 7 ) - 7 LegendreQ(n2, 5/7 I 7 ))||| (-1) | ) |-2 3 | ) ----------------------------------------------------------------------------------|||} | / | | / (n2 + 1) ||| |----- | |----- (n2 + 2) (-2/3) ||| \n1 = 0 \ \n2 = 0 /// "A166078" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n n | \ -I (7 I) 7 (7 LegendreP(n1 + 1, 3/7 I 7 ) I + 7 LegendreP(n1, 3/7 I 7 ))| {2 , 2 | ) ---------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- 2 | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n | \ -I (7 I) 7 (7 LegendreQ(n1 + 1, 3/7 I 7 ) I + 7 LegendreQ(n1, 3/7 I 7 ))| 2 | ) ---------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- 2 | \n1 = 0 / "A166135" LREtools/SearchTable: "SearchTable successful" n {(-1) ((11 n + 9) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (-9 n - 9) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n) (2 n + 1)/((n + 1) (2 n + 3) (5 n + 3))} "A166228" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- / n1 \| |----- / n1 \| n n | \ | (-1) (3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3))|| n | \ | (-1) (3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3))|| {(-1) , (-1) | ) |- --------------------------------------------------||, (-1) | ) |- --------------------------------------------------||} | / \ n1 + 2 /| | / \ n1 + 2 /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A166229" LREtools/SearchTable: "SearchTable successful" /3 n \ /3 n \ |--- + 1/2| |--- + 1/2| \ 2 / 1/2 1/2 1/2 \ 2 / 1/2 1/2 1/2 2 (2 LegendreQ(n + 1, 2 ) - 2 LegendreQ(n, 2 )) 2 (-2 LegendreP(n + 1, 2 ) + 2 LegendreP(n, 2 )) {-----------------------------------------------------------------, - ------------------------------------------------------------------} n n "A166286" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A166287" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A166289" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166290" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166292" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A166294" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A166296" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - _Z - _Z + 1, index = 1) , RootOf(2 _Z - _Z - _Z + 1, index = 2) , RootOf(2 _Z - _Z - _Z + 1, index = 3) } "A166297" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166300" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166302" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A166359" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" /n - 1 \ |----- 2 | | \ (n1 + 1) ((4 n1 + 2 n1 - 1) BesselJ(2 n1 + 1, -2) - BesselJ(2 n1 - 1, -2))| {(2 n + 1) | ) ---------------------------------------------------------------------------|, | / n1 (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / /n - 1 \ |----- 2 | | \ (n1 + 1) ((4 n1 + 2 n1 - 1) BesselY(2 n1 + 1, -2) - BesselY(2 n1 - 1, -2))| (2 n + 1) | ) ---------------------------------------------------------------------------|, 2 n + 1} | / n1 (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / "A166474" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 | 2 | 2 {(n + 2) (n + 1) |- ----| n! hypergeom([-n - 1, 1 + ----], [2], 2)} \ 2 / 2 "A166554" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A166587" LREtools/SearchTable: "SearchTable successful" hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4) {---------------------------------------------------------------} n "A166588" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 /{ n1 \ ----- |{ | ----- |{ 2 binomial(n1, ----) | \ |{ (2 n1 - 2) | \ |{ 2 | {1, ) |{ 2 |, ) |{ -------------------- n1::even|} / |{ ---------------------------------------- n1::odd | / |{ n1 + 2 | ----- |{ n1 | ----- |{ | n1 = 0 |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | n1 = 0 \{ 0 n1::odd / \{ 2 / "A166677" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A166680" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A166694" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166696" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A166697" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A166741" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n - 1) 2 2 { 2 GAMMA(n/2 - 1/2 - RootOf(4 _Z + 4 _Z + 5)) GAMMA(n/2 + 1/2 + RootOf(4 _Z + 4 _Z + 5)) n::odd { n 2 2 { 2 GAMMA(n/2 - RootOf(_Z + 1)) GAMMA(n/2 + RootOf(_Z + 1)) n::even} { { 0 n::odd "A166748" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even {{ , { (n - 1) 2 2 { 2 GAMMA(n/2 - 1/2 - RootOf(4 _Z + 4 _Z + 37)) GAMMA(n/2 + 1/2 + RootOf(4 _Z + 4 _Z + 37)) n::odd { n 2 2 { 2 GAMMA(n/2 - 3 RootOf(_Z + 1)) GAMMA(n/2 + 3 RootOf(_Z + 1)) n::even} { { 0 n::odd "A167022" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A167449" LREtools/SearchTable: "SearchTable not successful" {} "A167479" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) | {(-2) , (-2) | ) ------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (n1 + 2) (-2) | \n1 = 0 / "A167481" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(-2) , (-2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-2) | \n1 = 0 / "A167539" memory used=84241.1MB, alloc=2423.5MB, time=617.38 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A167550" n {4 n!, n!} "A167559" n n! 4 n! (2 n + 1) {1/2 -------, 1/2 ---------------} n + 3/2 n + 3/2 "A167571" n (n + 1) n! (n + 1) 4 n! {----------, -------------} 2 n + 3 2 n + 3 "A167576" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | 2 (-1) binomial(2 n1, n1) n1! || {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A167577" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), /n - 1 \ |----- / n1 \| n | \ | 2 (-1) binomial(2 n1, n1) n1! || (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) | ) |---------------------------------------------------------------------||} | / \(n1 - 1/2) (2 n1 + 3) (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A167578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) /n - 1 \ |----- / n1 \| | \ | 2 (-1) binomial(2 n1, n1) n1! || | ) |-------------------------------------------------------------------------------------------||} | / \(2 n1 - 3) (2 n1 - 1) (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A167588" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), /n - 1 \ |----- / n1 \| n | \ | 2 (-1) binomial(2 n1, n1) n1! || (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) | ) |----------------------------------------------------------||} | / \(2 n1 + 3) (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A167589" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) /n - 1 \ |----- / n1 \| | \ | 2 (-1) binomial(2 n1, n1) n1! || | ) |--------------------------------------------------------------------------------||} | / \(n1 - 1/2) (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A167635" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A167636" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 5 | | 5 | {|- 1/2 - ----| , |- 1/2 + ----| } \ 2 / \ 2 / "A167638" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A167639" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 5 | | 5 | {|- 1/2 - ----| , |- 1/2 + ----| } \ 2 / \ 2 / "A167660" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | n n | \ binomial(2 n1, n1) (3 n1 + 3 n1 - 3 n1 - 2)| {(-1/2) (n - 1), (-1/2) (n - 1) | ) ---------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) n1 (2 n1 - 2) | \n1 = 0 / "A167713" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) | n n | \ 2 (2 n1 + 1) binomial(2 n1, n1)| {16 , 16 | ) ------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A167760" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \ \\ |----- | |----- | |----- n2 | || n n | \ n1 | n | \ | n1 | \ (-1) | || {(-1) n, (-1) n | ) (-(-1) n1!)|, (-1) n | ) |-(-1) | ) ---------| n1!||} | / | | / | | / (n2 + 1)!| || |----- | |----- | |----- | || \n1 = 0 / \n1 = 0 \ \n2 = 0 / // "A167859" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 2) 2 2| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {4 , 4 | ) --------------------------------------------|} | / 2 | |----- (n1 + 1) | \n1 = 0 / "A167867" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) 3 3| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {2 , 2 | ) ------------------------------------------|} | / 3 | |----- (n1 + 1) | \n1 = 0 / "A167868" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) 3 3| n n | \ 3 (2 n1 + 1) binomial(2 n1, n1) | {3 , 3 | ) ------------------------------------------|} | / 3 | |----- (n1 + 1) | \n1 = 0 / "A167869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 2) 3 3| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {4 , 4 | ) --------------------------------------------|} | / 3 | |----- (n1 + 1) | \n1 = 0 / "A167870" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 3 3| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {16 , 16 | ) --------------------------------------------|} | / 3 | |----- (n1 + 1) | \n1 = 0 / "A167871" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-6 n1 - 6) 3 3| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {64 , 64 | ) --------------------------------------------|} | / 3 | |----- (n1 + 1) | \n1 = 0 / "A167892" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 2 \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) -------------------------------------------} / 2 2 2 ----- (n1 + 3) (n1 + 2) (n1 + 1) n1 = 0 "A167893" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 3 3 3 \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, ) -------------------------------------------} / 3 3 3 ----- (n1 + 3) (n1 + 2) (n1 + 1) n1 = 0 "A167987" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 2 {1, ) (-2) n1! (n1 + 1) (n1 + 2) ((2 n1 + 3) (4 n1 + 12 n1 + 3) BesselJ(n1 + 1/2, -1) + (4 n1 + 16 n1 + 14) BesselJ(n1 - 1/2, -1)), / ----- n1 = 0 n - 1 ----- \ n1 2 2 ) (-2) n1! (n1 + 1) (n1 + 2) ((2 n1 + 3) (4 n1 + 12 n1 + 3) BesselY(n1 + 1/2, -1) + (4 n1 + 16 n1 + 14) BesselY(n1 - 1/2, -1))} / ----- n1 = 0 "A168049" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168051" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168055" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168058" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168073" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168076" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} n (n - 1) "A168452" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 2 (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {--------------------------------------, binomial(2 n, n) ( 2 2 (n + 1) (n + 3) (n + 2) 5 4 3 2 2 (4 n + 44 n + 200 n + 456 n + 510 n + 223) (2 n + 1) hypergeom([1/2, 1/2, -n - 1, -n - 1], [1, -n - 1/2, -n - 1/2], 1) 3 2 4 / 2 3 2 - 8 (2 n + 16 n + 47 n + 49) (n + 1) hypergeom([1/2, 1/2, -n, -n], [1, -n + 1/2, -n + 1/2], 1)) / ((n + 1) (n + 2) (n + 3) (n + 4))} / "A168490" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A168492" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A168494" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A168503" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A168505" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A168506" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A168592" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" LREtools/SearchTable: "SearchTable successful" (8 n + 3) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 7) hypergeom([-1/2, -n], [1], -8) {----------------------------------------------------------------------------------------, n + 2 (6 n + 5) hypergeom([1/2, -2 n - 2], [1], 4) + (-6 n - 3) hypergeom([1/2, -2 n], [1], 4) ----------------------------------------------------------------------------------------} (4 n + 3) (n + 2) "A168595" LREtools/SearchTable: "SearchTable successful" {(4 n (4 n + 7) (n - 5) (4 n + 5) hypergeom([-n - 3, -2 n - 5], [-2 n - 7/2], 1/4) 4 3 2 / 2 + (n + 431 n + 1481 n + 1669 n + 594) hypergeom([-n - 2, -2 n - 3], [-2 n - 3/2], 1/4)) binomial(4 n, 2 n) (4 n + 1) (4 n + 3) / ((n + 1) / (n + 2) (2 n + 1) (2 n + 3) (7 n + 11))} "A168597" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(7*n^2+28*n+27)*E^2-3*(2*n+3)*(7*n^2+28*n+27)*E+27*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" 2 {hypergeom([1/2, -n], [1], 4) } "A169714" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A169715" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A169895" 4 3 2 n (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) binomial(2 n, n) (103 n + 2418 n + 21041 n + 80478 n + 114168) {4 (2 n - 37), ---------------------------------------------------------------------------------------------------------} (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A170941" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A171074" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 1 "A171155" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A171199" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A171215" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A171416" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A171556" binomial(2 n, n) {1, ----------------} n + 1 "A171853" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) , 2 n + 1} "A172025" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 4) (-1/2) | \n1 = 0 / "A172060" n {4 , binomial(2 n, n) (2 n + 1)} "A172061" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 11)| {(-1/2) , (-1/2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 1) (-1/2) | \n1 = 0 / "A172062" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 13)| {(-1/2) , (-1/2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 1) (-1/2) | \n1 = 0 / "A172063" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (2 n1 + 7) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) --------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 7) (n1 + 6) (n1 + 4) (n1 + 1) (-1/2) | \n1 = 0 / "A172064" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (2 n1 + 7) binomial(2 n1, n1) (3 n1 + 17)| {(-1/2) , (-1/2) | ) --------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 1) (-1/2) | \n1 = 0 / "A172065" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 9) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 19)| {(-1/2) , (-1/2) | ) -------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 9) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 1) (-1/2) | \n1 = 0 / "A172066" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 9) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(-1/2) , (-1/2) | ) -------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 10) (n1 + 9) (n1 + 8) (n1 + 6) (n1 + 1) (-1/2) | \n1 = 0 / "A172067" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 11) (2 n1 + 9) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 23)| {(-1/2) , (-1/2) | ) -------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 11) (n1 + 10) (n1 + 9) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 1) (-1/2) | \n1 = 0 / "A172361" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A172634" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A172642" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A172651" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A172660" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A172675" memory used=85007.8MB, alloc=2423.5MB, time=622.90 LREtools/SearchTable: "SearchTable not successful" {} "A172743" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A173038" 2 {1, n! (n - n - 4)} "A173103" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 4 3 2 3 | 3 4 2 | \ (-1) (n1!) (n1 + 2 n1 - 4 n1 - 3 n1 + 7) (n1 + 1) n1 (n1 - 1) | (n!) (n - 4 n - n + 3) | ) ----------------------------------------------------------------------| | / 3 4 2 4 2 | 3 4 2 |----- ((n1 + 1)!) ((n1 + 1) - 4 (n1 + 1) - n1 + 2) (n1 - 4 n1 - n1 + 3)| (n!) (n - 4 n - n + 3) \n1 = 0 / 3 {-------------------------, ---------------------------------------------------------------------------------------------------------, (n!) 3 3 n (n - 1) (n - 2) n (n - 1) (n - 2) /n - 1 |----- 4 2 | \ n1 2 4 3 2 (n - 4 n - n + 3) | ) (-1) (n1!) (n1 + 2 n1 - 4 n1 - 3 n1 + 7) | / |----- \n1 = 0 /n1 - 1 \ |----- / n2 4 3 2 5 4 3 2 \| | \ | (-1) (n2 + 1) (n2 + 4 n2 + 2 n2 - 5 n2 - 1) (n2 + 6 n2 + 6 n2 - 13 n2 - 7 n2 + 2) n2! || 3 / | ) |- ------------------------------------------------------------------------------------------------|| (n1 + 1) n1 (n1 - 1) / ( | / | 4 3 2 4 3 2 2|| / |----- \ ((n2 + 1) + 2 (n2 + 1) - 4 (n2 + 1) - 3 n2 + 4) (n2 + 2 n2 - 4 n2 - 3 n2 + 7) ((n2 + 1)!) /| \n2 = 0 / \ | 3 4 2 4 2 | / 3 ((n1 + 1)!) ((n1 + 1) - 4 (n1 + 1) - n1 + 2) (n1 - 4 n1 - n1 + 3))| / (n (n - 1) (n - 2))} | / | / "A173104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 n1 3 4 3 2 | 4 4 2 | \ (n1 + 1) (-1) (n1!) (n1 + 2 n1 - 4 n1 - 3 n1 + 7) n1 (n1 - 1) | (n + 1) (n!) (n - 4 n - n + 3) | ) ----------------------------------------------------------------------| | / 4 4 2 4 2 | 4 4 2 |----- ((n1 + 1)!) ((n1 + 1) - 4 (n1 + 1) - n1 + 2) (n1 - 4 n1 - n1 + 3)| (n + 1) (n!) (n - 4 n - n + 3) \n1 = 0 / {---------------------------------, -----------------------------------------------------------------------------------------------------------------, 3 3 n (n - 1) (n - 2) n (n - 1) (n - 2) /n - 1 |----- 4 4 2 | \ 4 n1 3 4 3 2 (n + 1) (n!) (n - 4 n - n + 3) | ) (n1 + 1) (-1) (n1!) (n1 + 2 n1 - 4 n1 - 3 n1 + 7) | / |----- \n1 = 0 /n1 - 1 \ |----- / n2 2 2 4 3 2 5 4 3 2 \| | \ | (-1) (n2 + 1) (n2!) (n2 + 4 n2 + 2 n2 - 5 n2 - 1) (n2 + 6 n2 + 6 n2 - 13 n2 - 7 n2 + 2)|| / 4 | ) |- --------------------------------------------------------------------------------------------------|| n1 (n1 - 1) / (((n1 + 1)!) | / | 4 3 2 4 3 2 3 || / |----- \ ((n2 + 1) + 2 (n2 + 1) - 4 (n2 + 1) - 3 n2 + 4) (n2 + 2 n2 - 4 n2 - 3 n2 + 7) ((n2 + 1)!) /| \n2 = 0 / \ | 4 2 4 2 | / 3 ((n1 + 1) - 4 (n1 + 1) - n1 + 2) (n1 - 4 n1 - n1 + 3))| / (n (n - 1) (n - 2))} | / | / "A173184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- n2 | \ \ | \ (-1) | {1, ) (n1 + 1) n1!, ) (n1 + 1) n1! | ) ------------------|} / / | / (n2 + 2) (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A173227" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) (n1 + 1) n1! (LaguerreL(n1 + 1, -1) - LaguerreL(n1, -1))} / ----- n1 = 0 "A173314" {1, n!} "A173316" {1, n!} "A173317" {1, n!} "A173319" {1, n!} "A173321" {1, n!} "A173322" {1, n!} "A173323" {1, n!} "A173324" {1, n!} "A173516" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 | {(n + 1) 3 n!, (n + 1) 3 n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A173781" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n { 3 (n + 2) (4 n + 5) GAMMA(n/3 + 3/2) GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/6) { ----------------------------------------------------------------------- irem(n, 3) = 0 { 11 17 { (2 n + 1) (2 n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + --) GAMMA(n/3 + --) { 12 12 { { (n - 1) {{ 6 3 (n + 1) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/6) , { --------------------------------------------------------------------- irem(n, 3) = 1 { 13 { (2 n + 1) GAMMA(n/3 + 4/3) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) { 12 { { n { 3 GAMMA(1/6 + n/3) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/6) { ----------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1) GAMMA(n/3 + 1/4) GAMMA(n/3 + 3/4) { 2 n { 9 binomial(---, n/3) binomial(2 n, n) { 3 { ------------------------------------- irem(n, 3) = 0 { 4 n { binomial(---, n/3) { 3 { { 2 n { binomial(2 n + 4, n + 2) binomial(--- + 4/3, n/3 + 2/3) (n + 2) (4 n + 5) { 3 { ------------------------------------------------------------------------- irem(n, 3) = 1, { 4 n { binomial(--- + 8/3, n/3 + 2/3) (2 n + 3) (2 n + 1) { 3 { { 2 n { 6 (n + 1) binomial(--- + 2/3, n/3 + 1/3) binomial(2 n + 2, n + 1) { 3 { ----------------------------------------------------------------- irem(n, 3) = 2 { 4 n { (2 n + 1) binomial(--- + 4/3, n/3 + 1/3) { 3 { n { 6 3 (n + 1) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/6) { --------------------------------------------------------------- irem(n, 3) = 0 { 13 { (2 n + 1) GAMMA(n/3 + 4/3) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) { 12 { { (n - 1) { 9 3 GAMMA(1/6 + n/3) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/6) } { ------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 1) GAMMA(n/3 + 1/4) GAMMA(n/3 + 3/4) { { (n + 1) { 3 (n + 2) (4 n + 5) GAMMA(n/3 + 3/2) GAMMA(n/3 + 5/6) GAMMA(n/3 + 7/6) { ----------------------------------------------------------------------------- irem(n, 3) = 2 { 11 17 { (2 n + 1) (2 n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + --) GAMMA(n/3 + --) { 12 12 "A173936" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (2 n1 + 1) (2 n1 + 3) n1!\| {(-2) n!, (-2) n! | ) |-1/2 -------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A173992" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A173993" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 3 } "A173998" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- n1 ----- n1 \ 3 (7 LegendreP(n1 + 1, 7/3) - 3 LegendreP(n1, 7/3)) \ 3 (7 LegendreQ(n1 + 1, 7/3) - 3 LegendreQ(n1, 7/3)) {1, ) -----------------------------------------------------, ) -----------------------------------------------------} / n1 + 2 / n1 + 2 ----- ----- n1 = 0 n1 = 0 "A174013" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A174015" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A174016" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A174107" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A174123" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) hypergeom([1/2, -n1 - 1, -n1 - 1], [1, 1], 4)} / ----- n1 = 0 "A174169" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A174171" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A174193" 4 3 2 binomial(2 n, n) (n - 5 n + 10 n + 14 n - 8) {1, -----------------------------------------------} (n + 2) (n + 1) (2 n - 1) "A174227" LREtools/SearchTable: "SearchTable successful" n 3 2 2 10 ((32 n - 624 n + 2614 n - 1875) hypergeom([-5/2, -n - 1], [1], -2/5) - 2 n (16 n - 272 n + 837) hypergeom([-5/2, -n], [1], -2/5)) {----------------------------------------------------------------------------------------------------------------------------------------} n "A174297" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (3 n - 2) hypergeom([1/2, -n - 1], [1], 4) + (9 n - n - 2) hypergeom([1/2, -n], [1], 4) {------------------------------------------------------------------------------------------------} n (n - 1) "A174318" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) n1| {1, n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A174347" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / /3 n1\ \ n - 1 | |----| | ----- | \ 2 / 1/2 1/2 1/2 | \ | 2 (-2 LegendreP(n1 + 1, 2 ) + LegendreP(n1, 2 ))| {1, ) |- -------------------------------------------------------------|, / \ n1 + 2 / ----- n1 = 0 / /3 n1\ \ n - 1 | |----| | ----- | \ 2 / 1/2 1/2 1/2 | \ | 2 (-2 LegendreQ(n1 + 1, 2 ) + LegendreQ(n1, 2 ))| ) |- -------------------------------------------------------------|} / \ n1 + 2 / ----- n1 = 0 "A174399" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A174403" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A174662" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- (n2 + 1) | \ (-n1) \ (-n1) | \ 2 (n2 + 1) n2!| {1, ) 2 n1! (n1 + 1) (n1 + 2), ) 2 (n1 + 1) (n1 + 2) | ) ----------------------| n1!} / / | / (n2 + 3) (n2 + 1)! | ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A174764" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) ((n + 2) HermiteH(n + 1, 1/2 I 2 ) - 2 I 2 (n + 1) HermiteH(n, 1/2 I 2 )) (n + 1)} "A174783" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n2 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n2::even||| n - 1 |n1 - 1 | |{ n2 ||| ----- |----- | |{ (n2 + 1) binomial(n2, ----) ||| n \ n1 | \ | n2 |{ 2 ||| {1, (-1) , ) (-1) | ) |-(-1) |{ |||, / | / | |{ (2 n2 - 2) ||| ----- |----- | |{ 2 (n2 + 1) ||| n1 = 0 |n2 = 0 | |{ ---------------------------------------- n2::odd ||| | | |{ n2 ||| | | |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n2 \\\ n - 1 |n1 - 1 | |{ 4 binomial(n2, ----) (n2 + 1) ||| ----- |----- | |{ 2 ||| \ n1 | \ | n2 |{ ----------------------------- n2::even||| ) (-1) | ) |-(-1) |{ n2 + 2 |||} / | / | |{ ||| ----- |----- | |{ n2 ||| n1 = 0 |n2 = 0 | |{ 2 binomial(n2 + 1, ---- + 1/2) n2::odd ||| \ \ \{ 2 /// "A174808" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A174810" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A175167" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (3 n + 2) n { 2 { 1/2 ------------------------------ n::even { (n + 1) (n + 2) {{ , { 3 n { binomial(--- - 3/2, n/2 - 1/2) (3 n + 1) (3 n - 1) { 2 { 3/4 -------------------------------------------------- n::odd { (n + 2) n { (-n) 3 n 3 n { 6 4 (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------- n::even { (n + 2) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 4 2 n binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { --------------------------------------------------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) "A175912" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) (n + 6)} "A175934" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A175939" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1/2) (n + 14/3)} "A175962" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n {(3/2) (n - 22), /n - 1 \ |----- / (-n1 - 1) n1 2 2 \| n | \ |2 3 8 ((10 n1 - 110 n1 + 28) LegendreP(n1 + 1, 3/2) + (-15 n1 + 45 n1 - 42) LegendreP(n1, 3/2))|| (3/2) (n - 22) | ) |------------------------------------------------------------------------------------------------------------||, | / \ n1 (n1 - 21) (n1 + 2) (2 n1 - 44) /| |----- | \n1 = 0 / /n - 1 \ |----- / (-n1 - 1) n1 2 2 \| n | \ |2 3 8 ((10 n1 - 110 n1 + 28) LegendreQ(n1 + 1, 3/2) + (-15 n1 + 45 n1 - 42) LegendreQ(n1, 3/2))|| (3/2) (n - 22) | ) |------------------------------------------------------------------------------------------------------------||} | / \ n1 (n1 - 21) (n1 + 2) (2 n1 - 44) /| |----- | \n1 = 0 / "A176006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- ----- \ LegendreP(n1 + 1, 3) - 3 LegendreP(n1, 3) \ LegendreQ(n1 + 1, 3) - 3 LegendreQ(n1, 3) {1, ) -----------------------------------------, ) -----------------------------------------} / n1 / n1 ----- ----- n1 = 0 n1 = 0 "A176085" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) || {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) (2 + 5 ) binomial(2 n2, n2)||} | / | / || |----- |----- || \n1 = 0 \n2 = 0 // "A176097" LREtools/SearchTable: "SearchTable successful" 2 {(n + 1) (hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) - 8 hypergeom([-n, -n, -n], [1, 1], -1))} "A176280" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A176287" memory used=85786.8MB, alloc=2455.5MB, time=628.53 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A176332" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(1/2 - 1/2 I) , (1/2 + 1/2 I) , /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n | \ | \ (1/2 + 1/2 I) (2 n2 + 1) binomial(2 n2, n2) (5 n2 + 7)|| (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) ---------------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A176408" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A176479" LREtools/SearchTable: "SearchTable successful" (n + 1) (LegendreP(n + 1, 3) - 3 LegendreP(n, 3)) (n + 1) (LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3)) {-------------------------------------------------, -------------------------------------------------} n n "A176604" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176605" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176606" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176607" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176609" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176610" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176611" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176612" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176645" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176648" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176675" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176677" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176678" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176697" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176732" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ n1 / {n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957), n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957) | ) (-1) / ( | / / |----- \n1 = 0 6 5 4 3 2 (n1 + 1)! ((n1 + 1) + 27 (n1 + 1) + 280 (n1 + 1) + 1415 (n1 + 1) + 3634 (n1 + 1) + 4429 n1 + 6386) \ | 6 5 4 3 2 | (n1 + 27 n1 + 280 n1 + 1415 n1 + 3634 n1 + 4429 n1 + 1957))|} | | / "A176733" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 7 6 5 4 3 2 {n! (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700), n! /n - 1 |----- 7 6 5 4 3 2 | \ n1 / (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700) | ) (-1) / ((n1 + 1)! | / / |----- \n1 = 0 7 6 5 4 3 2 ((n1 + 1) + 35 (n1 + 1) + 490 (n1 + 1) + 3535 (n1 + 1) + 14084 (n1 + 1) + 30681 (n1 + 1) + 33375 n1 + 47075) \ | 7 6 5 4 3 2 | (n1 + 35 n1 + 490 n1 + 3535 n1 + 14084 n1 + 30681 n1 + 33375 n1 + 13700))|} | | / "A176734" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 8 7 6 5 4 3 2 {n! (n + 44 n + 798 n + 7756 n + 43939 n + 147532 n + 284066 n + 283072 n + 109601), n! /n - 1 |----- 8 7 6 5 4 3 2 | \ n1 / (n + 44 n + 798 n + 7756 n + 43939 n + 147532 n + 284066 n + 283072 n + 109601) | ) (-1) / ((n1 + 1)! | / / |----- \n1 = 0 8 7 6 5 4 3 2 ((n1 + 1) + 44 (n1 + 1) + 798 (n1 + 1) + 7756 (n1 + 1) + 43939 (n1 + 1) + 147532 (n1 + 1) + 284066 (n1 + 1) + 283072 n1 + 392673) \ | 8 7 6 5 4 3 2 | (n1 + 44 n1 + 798 n1 + 7756 n1 + 43939 n1 + 147532 n1 + 284066 n1 + 283072 n1 + 109601))|} | | / "A176735" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 9 8 7 6 5 4 3 2 {n! (n + 54 n + 1230 n + 15456 n + 117579 n + 558642 n + 1646714 n + 2878284 n + 2673321 n + 986410), n! /n - 1 |----- 9 8 7 6 5 4 3 2 | \ n1 / 9 (n + 54 n + 1230 n + 15456 n + 117579 n + 558642 n + 1646714 n + 2878284 n + 2673321 n + 986410) | ) (-1) / ((n1 + 1)! ((n1 + 1) | / / |----- \n1 = 0 8 7 6 5 4 3 2 + 54 (n1 + 1) + 1230 (n1 + 1) + 15456 (n1 + 1) + 117579 (n1 + 1) + 558642 (n1 + 1) + 1646714 (n1 + 1) + 2878284 (n1 + 1) + 2673321 n1 \ | 9 8 7 6 5 4 3 2 | + 3659731) (n1 + 54 n1 + 1230 n1 + 15456 n1 + 117579 n1 + 558642 n1 + 1646714 n1 + 2878284 n1 + 2673321 n1 + 986410))|} | | / "A176736" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 10 9 8 7 6 5 4 3 2 {n! (n + 65 n + 1815 n + 28590 n + 280413 n + 1782207 n + 7396325 n + 19664350 n + 31777851 n + 27845293 n + 9864101), n! /n - 1 |----- 10 9 8 7 6 5 4 3 2 | \ n1 / (n + 65 n + 1815 n + 28590 n + 280413 n + 1782207 n + 7396325 n + 19664350 n + 31777851 n + 27845293 n + 9864101) | ) (-1) / ( | / / |----- \n1 = 0 10 9 8 7 6 5 4 (n1 + 1)! ((n1 + 1) + 65 (n1 + 1) + 1815 (n1 + 1) + 28590 (n1 + 1) + 280413 (n1 + 1) + 1782207 (n1 + 1) + 7396325 (n1 + 1) 3 2 + 19664350 (n1 + 1) + 31777851 (n1 + 1) + 27845293 n1 + 37709394) \ | 10 9 8 7 6 5 4 3 2 | (n1 + 65 n1 + 1815 n1 + 28590 n1 + 280413 n1 + 1782207 n1 + 7396325 n1 + 19664350 n1 + 31777851 n1 + 27845293 n1 + 9864101))|} | | / "A176749" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176750" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176751" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176752" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176753" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176754" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176755" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176756" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176757" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176759" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176806" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A176826" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176828" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176829" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176830" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176832" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176855" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176856" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176857" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176858" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176859" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176952" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176953" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176954" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176956" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176958" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176959" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176962" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176966" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A176967" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177010" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (10 n - 7) LegendreP(n + 1, 3) + (-58 n + 11 n + 21) LegendreP(n, 3) {-----------------------------------------------------------------------------, n (n - 1) (n - 2) 2 (n + 1) (10 n - 7) LegendreQ(n + 1, 3) + (-58 n + 11 n + 21) LegendreQ(n, 3) -----------------------------------------------------------------------------} n (n - 1) (n - 2) "A177110" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177111" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177113" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177115" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177117" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177118" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177122" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177123" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177124" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177125" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177126" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177127" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177128" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177129" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177130" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177131" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177162" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177163" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177165" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177166" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177167" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177168" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177169" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177170" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177171" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177172" memory used=86573.9MB, alloc=2487.5MB, time=634.17 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177175" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177177" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177178" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177179" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177180" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177181" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177182" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177183" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177185" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177197" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177198" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177200" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177203" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A177249" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A177258" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A177259" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177265" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- n2 | \ \ | \ (-1) | {1, ) (n1 + 2) (n1 + 1) n1!, ) (n1 + 2) (n1 + 1) n1! | ) ------------------|} / / | / (n2 + 2) (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A177316" LREtools/SearchTable: "SearchTable successful" {hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1)} "A177322" LREtools/SearchTable: "SearchTable successful" 2 2 {((20 n + 28 n + 10) hypergeom([-2 n - 2, 2 n + 3, -n - 1], [1, 1], 1) - (2 n + 1) hypergeom([-2 n, -n, 2 n + 1], [1, 1], 1)) binomial(2 n, n) / 2 (2 n + 1) / ((n + 1) (48 n + 66 n + 23))} / "A177373" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 4 GAMMA(n/2 + 3/2) GAMMA(n/2 + 3/4) { 2 4 GAMMA(n/2 + 1) GAMMA(n/2 + 1/4) n::even { ------------------------------------ n::even { {{ n + 1 , { (2 n + 2) } { { 2 GAMMA(n/2 + 3/2) GAMMA(n/2 + 3/4) { (2 n - 2) { -------------------------------------------- n::odd { 2 2 GAMMA(n/2 + 1) GAMMA(n/2 + 1/4) n::odd { n + 1 "A177452" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (n1 + 3) {1, ) ------------------------------------------------------------} / (n1 + 5) (n1 + 4) (n1 + 1) ----- n1 = 0 "A177790" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A177792" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A177840" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! (n + 1) ((2 n + 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-2) n! (n + 1) ((2 n + 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A178061" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \ \\ | | |----- | || | | n2 | \ (n3 - 1) (n3 + 2)| || | | (-1) (n2 + 1) | ) -----------------| n2!|| n - 1 n - 1 |n1 - 1 | | / (n3 + 1)! | || n - 1 ----- ----- |----- | |----- | || ----- \ n1 \ n1 | \ | \n3 = 0 / || \ {1, ) (-1) n1!, ) (-1) n1! | ) |- ----------------------------------------------||, ) n1!} / / | / \ (n2 + 1)! /| / ----- ----- |----- | ----- n1 = 0 n1 = 0 \n2 = 0 / n1 = 0 "A178072" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(-1/2 I 2 ) , (1/2 I 2 ) } "A178074" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A178076" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A178078" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 6 | |6 | {|- ----| , |----| } \ 3 / \ 3 / "A178080" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(-1/3 I 6 ) , (1/3 I 6 ) } "A178113" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=87473.0MB, alloc=2487.5MB, time=640.16 { (n/2) { (-1) (3 hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) n::even { {{ (n/2 + 1/2) , { 4 (-1) ((n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + (- n/2 + 1/2) hypergeom([1/2, - n/2 - 1/2], [1], 4)) { --------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { 4 (-1) ((n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + (- n/2 + 1/2) hypergeom([1/2, - n/2 - 1/2], [1], 4)) { --------------------------------------------------------------------------------------------------------------------- n::even { n + 1 } { { (n/2 - 1/2) { (-1) (3 hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) n::odd "A178114" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A178520" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A178578" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A178594" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) 2 | n n | \ 6 ((n1 + 1) (17 n1 - 8) LegendreP(n1 + 1, 3) + (-99 n1 - 3 n1 + 24) LegendreP(n1, 3))| {6 , 6 | ) -----------------------------------------------------------------------------------------------|, | / n1 (n1 - 1) (n1 - 2) | |----- | \n1 = 0 / /n - 1 \ |----- (-n1 - 1) 2 | n | \ 6 ((n1 + 1) (17 n1 - 8) LegendreQ(n1 + 1, 3) + (-99 n1 - 3 n1 + 24) LegendreQ(n1, 3))| 6 | ) -----------------------------------------------------------------------------------------------|} | / n1 (n1 - 1) (n1 - 2) | |----- | \n1 = 0 / "A178669" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" {(n + 3) (n + 2) (n + 1) n!, / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | |n - 1 { \ 2 / / n1 \ | |----- { 1/4 (-2) |---- - 1/2|! (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1::odd | | \ { \ 2 / | (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / n1 / n1 \ | | { 1/4 (-1/2) binomial(n1, ----) |----|! (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1::even| |n - 1 { 2 \ 2 / | |----- { | | \ { 0 n1::odd | (n + 3) (n + 2) (n + 1) n! | ) -------------------------------------------------------------------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A178790" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ ) (2 n1 + 3) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) / ----- 1 n1 = 0 {-----, -----------------------------------------------------------------------------} n + 1 n + 1 "A178791" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 ) (-1) (2 n1 + 3) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) / ----- 1 n1 = 0 {-----, ------------------------------------------------------------------------------------} n + 1 n + 1 "A178792" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A178807" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A178808" LREtools/SolveLRE: "Reduced the order of" (n+3)*(2*n+3)*(n+4)^2*E^3-(n+3)*(2*n+3)*(35*n^2+211*n+310)*E^2+(2*n+7)*(n+2)*(35*n^2+139*n+130)*E-(2*n+ 7)*(n+2)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E+n+1 LREtools/SearchTable: "SearchTable successful" 2 2 {LegendreP(n + 1, 3) - 6 LegendreP(n, 3) LegendreP(n + 1, 3) + LegendreP(n, 3) , 2 2 LegendreQ(n + 1, 3) - 6 LegendreQ(n, 3) LegendreQ(n + 1, 3) + LegendreQ(n, 3) , LegendreP(n + 1, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) + LegendreP(n, 3) LegendreQ(n, 3)} "A178824" LREtools/SearchTable: "SearchTable successful" hypergeom([-n, -n, -n, -n], [1, 1, 1], 1) {-----------------------------------------} n + 1 "A178834" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n n (-1) ((n + 7 n + 8) hypergeom([1/2, -n - 1], [1], 4) - 3 (n + 1) (n + 4) hypergeom([1/2, -n], [1], 4)) {(-1) , 3 , --------------------------------------------------------------------------------------------------------} n + 2 "A179089" LREtools/SolveLRE: "Reduced the order of" (n+3)*(2*n+3)*(n+4)^2*E^3-(7*n+25)*(2*n+3)*(n+3)*(n+2)*E^2-3*(7*n+10)*(n+3)*(n+2)*(2*n+7)*E+27*(2*n+7)* (n+2)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" {-(hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A179176" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ n2 {(- 1/3 - 2/3 I 2 ) , (- 1/3 + 2/3 I 2 ) , (- 1/3 - 2/3 I 2 ) | ) (- 1/3 + 2/3 I 2 ) (- 1/3 - 2/3 I 2 ) | ) (-1) | / | / |----- |----- \n1 = 0 \n2 = 0 1/2 (-n2 - 1) (- 1/3 + 2/3 I 2 ) (2 n2 + 7) \\ || 2 || ((13 n2 + 55 n2 + 54) hypergeom([1/2, -n2 - 1], [1], 4) - 3 (5 n2 + 14) (n2 + 1) hypergeom([1/2, -n2], [1], 4))/((n2 + 2) (n2 + 3) (n2 + 4))||} || || // "A179190" LREtools/SearchTable: "SearchTable successful" n n (-2 I) ((3 n + 1) LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (-2 I) ((3 n + 1) LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- -------------------------------------------------------------------, - -------------------------------------------------------------------} (n - 1) n (n - 1) n "A179191" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) LegendreP(n, I), (-2 I) LegendreQ(n, I)} "A179508" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 3)|| {(-1) n!, (-1) n! | ) |- ---------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A179521" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n - 1 ----- \ 1 {1, ) ------} / n1 + 1 ----- n1 = 0 "A179533" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A179540" LREtools/SearchTable: "SearchTable successful" n {(n + 1) (-2) n! LaguerreL(n + 1, -n - 3/2, -1/2)} "A179648" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A180016" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A180064" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 4 binomial(n, n/2) ((n/2)!) n::even {{ 2 , { } { 4 ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { ----------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 2 { (n + 1) "A180189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) (-1) | {(n + 1) n!, (n + 1) n! | ) ---------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A180191" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, n! (n + 2), n! (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2)| |----- | \n1 = 0 / "A180195" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ n1 2 | {(-1) , (-1) | ) (-(-1) (n1 + 1) n1! (n1 + 3 n1 + 3))|} | / | |----- | \n1 = 0 / "A180218" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A180254" LREtools/SearchTable: "SearchTable successful" n {(24/5) GAMMA(n + 2/3) hypergeom([7/6, 2/3 - n], [4/3], -1/4)} "A180255" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ n1 + 1 | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A180282" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (-1) ((2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) {1, --------------------------------------------------------------------------------------------} n + 2 "A180283" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 (-1) ((7 n + 28 n + 27) hypergeom([1/2, -n - 1], [1], 4) - 3 (2 n + 5) (n + 1) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A180364" LREtools/SearchTable: "SearchTable successful" 2 2 {(n + 1) LegendreP(n + 1, 3) + (4 n + n - 1) LegendreP(n, 3), (n + 1) LegendreQ(n + 1, 3) + (4 n + n - 1) LegendreQ(n, 3)} "A180399" LREtools/SearchTable: "SearchTable successful" / 1/2 \n | 3 13 | {- |- ------- + 9/2| \ 2 / / 1/2 1/2 \ | 3 13 1/2 3 13 | |(18 n + 18) hypergeom([1/3, - 4/3 - n], [2/3], 13/2 + -------) + (3 + 13 ) (33 n + 20) hypergeom([1/3, - 1/3 - n], [2/3], 13/2 + -------)| \ 2 2 / 1/2 (13 - 3)/((3 n - 2) (3 n + 1))} "A180400" LREtools/SearchTable: "SearchTable successful" / 1/2 \n 1/2 | 3 13 | 3 13 {|- ------- + 9/2| hypergeom([2/3, 1/3 - n], [4/3], 13/2 + -------)} \ 2 / 2 "A180473" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / |n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 5 | | 5 | | 5 | | \ {|- 3/2 - ----| , |- 3/2 + ----| , |- 3/2 - ----| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 5 | ||| | |----- |- 3/2 + ----| ((37 n2 + 59) LegendreP(n2 + 1, 3) + (-7 n2 - 9) LegendreP(n2, 3))||| | 1/2 (-n1 - 1) / 1 \(-n1) | \ \ 2 / ||| |-2 (3 + 5 ) |- --------| | ) ------------------------------------------------------------------------------------------|||, | | 1/2 | | / (n2 + 3) (n2 + 2) ||| | \ 5 - 3/ |----- ||| \ \n2 = 0 /// / |n - 1 / 1/2\n |----- | 5 | | \ |- 3/2 - ----| | ) \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 5 | ||| | |----- |- 3/2 + ----| ((37 n2 + 59) LegendreQ(n2 + 1, 3) + (-7 n2 - 9) LegendreQ(n2, 3))||| | 1/2 (-n1 - 1) / 1 \(-n1) | \ \ 2 / ||| |-2 (3 + 5 ) |- --------| | ) ------------------------------------------------------------------------------------------|||} | | 1/2 | | / (n2 + 3) (n2 + 2) ||| | \ 5 - 3/ |----- ||| \ \n2 = 0 /// "A180564" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {-I (-1/2 I) (HermiteH(n + 1, 1/2 I) - 2 I (n + 3) HermiteH(n, 1/2 I))} "A180717" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A180879" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=88331.0MB, alloc=2487.5MB, time=646.02 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /5 n\ { |---| { \ 2 / { (n/2) { 2 (n + 2) { 12 2 binomial(n, n/2) (n + 1) (n + 3) { 1/4 -------------------------------- n::even { ------------------------------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { , { /5 n \ { (n/2 + 1/2) { |--- - 5/2| { 4 2 binomial(n + 1, n/2 + 1/2) (n + 2) { \ 2 / { ------------------------------------------------- n::odd { 2 (n + 1) (n + 3) { n + 3 { 3/4 -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { n { 4 (n + 2) (n + 5) { 8 binomial(n, n/2) (n + 5) (n + 3) (n + 1) { 1/8 -------------------------------- n::even { ------------------------------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) { , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) (n + 2) (n + 5) { 2 (n + 1) (n + 3) (n + 5) { -------------------------------------------- n::odd { 1/4 -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A180898" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" {(8 n + 3) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7) hypergeom([-1/2, -n], [1], -4)} "A180907" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A180967" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A181067" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + 4 hypergeom([-n, -n, -n], [1, 1], -1)} "A181072" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A181147" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n (-1) {1/2 --------------------------, (n + 1/2) binomial(2 n, n) /n - 1 \ |----- | n | \ / (n1 + 3/2) (2 n1 + 1) (105 n1 + 149) binomial(2 n1, n1) binomial(2 n1 + 2, n1 + 1) hypergeom([1/2, -n1 - 1], [1], 4)\| (-1) | ) |- --------------------------------------------------------------------------------------------------------------------|| | / \ (n1 + 1) (4 n1 + 6) /| |----- | \n1 = 0 / 1/2 ---------------------------------------------------------------------------------------------------------------------------------------} (n + 1/2) binomial(2 n, n) "A181197" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 8)|| {(-1) , (-1) | ) |- -----------------------------------------------------------------------------||} | / | 2 2 || |----- \ (2 n1 + 3) (2 n1 + 1) (n1 + 2) (n1 + 1) /| \n1 = 0 / "A181226" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" /n - 1 |----- n n / 2 47 \ n 2 n 2 | \ (n1 + 1) {(1/2) , (1/2) |n - -- n|, (1/2) (567 n - 1323 n - 1208), (1/2) (567 n - 1323 n - 1208) | ) 2 ( \ 27 / | / |----- \n1 = 0 5 4 3 2 (4032 n1 + 6832 n1 - 33126 n1 - 77278 n1 - 51865 n1 - 12649) hypergeom([-1/2, -n1 - 1], [1], -4) 4 3 2 / - (4032 n1 + 4816 n1 - 35282 n1 - 59714 n1 - 23759) (n1 + 1) hypergeom([-1/2, -n1], [1], -4)) / ((n1 + 2) (n1 + 3) / \ | 2 2 | (567 (n1 + 1) - 1323 n1 - 2531) (1134 n1 - 2646 n1 - 2416))|} | | / "A181237" (3 n + 1) (3 n + 2) binomial(2 n, n) binomial(3 n, n) {1, -----------------------------------------------------} 2 (n + 1) "A181418" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([-n, -n, -n], [1, 1], -1)} "A181517" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A181545" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A181546" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A181553" memory used=89158.0MB, alloc=2487.5MB, time=651.54 LREtools/SearchTable: "SearchTable successful" n / 23 22 21 20 19 18 {96 |(8388608 n - 113351065600 n + 701937957732352 n - 2645572480375193600 n + 6797456218208532430848 n - 12639261602675399688192000 n \ 17 16 15 + 17622439600403271038923046912 n - 18838470006473288822534181683200 n + 15658143752693208777659552186564608 n 14 13 12 - 10203486738857517130216366425765478400 n + 5233400868278191472644872904093770596352 n - 2113447345197219692293595380053934883020800 n 11 10 + 670021079847668137382234158360402110102007808 n - 165720032179789496548678975591531793029656524800 n 9 8 + 31668129652138793981170275153372658495686079409152 n - 4611324493208548599832732667898654471912011903014400 n 7 6 + 502076730696201188825807243355909439415381830518592128 n - 39839214288254113753094302851499254586636231119250939200 n 5 4 + 2224126361449268769708886142371107155792485733485834835232 n - 83126213690251173810942309240556695550652187769445859738000 n 3 2 + 1932200638544482316336129023991405034522259946602479745191000 n - 24813296178403181589468789984437263320819252547715057876587500 n + 141462419201341766798935175544001932560159811407401732157298750 n - 208598962028943298655513690015882834796677328396232505654859375) (n + 1) -1 24 23 22 21 hypergeom([-47/2, -n - 1], [1], --) + (-8388608 n + 113145544704 n - 699276671844352 n + 2629809487715762176 n 24 20 19 18 17 - 6740750052292019355648 n + 12500510860295848249524224 n - 17377367716120783632802906112 n + 18514797874715937266444153389056 n 16 15 14 - 15331408531085399229619223894622208 n + 9947935565333995594019551807423381504 n - 5077320211928753846265133686923282890752 n 13 12 + 2038754422066706346234066902082312020197376 n - 642023896731465682669079209067246449195102208 n 11 10 + 157531677052156363911712470968669117821512597504 n - 29813107415020858924898533984293350056695194954752 n 9 8 + 4289424087792120146531695181117163065017208888805376 n - 459957766179991729267862963875288389220531084982771328 n 7 6 + 35772397848414876053851009139103772100999746656281944064 n - 1942771116435774312614031218894085479298307314150058624032 n 5 4 + 69739153767816510501024838568532851179140520511486523806016 n - 1519257944100014576802302388858118121376668517614289433635000 n 3 2 + 17270383740631311043486447387732427581967957154264487280508000 n - 71513721702841016740061491451990954832681044490810500427498750 n - 30898385124879128895521887931521776769638577242023823369397500 n + 92613589445855585931282784235505722657554239339886265910656250) -1 \ hypergeom([-47/2, -n], [1], --)|} 24 / "A181618" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) hypergeom([1/2, -n - 1], [1], 4)} "A181641" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A181665" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A181734" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A181749" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A181768" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-n + 1) hypergeom([-1/2, -n], [1], -4) {------------------------------------------------------------------------------------} n "A181914" LREtools/SearchTable: "SearchTable successful" n 2 (n + 1) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) {----------------------------------------------------------} n + 2 "A181933" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-1/2 - 1/2 I) , (-1/2 + 1/2 I) , /n - 1 /n1 - 1 \\ |----- |----- / (n2 + 1) \|| n | \ | \ | (1 + I) (2 n2 + 1) (5 n2 + 8) binomial(2 n2, n2)||| (-1/2 - 1/2 I) | ) (-1 + I) exp(-1/2 I n1 Pi) | ) |- --------------------------------------------------------|||} | / | / \ (n2 + 1) (n2 + 2) /|| |----- |----- || \n1 = 0 \n2 = 0 // "A182027" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n binomial(n, n/2) { 2 4 { ------------------ n::even { ------------------ n::even { n - 1 { n binomial(n, n/2) {{ , { } { (n + 1) binomial(n + 1, n/2 + 1/2) { (2 n - 2) { 1/2 ---------------------------------- n::odd { 4 2 { n { ---------------------------------- n::odd { (n - 1) binomial(n - 1, n/2 - 1/2) "A182037" LREtools/SearchTable: "SearchTable successful" n n (-I) n! ((3 n + 1) LegendreP(n, I) + (n + 1) LegendreP(n + 1, I) I) (-I) n! ((3 n + 1) LegendreQ(n, I) + (n + 1) LegendreQ(n + 1, I) I) {- --------------------------------------------------------------------, - --------------------------------------------------------------------} (n - 1) n (n - 1) n "A182053" memory used=89913.0MB, alloc=2487.5MB, time=657.32 memory used=90491.6MB, alloc=2487.5MB, time=661.27 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A182186" n (2 n + 1) binomial(2 n, n) (3 n + 5) {4 (3 n + 11), ------------------------------------} n + 1 "A182333" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 4 3 2 { ((n/2)!) (n + 6 n + 9 n + 2 n + 2) { -------------------------------------- n::even {{ n , { { 2 2 2 { 1/2 ((n/2 - 1/2)!) (n + 4 n + 5) (n + 2 n + 2) n::odd { /-1\(n/2) 2 2 2 { |--| binomial(n, n/2) ((n/2)!) (2 n + 6 n + 6) n::even { \16/ { , { /-1\(n/2 - 1/2) 2 2 2 { -|--| binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n (n + n - 1) n::odd { \16/ { (n/2) 2 2 { 2 (-1) ((n/2)!) (n + n - 1) { - ---------------------------------- n::even { n { , { (n/2 + 1/2) 2 2 { 4 (-1) ((n/2 + 1/2)!) (n + 3 n + 3) { ------------------------------------------------ n::odd { 2 { (n + 1) { (-n) 2 2 2 2 { 1/2 4 binomial(n, n/2) ((n/2)!) (n + 2 n + 2) (n + 4 n + 5) n::even { { (-2 n - 2) 2 2 4 3 2 } { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 6 n + 9 n + 2 n + 2) { ------------------------------------------------------------------------------------ n::odd { n "A182386" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) || {(-1) n!, (-1) n! | ) |- ---------||} | / \ (n1 + 1)!/| |----- | \n1 = 0 / "A182401" LREtools/SearchTable: "SearchTable successful" n 3 ((4 n - 1) hypergeom([-1/2, -n - 1], [1], -4/3) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4/3)) {-------------------------------------------------------------------------------------------------} n + 2 "A182421" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182430" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 - 1) n1!| {(1/2) n!, (1/2) n! | ) ----------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A182454" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A182486" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n n n | \ | \ (1/2 + 1/2 I) binomial(2 n2, n2)|| {(1/2 - 1/2 I) , (1/2 + 1/2 I) , (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) -----------------------------------------||} | / | / n2 + 1 || |----- |----- || \n1 = 0 \n2 = 0 // "A182520" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 2 3 ((n + 2) (n + n + 1) hypergeom([-n - 1], [n + 2], 1) - (n + 1) hypergeom([-n], [n + 1], 1)) binomial(2 n, n) (2 n + 1) {------------------------------------------------------------------------------------------------------------------------} n (n + 1) "A182525" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 n1! | {(1/2) n! binomial(2 n, n) n, (1/2) n! binomial(2 n, n) n | ) ---------------------------------------|} | / n1 binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A182541" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A182542" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 , ----------------------------------------------} (n + 4) (n + 2) (n + 1) "A182543" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A182555" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 (n + 3), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A182584" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 3 n 3 n { 4 (2 n - 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------------- n::even { (3 n - 1) binomial(n, n/2) {{ , { (-2 n + 2) 3 n 3 n { 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { --------------------------------------------------------------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 3 n { binomial(---, n/2) n::even { 2 { { 3 n } { binomial(--- + 3/2, n/2 + 1/2) (2 n - 1) (n + 1) { 2 { ------------------------------------------------ n::odd { (3 n + 1) (3 n - 1) "A182626" LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 1) LegendreQ(n + 1, 3) + (-5 n - 3) LegendreQ(n, 3)) (-1) ((5 n + 3) LegendreP(n, 3) + (-n - 1) LegendreP(n + 1, 3)) {----------------------------------------------------------------, - ----------------------------------------------------------------} n n "A182661" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A182827" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! LegendreP(n, 1/2), (-2) n! LegendreQ(n, 1/2)} "A182879" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182881" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182883" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182884" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182887" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182889" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 2 _Z - 2 _Z - 2 _Z + 1 "A182892" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - _Z - 2, index = 1) , RootOf(_Z - 2 _Z - _Z - 2, index = 2) , RootOf(_Z - 2 _Z - _Z - 2, index = 3) } "A182894" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2 _Z - 4, index = 1) , RootOf(_Z - 2 _Z - 2 _Z - 4, index = 2) , RootOf(_Z - 2 _Z - 2 _Z - 4, index = 3) } "A182897" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n 1/2 n 1/2 n | 5 | | 5 | {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , |3/2 - ----| , |3/2 + ----| } \ 2 / \ 2 / "A182899" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n 1/2 n 1/2 n | 5 | | 5 | {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , |3/2 - ----| , |3/2 + ----| } \ 2 / \ 2 / "A182901" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(-1) } "A182902" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A182904" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A182905" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1 - 3 ) , (1 + 3 ) } "A183069" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 2 \ (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 22 n1 + 17) (n1 + 3/2) binomial(2 n1 + 2, n1 + 1) ) ------------------------------------------------------------------------------------------ / 2 2 ----- (n1 + 1) (n1 + 2) (4 n1 + 6) 1 n1 = 0 {1/2 --------------------------, 1/2 -------------------------------------------------------------------------------------------------} (n + 1/2) binomial(2 n, n) (n + 1/2) binomial(2 n, n) "A183160" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) binomial(3 n1, n1) (14 n1 + 9)| {(-1/4) , (-1/4) | ) -----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (-1/4) | \n1 = 0 / "A183161" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A183204" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([-n, -n, -n, n + 1], [1, 1, -2 n], 1)} "A183249" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A183256" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 6 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A183257" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {1} "A183876" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A183893" memory used=91335.5MB, alloc=2487.5MB, time=667.07 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/ReduceToOrderTwo: "Only implemented for absolutely irreducible operators of order 3 or 4 whose coefficients are of type ratpoly(rational)" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A183894" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/ReduceToOrderTwo: "Only implemented for absolutely irreducible operators of order 3 or 4 whose coefficients are of type ratpoly(rational)" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A184018" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A184120" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A184185" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A184459" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A184539" (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) 4 3 2 {----------------------------------------------, 3 n + 22 n + 71 n + 100 n + 50} (n + 3) (n + 2) (n + 1) "A184596" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A184877" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 2 {{ , { 2 ((n/2)!) n::even} { (-n + 1) 2 2 2 { { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 0 n::odd "A184958" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- n1 | |----- | |----- n1 | | \ 1 | | \ (-1) | | \ n1! | | \ (-1) n1!| {n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n! | ) ----------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / \n1 = 0 / "A185009" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 2 {1, ) (n1 + 1) n1! (n1 + 4 n1 + 5)} / ----- n1 = 0 "A185010" n n (-1) binomial(2 n, n) 3 binomial(2 n, n) {----------------------, -------------------} n + 1 n + 1 "A185020" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n (1 - 3 ) binomial(2 n, n) (1 + 3 ) binomial(2 n, n) {----------------------------, ----------------------------} n + 1 n + 1 "A185087" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A185089" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A185106" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \\ \ | | |----- (-n3 - 1) || | | | n2 | \ 3 || | | | (n2 + 1) 3 n2! | ) ---------------------------|| | | |n1 - 1 | / (n3 + 2) (n3 + 3) (n3 + 1)!|| | | |----- |----- || | | (-n1 - 1) | \ \n3 = 0 /| | | 2 (n1 + 1) | ) -----------------------------------------------------| n1!| |n - 1 | / (n2 + 1)! | | |----- |----- | | n n n | \ \n2 = 0 / | {2 n!, 3 n!, 2 n! | ) --------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A185108" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 2) (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A185109" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 2) (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A185132" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 n1 \| n n | \ | 4 8 (-1) (2 n1 hypergeom([-1/2, -n1 - 1], [1], -2) + (-2 n1 - 1) hypergeom([-1/2, -n1], [1], -2))|| {(-1/4) , (-1/4) | ) |- -----------------------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A185251" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n n { 0 n::even { 4 {2 n, { , { ---------------- n::even} { n binomial(n - 1, n/2 - 1/2) n::odd { binomial(n, n/2) { { 0 n::odd "A185252" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n n { 0 n::even { 4 {2 n, { , { ---------------- n::even} { n binomial(n - 1, n/2 - 1/2) n::odd { binomial(n, n/2) { { 0 n::odd "A185308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A185309" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {(-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I) 2 I} "A185369" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A185387" n! {1, ----} n "A185619" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A185655" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 1) binomial(2 n1, n1) (7 n1 + 12) (3 n1 + 4) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) ) -------------------------------------------------------------------------------------------- / 2 ----- (n1 + 1) (n1 + 2) (4 n1 + 6) 1 n1 = 0 {----------------------------, ---------------------------------------------------------------------------------------------------} (2 n + 1) n binomial(2 n, n) (2 n + 1) n binomial(2 n, n) "A185965" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A185966" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186031" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 { { 4 binomial(n, n/2) binomial(2 n, n) { (4 n - 4) { ------------------- n::even {----------------, { 2 , { 2 } n + 1 { --------------------------------------- n::odd { (n + 2) { 2 2 2 { { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A186195" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ (-3) (3 LegendreP(n1 + 1, 1/3) - LegendreP(n1, 1/3))| n | \ (-3) (3 LegendreQ(n1 + 1, 1/3) - LegendreQ(n1, 1/3))| {(-2) , (-2) | ) ------------------------------------------------------|, (-2) | ) ------------------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- n1 (-2) | |----- n1 (-2) | \n1 = 0 / \n1 = 0 / "A186239" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186240" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186241" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186334" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186335" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186338" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A186341" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n n {(-1) , 2 } "A186359" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-I) | | \ I | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A186374" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) 2 | n | \ 2 (n1 + 1) (n1 - 3 n1 + 1) n1!| (1/2) n! (n - 2) | ) ---------------------------------------| | / n1 (n1 - 1) (2 n1 - 4) (n1 + 1)! | n |----- | (1/2) n! (n - 2) \n1 = 0 / {-----------------, ------------------------------------------------------------------} n n "A186375" LREtools/SearchTable: "SearchTable successful" n {4 hypergeom([1/2, -n, -n], [1, 1], 1)} "A186638" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A186639" n {(-2) (n - 2), n!} "A186648" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n { (4 n - 4) { 2 {4 , binomial(2 n, n), { 2 , { binomial(n, n/2) n::even} { ------------------------------ n::odd { { 2 2 { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A186738" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A186739" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I) {-----------------------------------------------------------------------------} n "A186760" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | | \ 1 | | \ n1! | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A186763" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ (-1) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A186768" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- n1 | |----- | |----- n1 | | \ 1 | | \ (-1) | | \ n1! | | \ (-1) n1!| {n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n! | ) ----------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / \n1 = 0 / "A186810" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) (n1 + 1) n1! LaguerreL(n1 + 1, 1)} / ----- n1 = 0 "A186828" memory used=92173.7MB, alloc=2487.5MB, time=672.83 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + _Z + 1, index = 1) , RootOf(_Z - _Z + _Z + 1, index = 2) , RootOf(_Z - _Z + _Z + 1, index = 3) } "A186858" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A186859" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A186940" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(3 _Z + _Z - 3 _Z + 1, index = 1) , RootOf(3 _Z + _Z - 3 _Z + 1, index = 2) , RootOf(3 _Z + _Z - 3 _Z + 1, index = 3) } "A186996" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A186997" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 binomial(2 n, n) {-------------------, n + 1 2 (2 n + 5) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) + (-14 n - 42 n - 30) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) --------------------------------------------------------------------------------------------------------------------------------------------} (n + 2) (2 n + 1) "A187044" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A187071" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n {(1/5 - 3/5 I) , (1/5 + 3/5 I) , (1/5 - 3/5 I) /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || | \ (-n1) | \ (1/5 + 3/5 I) ((37 n2 + 53) LegendreP(n2 + 1, 3) + (-7 n2 - 7) LegendreP(n2, 3))|| | ) (1/2 + 3/2 I) (-4/5 - 3/5 I) | ) -----------------------------------------------------------------------------------------||, | / | / n2 + 2 || |----- |----- || \n1 = 0 \n2 = 0 // n (1/5 - 3/5 I) /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || | \ (-n1) | \ (1/5 + 3/5 I) ((37 n2 + 53) LegendreQ(n2 + 1, 3) + (-7 n2 - 7) LegendreQ(n2, 3))|| | ) (1/2 + 3/2 I) (-4/5 - 3/5 I) | ) -----------------------------------------------------------------------------------------||} | / | / n2 + 2 || |----- |----- || \n1 = 0 \n2 = 0 // "A187151" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n { (4 n - 4) { 2 {4 , binomial(2 n, n), { 2 , { binomial(n, n/2) n::even} { ------------------------------ n::odd { { 2 2 { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A187246" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ 2 | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A187249" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A187254" LREtools/SolveLRE: "Reduced the order of" (n+5)*(2*n+3)*(n+6)*E^3-(2*n+5)*(13*n+37)*(n+2)*E^2-(13*n+15)*(2*n+3)*(n+2)*E+(2*n+5)*(n-1)*(n-2) "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" n 1/2 2 1/2 1/2 2 (-1) (%5 LegendreP(n, 3 I) + 2 I 3 %4 (n + 1) LegendreP(n, 3 I) %2 + %3 %2 ) {- ---------------------------------------------------------------------------------------, 2 2 2 (n - 2) (n - 1) n (n + 3) (n + 2) n 1/2 2 1/2 1/2 2 (-1) (%5 LegendreQ(n, 3 I) + 2 I 3 %4 (n + 1) LegendreQ(n, 3 I) %1 + %3 %1 ) n - ---------------------------------------------------------------------------------------, - (-1) 2 2 2 (n - 2) (n - 1) n (n + 3) (n + 2) 1/2 1/2 1/2 1/2 1/2 1/2 (%5 LegendreP(n, 3 I) LegendreQ(n, 3 I) + 3 %4 (n + 1) LegendreQ(n, 3 I) %2 I + 3 %4 (n + 1) %1 LegendreP(n, 3 I) I + %3 %2 %1) / 2 2 2 / ((n - 2) (n - 1) n (n + 3) (n + 2) )} / 1/2 %1 := LegendreQ(n + 1, 3 I) 1/2 %2 := LegendreP(n + 1, 3 I) 6 5 4 3 2 %3 := -2 n - 6 n + 17 n + 44 n + 39 n + 16 n + 108 5 4 3 2 %4 := 2 n + 3 n - 22 n - 39 n - 16 n - 108 6 5 4 3 2 %5 := -2 n - 4 n - 7 n - 170 n - 597 n - 696 n - 324 "A187256" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A187257" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A187258" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A187260" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A187306" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((n + 3) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) {(-1) , -----------------------------------------------------------------------------------------} n + 2 "A187430" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) n {--------------------------, (2 n + 1) (-1) binomial(2 n, n) (n + 1) (n + 2) (2 hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) - 3 hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4))/((n + 2) (5 n + 3))} "A187444" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 64 (3 n + 2) { 3 n { 1/4 ------------------------------------ n::even { 16 binomial(3 n, ---) (3 n + 1) { 3 n { 2 { (n + 1) (3 n + 1) binomial(3 n, ---) { ------------------------------- n::even { 2 {{ 3 n + 2 , { } { { (6 n - 6) { 3 n { 2 2 (3 n - 1) (3 n + 1) { 2 binomial(3 n + 3, --- + 3/2) n::odd { -------------------------------------------------- n::odd { 2 { 3 n { n (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) { 2 "A187536" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 \ (2 n1 + 1) ((2 n1)!) {1, ) ----------------------} / 3 ----- (n1 + 1) (n1!) n1 = 0 "A187538" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2\| n n | \ | (-1) (2 n1 + 1) ((2 n1)!) || {(-1) , (-1) | ) |- -----------------------------||} | / | 3 || |----- \ (n1 + 1) (n1!) /| \n1 = 0 / "A187539" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A187540" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A187543" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 2 ((2 n)!) ((2 n)!) hypergeom([1/2, 1/2, -n, -n], [1, -n + 1/2, -n + 1/2], 1) {---------, -------------------------------------------------------------------} 3 3 (n!) (n!) "A187638" LREtools/SearchTable: "SearchTable successful" 2 2 {binomial(2 n, n) ((n + 1) (2 n + 1) hypergeom([1/2, 1/2, -n - 1, -n - 1], [1, -n - 1/2, -n - 1/2], 1) 3 2 / 2 + (-4 n - 8 n - 8 n - 2) hypergeom([1/2, 1/2, -n, -n], [1, -n + 1/2, -n + 1/2], 1)) / (n (n - 1))} / "A187741" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { (n/2 + 1) (n/2)! n::even { 2 (4 n + 4) binomial(n, n/2) (n/2)! n::even {{ , { } { 2 (n/2 + 1/2)! n::odd { (-n + 1) { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A187764" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([1/2, -n - 1], [n + 2], -4) + (-7 n - 4) hypergeom([1/2, -n], [n + 1], -4)) binomial(2 n, n) {-----------------------------------------------------------------------------------------------------------------} n (3 n + 1) "A187830" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (2 I 2 (n + 1) (n + 4) HermiteH(n, 1/2 I 2 ) + (n + 6) (n + 3) HermiteH(n + 1, 1/2 I 2 ))} "A187916" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n (2 n + 1) binomial(2 n, n) { { 4 {--------------------------, { 4 binomial(n - 1, n/2 - 1/2) n , { -------------------------------- n::even} (n + 1) (n + 2) { ------------------------------ n::odd { (n + 1) (n + 3) binomial(n, n/2) { (n + 1) (n + 3) { { 0 n::odd "A187925" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188289" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(-1) , (-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A188312" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188314" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188350" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A188441" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) {1, ) -----------------------------------------------------------} / 2 ----- (n1 + 1) n1 = 0 "A188442" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1 - 5 ) , (5 + 1) , (1 - 5 ) | ) (5 + 1) (1 - 5 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) || | \ (-1) (5 + 1) ((3 n2 + 4) hypergeom([1/2, -n2 - 1], [1], 4) + (-n2 + 4) hypergeom([1/2, -n2], [1], 4))|| | ) -------------------------------------------------------------------------------------------------------------------||} | / n2 (n2 + 2) || |----- || \n2 = 0 // "A188460" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188464" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188476" memory used=93002.9MB, alloc=2487.5MB, time=678.87 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2 _Z - 4, index = 1) , RootOf(_Z - 2 _Z - 2 _Z - 4, index = 2) , RootOf(_Z - 2 _Z - 2 _Z - 4, index = 3) } "A188482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - 4 _Z - 1, index = 1) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 2) , RootOf(_Z - 3 _Z - 4 _Z - 1, index = 3) } "A188622" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) (3 n1 + 1)| {2 , 2 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A188648" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188675" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) {1, ) ----------------------------------------} / (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A188676" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) /| |----- | \n1 = 0 / "A188678" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 1) (2 n1 + 3) (2 n1 + 1) /| |----- | \n1 = 0 / "A188679" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 2 \ (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) {1, ) -------------------------------------------} / 2 2 ----- (2 n1 + 1) (n1 + 1) n1 = 0 "A188680" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2 2\| n n | \ | (-1) (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) || {(-1) , (-1) | ) |- --------------------------------------------------||} | / | 2 2 || |----- \ (2 n1 + 1) (n1 + 1) /| \n1 = 0 / "A188682" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 2 \ (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) {1, ) -------------------------------------------} / 2 2 ----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) n1 = 0 "A188683" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2 2\| n n | \ | (-1) (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) || {(-1) , (-1) | ) |- --------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (2 n1 + 3) (2 n1 + 1) /| \n1 = 0 / "A188684" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 2 2 \ (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) {1, ) -------------------------------------------} / 2 2 2 ----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) n1 = 0 "A188685" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2 2\| n n | \ | (-1) (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1) || {(-1) , (-1) | ) |- --------------------------------------------------||} | / | 2 2 2 || |----- \ (n1 + 1) (2 n1 + 3) (2 n1 + 1) /| \n1 = 0 / "A188686" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188687" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188805" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | {|- 3/2 - ----| n!, |- 3/2 + ----| n!} \ 2 / \ 2 / "A188860" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((n + 1) (2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (-2 n - 1) hypergeom([1/2, -n], [1], 4)) {3 (n + 2), -----------------------------------------------------------------------------------------------------} n "A188911" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188912" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188913" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A188914" 2 {1, (n + 1) n!} "A188918" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1)|| {(-1) , (-1) | ) |- ------------------------------------------------------------------||} | / | 2 || |----- \ (n1 + 1) /| \n1 = 0 / "A188946" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A189053" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { (2 n2 - 2) ||| | | | { 2 (n2 + 1) ||| | | | { 1/2 ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| n n 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(-2) , 2 , (5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) --------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { 2 binomial(n2, ----) (n2 + 1) ||| | | | { 2 ||| | | | { ----------------------------- n2::even||| |n - 1 | |n1 - 1 { n2 + 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (5 ) | ) |1/5 5 (-1) | ) -----------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A189054" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A189162" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 (2 n + 3), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A189177" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A189244" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A189390" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 (2 n + 1), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A189391" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 , { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A189538" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) {{ 2 (-2) ((2 n + 5) hypergeom([3/4, - n/2 - 3/4], [3/2], 9) + 4 n hypergeom([3/4, - n/2 + 1/4], [3/2], 9)) GAMMA(n/2 + 5/4) { ------------------------------------------------------------------------------------------------------------------------------ , n::even { GAMMA(n/2 + 2) (n/2 - 1/2) 8 (-2) ((2 n + 3) hypergeom([3/4, - n/2 - 1/4], [3/2], 9) + (4 n + 2) hypergeom([3/4, - n/2 + 3/4], [3/2], 9)) GAMMA(n/2 + 3/4) ------------------------------------------------------------------------------------------------------------------------------------------ , GAMMA(n/2 + 3/2) { , { n::odd { { (n/2) 8 (-2) ((2 n + 3) hypergeom([3/4, - n/2 - 1/4], [3/2], 9) + (4 n + 2) hypergeom([3/4, - n/2 + 3/4], [3/2], 9)) GAMMA(n/2 + 3/4) ------------------------------------------------------------------------------------------------------------------------------------ , n::even GAMMA(n/2 + 3/2) (n/2 + 1/2) 2 (-2) ((2 n + 5) hypergeom([3/4, - n/2 - 3/4], [3/2], 9) + 4 n hypergeom([3/4, - n/2 + 1/4], [3/2], 9)) GAMMA(n/2 + 5/4) } ------------------------------------------------------------------------------------------------------------------------------------ , n::odd GAMMA(n/2 + 2) "A189766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) (2 n1 + 3) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1)} / ----- n1 = 0 "A189772" LREtools/SearchTable: "SearchTable successful" n {-(-1/2 + 1/2 I) n! hypergeom([1 + I, -n + 1], [2], 1 - I)} "A189790" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { (-n) 2 2 2 {(n + 1) n!, (n + 1) (-1) n!, { , { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even} { (n - 1) 2 2 { { 1/4 2 (n + 1) ((n/2 - 1/2)!) n::odd { 0 n::odd "A189791" LREtools/SearchTable: "SearchTable successful" n {(n + 1) 2 n! hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1)} "A189832" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n n | \ {(-1/2 - 1/2 I) , (-1/2 + 1/2 I) , (-1/2 - 1/2 I) | ) (-1 + I) exp(-1/2 I n1 Pi) | / |----- \n1 = 0 /n1 - 1 \\ |----- / (n2 + 1) \|| | \ | (1 + I) ((7 n2 + 11) hypergeom([1/2, -n2 - 1], [1], 4) + (-9 n2 - 9) hypergeom([1/2, -n2], [1], 4))||| | ) |- -----------------------------------------------------------------------------------------------------------|||} | / \ n2 + 2 /|| |----- || \n2 = 0 // "A189849" LREtools/SearchTable: "SearchTable successful" n 2 {(-4) (n!) LaguerreL(n, -n - 1/2, -1/2)} "A189886" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 3 n {(1/6 RootOf(_Z + 3 _Z + 10, index = 1) - 1/6 RootOf(_Z + 3 _Z + 10, index = 1) + 2/3) n!, 3 2 3 n (1/6 RootOf(_Z + 3 _Z + 10, index = 2) - 1/6 RootOf(_Z + 3 _Z + 10, index = 2) + 2/3) n!, 3 2 3 n (1/6 RootOf(_Z + 3 _Z + 10, index = 3) - 1/6 RootOf(_Z + 3 _Z + 10, index = 3) + 2/3) n!} "A189911" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 3) { ------------------------ n::even { binomial(n, n/2) (n + 2) n::even { (n + 1) binomial(n, n/2) {{ , { } { binomial(n + 1, n/2 + 1/2) (n/2 + 3/2) n::odd { (2 n - 2) { 2 2 (n + 2) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A189912" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 3) hypergeom([1/2, -n - 1], [1], 4) + (n - 1) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------} n + 2 "A189924" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| /n - 1 \ |n - 1 | / (n2 + 2) (n2 + 1)!|| |----- | |----- |----- || | \ (n1 + 1) n1! | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ----------------------------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A189940" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (n + 1) (HermiteH(n + 1, 1/2 I 2 ) + 2 HermiteH(n, 1/2 I 2 ) I)} "A189944" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190155" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190156" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-2) } "A190159" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z + 2, index = 1) , RootOf(_Z - _Z - _Z + 2, index = 2) , RootOf(_Z - _Z - _Z + 2, index = 3) } "A190160" memory used=93836.4MB, alloc=2488.9MB, time=684.66 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" 2 {n + n - 4} "A190161" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A190162" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {n} "A190163" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A190165" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A190166" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A190168" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A190169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A190171" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z + _Z - _Z + _Z + 1 "A190253" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190254" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 13 | | 13 | {|3/2 + -----| , |- ----- + 3/2| } \ 2 / \ 2 / "A190255" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1/2) } "A190315" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190425" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) binomial(2 n, n) n::even { 2 { {{ 3 n 2 , { binomial(2 n + 2, n + 1) binomial(--- + 3/2, n/2 + 1/2) (n + 1) n { 2 { ------------------------------------------------------------------ n::odd { (3 n + 1) (2 n + 1) (3 n - 1) { (-n) 2 { 4 n binomial(2 n, n) binomial(3 n, n) { ------------------------------------------ n::even { 3 n { (3 n - 1) binomial(---, n/2) { 2 { } { (-2 n + 2) 2 { 2 binomial(2 n - 2, n - 1) (2 n - 1) (3 n - 2) binomial(3 n - 3, n - 1) { ---------------------------------------------------------------------------------- n::odd { 2 3 n { n binomial(--- - 3/2, n/2 - 1/2) { 2 "A190452" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190587" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2, index = 1) , RootOf(_Z - 2 _Z - 2, index = 2) , RootOf(_Z - 2 _Z - 2, index = 3) } "A190590" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190666" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (5 n + 8 n + 5) LegendreP(n + 1, 3) + (-n - 2 n - 3) LegendreP(n, 3) (5 n + 8 n + 5) LegendreQ(n + 1, 3) + (-n - 2 n - 3) LegendreQ(n, 3) {----------------------------------------------------------------------, ----------------------------------------------------------------------} (n + 2) (n + 3) (n + 2) (n + 3) "A190724" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" memory used=94657.3MB, alloc=2487.5MB, time=690.22 /n - 1 \ /n - 1 \ |----- | |----- | n n | \ (-n1 - 1) | n | \ (-n1 - 1) | {6 , 6 | ) 6 (LegendreP(n1 + 1, 3) - 7 LegendreP(n1, 3))|, 6 | ) 6 (LegendreQ(n1 + 1, 3) - 7 LegendreQ(n1, 3))|} | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / "A190725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / |n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 5 | | 5 | | 5 | | \ {|- 3/2 - ----| , |- 3/2 + ----| , |- 3/2 - ----| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 5 | ||| | |----- |- 3/2 + ----| ((7 n2 + 11) LegendreP(n2 + 1, 3) + (-n2 - 1) LegendreP(n2, 3))||| | 1/2 (-n1 - 1) / 1 \(-n1) | \ \ 2 / ||| |-2 (3 + 5 ) |- --------| | ) ---------------------------------------------------------------------------------------|||, | | 1/2 | | / n2 + 2 ||| | \ 5 - 3/ |----- ||| \ \n2 = 0 /// / |n - 1 / 1/2\n |----- | 5 | | \ |- 3/2 - ----| | ) \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 5 | ||| | |----- |- 3/2 + ----| ((7 n2 + 11) LegendreQ(n2 + 1, 3) + (-n2 - 1) LegendreQ(n2, 3))||| | 1/2 (-n1 - 1) / 1 \(-n1) | \ \ 2 / ||| |-2 (3 + 5 ) |- --------| | ) ---------------------------------------------------------------------------------------|||} | | 1/2 | | / n2 + 2 ||| | \ 5 - 3/ |----- ||| \ \n2 = 0 /// "A190726" LREtools/SearchTable: "SearchTable successful" {(3 (3 n + 5) (2 n + 3) (3 n + 1) (n + 2) hypergeom([-n - 1, 3 n + 6], [n + 4], -1) 2 - (105 n + 217 n + 104) (2 n + 1) (n + 3) hypergeom([-n, 3 n + 3], [n + 3], -1)) binomial(2 n, n)/((n + 1) (n + 2) (n + 3) (22 n + 29))} "A190729" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190733" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (2 I) ((n1 + 1) LegendreP(n1 + 1, I) I + LegendreP(n1, I)) I| {(-1) , (-1) | ) --------------------------------------------------------------|, | / n1 (n1 + 2) | |----- | \n1 = 0 / /n - 1 \ |----- n1 | n | \ (2 I) ((n1 + 1) LegendreQ(n1 + 1, I) I + LegendreQ(n1, I)) I| (-1) | ) --------------------------------------------------------------|} | / n1 (n1 + 2) | |----- | \n1 = 0 / "A190736" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 2 _Z + 1, index = 1) , RootOf(_Z - _Z + 2 _Z + 1, index = 2) , RootOf(_Z - _Z + 2 _Z + 1, index = 3) } "A190737" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z + _Z + 1, index = 1) , RootOf(_Z - 2 _Z + _Z + 1, index = 2) , RootOf(_Z - 2 _Z + _Z + 1, index = 3) } "A190788" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 2 _Z - 4 _Z + 8 _Z - 4 "A190823" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A190826" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190830" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190833" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A190835" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A190863" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190864" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - _Z - 4 "A190865" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190875" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190878" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A190901" n n n n {(-2) n!, 2 n!, (-1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n)} "A190917" LREtools/SearchTable: "SearchTable successful" (3 n + 5) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (3 n + 2) hypergeom([-n, -n, -n], [1, 1], -1) {---------------------------------------------------------------------------------------------------------} n + 2 "A190918" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A190920" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191237" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A191242" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191243" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A191307" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | /{ n1 \ |{ 1/2 --------------------------- n1::even| |{ 4 binomial(n1, ----) (n1 + 1) n1 | n - 1 |{ n1 | n - 1 |{ 2 | ----- |{ (n1 + 3) binomial(n1, ----) | ----- |{ -------------------------------- n1::even| \ |{ 2 | \ |{ (n1 + 2) (n1 + 4) | {1, ) |{ |, ) |{ |} / |{ (2 n1 - 2) | / |{ n1 | ----- |{ 2 (n1 + 1) | ----- |{ 2 binomial(n1 + 1, ---- + 1/2) (n1 + 1) | n1 = 0 |{ ---------------------------------------------- n1::odd | n1 = 0 |{ 2 | |{ n1 | |{ --------------------------------------- n1::odd | |{ (n1 + 2) (n1 + 4) binomial(n1 - 1, ---- - 1/2) | \{ n1 + 3 / \{ 2 / "A191309" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 (n + 1) { ------------------ n::even n { binomial(n, n/2) (n/2 + 1) n::even { n binomial(n, n/2) {2 , { , { } { binomial(n - 1, n/2 - 1/2) (n + 1) n::odd { (2 n + 2) { 2 2 (n + 2) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A191313" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 /{ n1 \ ----- |{ | ----- |{ 4 | n \ |{ n1 | \ |{ --------------------------- n1::even| {1, n, 2 , ) |{ 2 binomial(n1 - 1, ---- - 1/2) n1 |, ) |{ n1 |} / |{ 2 | / |{ (n1 + 1) binomial(n1, ----) | ----- |{ --------------------------------- n1::odd | ----- |{ 2 | n1 = 0 \{ n1 + 1 / n1 = 0 |{ | \{ 0 n1::odd / "A191317" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A191319" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /n1 - 1 / /{ n2 \\\ ----- |----- | |{ binomial(n2, ----) (2 n2 + 2) n2::even||| n \ n1 | \ | n2 |{ 2 ||| {1, (-1) , ) (-1) | ) |-(-1) |{ |||, / | / | |{ n2 ||| ----- |----- | |{ (n2 + 1) binomial(n2 + 1, ---- + 1/2) n2::odd ||| n1 = 0 \n2 = 0 \ \{ 2 /// / / /{ n2 \\\ | | |{ 4 ||| | | |{ 1/2 ------------------ n2::even||| n - 1 |n1 - 1 | |{ n2 ||| ----- |----- | |{ binomial(n2, ----) ||| \ n1 | \ | n2 |{ 2 ||| ) (-1) | ) |-(-1) |{ |||} / | / | |{ (2 n2 - 2) ||| ----- |----- | |{ 2 (n2 + 1) ||| n1 = 0 |n2 = 0 | |{ ------------------------------- n2::odd ||| | | |{ n2 ||| | | |{ n2 binomial(n2 - 1, ---- - 1/2) ||| \ \ \{ 2 /// "A191321" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" / / | | | | | | | | | | |n - 1 | / 1/2\n / 1/2 \n / 1/2\n |----- | | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) \ 2 / \ 2 / \ 2 / | / | |----- | |n1 = 0 | | | | | | | | | | | \ \ / /{ 0 irem(n2, 4) = 0\\\\ | |{ |||| | |{ 0 irem(n2, 4) = 1|||| | |{ |||| | |{ (-n2) 3 n2 3 n2 3 n2 n2 |||| | |{ 48 2 (3 n2 - 4) binomial(---- - 3, ---- - 3/2) binomial(---- - 3/2, ---- - 1/2) |||| |n1 - 1 |{ 2 4 4 4 |||| |----- / 1/2 \(-n2 - 1) |{ ------------------------------------------------------------------------------------ irem(n2, 4) = 2|||| | \ |5 | |{ n2 n2 |||| | ) |---- + 1/2| |{ n2 (n2 + 2) binomial(---- - 1, ---- - 1/2) ||||, | / \ 2 / |{ 2 4 |||| |----- |{ |||| |n2 = 0 |{ (-n2 - 1) 3 n2 3 n2 3 n2 n2 |||| | |{ 16 2 binomial(---- + 3/2, ---- + 3/4) binomial(---- + 3/4, ---- + 1/4) |||| | |{ 2 4 4 4 |||| | |{ ------------------------------------------------------------------------------- irem(n2, 4) = 3|||| | |{ n2 n2 |||| | |{ (n2 + 3) binomial(---- + 1/2, ---- + 1/4) |||| \ \{ 2 4 //// / / | | | | | | | | | | | | | | |n - 1 | / 1/2\n |----- | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) \ 2 / | / | |----- | |n1 = 0 | | | | | | | | | | | | | | | \ \ / /{ 0 irem(n2, 4) = 0\\\\ | |{ |||| | |{ / n2 \ |||| | |{ |---- - 1/4| |||| | |{ \ 4 / n2 n2 |||| / | |{ 3 (27/4) GAMMA(---- + 2/3) GAMMA(---- + 1/3) |||| | | |{ 4 4 |||| | | |{ -------------------------------------------------------- irem(n2, 4) = 1|||| | |n1 - 1 |{ n2 n2 |||| | |----- / 1/2 \(-n2 - 1) |{ GAMMA(---- + 3/2) GAMMA(---- + 1) |||| / 1/2\n | | \ |5 | |{ 4 4 |||| | 5 | | | ) |---- + 1/2| |{ ||||, |1/2 - ----| | | / \ 2 / |{ / n2 \ |||| \ 2 / | |----- |{ |---- + 1/2| |||| | |n2 = 0 |{ \ 4 / n2 13 n2 17 |||| | | |{ 4 (27/4) GAMMA(---- + --) GAMMA(---- + --) (n2 + 5) |||| | | |{ 4 12 4 12 |||| | | |{ --------------------------------------------------------------- irem(n2, 4) = 2|||| | | |{ n2 n2 |||| \ | |{ GAMMA(---- + 7/4) GAMMA(---- + 9/4) (3 n2 + 5) |||| | |{ 4 4 |||| | |{ |||| \ \{ 0 irem(n2, 4) = 3//// / / /{ 3 n2 n2 \\\\ | | |{ 6 binomial(----, ----) |||| | | |{ 4 4 |||| | | |{ ---------------------- irem(n2, 4) = 0|||| n - 1 | |n1 - 1 |{ n2 + 2 |||| ----- | |----- / 1/2 \(-n2 - 1) |{ |||| \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ 3 n2 n2 |||| ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 8 binomial(---- + 9/4, ---- + 3/4) |||| / | | / \ 2 / |{ 4 4 |||| ----- | |----- |{ ---------------------------------- irem(n2, 4) = 1|||| n1 = 0 | |n2 = 0 |{ 3 n2 + 5 |||| | | |{ |||| | | |{ 0 irem(n2, 4) = 2|||| | | |{ |||| \ \ \{ 0 irem(n2, 4) = 3//// / / | | | | | | | | | | | | | | |n - 1 | / 1/2\n |----- | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) \ 2 / | / | |----- | |n1 = 0 | | | | | | | | | | | | | | | \ \ / /{ / n2 \ \\\\ | |{ |----| |||| | |{ \ 4 / n2 13 n2 17 |||| | |{ 4 (27/4) GAMMA(---- + --) GAMMA(---- + --) (n2 + 5) |||| | |{ 4 12 4 12 |||| | |{ --------------------------------------------------------- irem(n2, 4) = 0|||| | |{ n2 n2 |||| | |{ GAMMA(---- + 7/4) GAMMA(---- + 9/4) (3 n2 + 5) |||| |n1 - 1 |{ 4 4 |||| |----- / 1/2 \(-n2 - 1) |{ |||| | \ |5 | |{ 0 irem(n2, 4) = 1|||| | ) |---- + 1/2| |{ ||||} | / \ 2 / |{ 0 irem(n2, 4) = 2|||| |----- |{ |||| |n2 = 0 |{ / n2 \ |||| | |{ |---- - 3/4| |||| | |{ \ 4 / n2 n2 |||| | |{ 3 (27/4) GAMMA(---- + 1/3) GAMMA(---- + 2/3) |||| | |{ 4 4 |||| | |{ -------------------------------------------------------- irem(n2, 4) = 3|||| | |{ n2 n2 |||| | |{ GAMMA(---- + 1) GAMMA(---- + 3/2) |||| \ \{ 4 4 //// "A191354" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191385" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A191386" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { n binomial(n, n/2) n::even { ------------------ n::even n { { n binomial(n, n/2) {2 , { binomial(n + 1, n/2 + 1/2) (n - 1) (n + 1) , { } { 1/2 ------------------------------------------ n::odd { (2 n - 2) { n { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A191388" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 3 _Z + 2 _Z + 1 "A191389" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { binomial(n, n/2) (3 n + 2) { 4 4 (3 n - 1) { -------------------------- n::even { -------------------------- n::even n { n + 2 { n (n + 1) binomial(n, n/2) {2 , { , { } { 2 binomial(n - 1, n/2 - 1/2) (3 n - 1) { (2 n + 2) { -------------------------------------- n::odd { 2 2 (3 n + 2) { n + 1 { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A191391" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { -------------------------- n::even { 2 binomial(n, n/2) n::even n { n (n + 1) binomial(n, n/2) { {2 , { , { binomial(n + 1, n/2 + 1/2) (n - 1) } { (2 n - 2) { ---------------------------------- n::odd { 2 2 { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A191393" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 2 %1 := _Z - 3 _Z + 4 _Z - _Z - 2 "A191394" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n2 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n2::even||| n - 1 |n1 - 1 | |{ n2 ||| ----- |----- | |{ (n2 + 1) binomial(n2, ----) ||| n n \ n1 | \ | n2 |{ 2 ||| {1, (-1) , 2 , ) (-1) | ) |-(-1) |{ |||, / | / | |{ (2 n2 - 2) ||| ----- |----- | |{ 2 (n2 + 1) ||| n1 = 0 |n2 = 0 | |{ ---------------------------------------- n2::odd ||| | | |{ n2 ||| | | |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n2 \\\ n - 1 |n1 - 1 | |{ 4 binomial(n2, ----) (n2 + 1) ||| ----- |----- | |{ 2 ||| \ n1 | \ | n2 |{ ----------------------------- n2::even||| ) (-1) | ) |-(-1) |{ n2 + 2 |||} / | / | |{ ||| ----- |----- | |{ n2 ||| n1 = 0 |n2 = 0 | |{ 2 binomial(n2 + 1, ---- + 1/2) n2::odd ||| \ \ \{ 2 /// "A191396" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n {1, (-1) , 2 , (-1) (n + 4/3), / / / /{ n2 \ \\\ | | | |{ 4 (n2 + 2) (3 n2 + 8) | ||| | | | |{ 1/2 --------------------------- n2::even| ||| | | | |{ n2 | ||| | | | |{ (n2 + 1) binomial(n2, ----) | ||| | | | |{ 2 | ||| | | | |{ | (3 n2 + 7)||| | | | |{ (2 n2 + 2) | ||| | | | |{ 2 (n2 + 3) (3 n2 + 5) | ||| | | | |{ 1/4 ------------------------------------- n2::odd | ||| | | |n1 - 1 |{ n2 | ||| | | |----- |{ (n2 + 2) binomial(n2 + 1, ---- + 1/2) | ||| | | n1 | \ \{ 2 / ||| | | (-1) (3 n1 + 5) (3 n1 + 10) | ) ------------------------------------------------------------------------||| |n - 1 | | / (3 n2 + 13) (3 n2 + 8) (3 n2 + 10) (3 n2 + 5) ||| |----- | |----- ||| n | \ | \n2 = 0 /|| (-1) (3 n + 4) | ) |- ---------------------------------------------------------------------------------------------------------------||, | / \ (3 n1 + 4) (3 n1 + 7) /| |----- | \n1 = 0 / n (-1) (3 n + 4) / / / /{ n2 \ \\\ | | | |{ 2 binomial(n2, ----) (3 n2 + 5) (n2 + 1) (n2 + 3) | ||| | | | |{ 2 | ||| | | | |{ ------------------------------------------------- n2::even| ||| | | | |{ n2 + 2 | ||| | | | |{ | (3 n2 + 7)||| | | | |{ n2 | ||| | | | |{ 4 binomial(n2 - 1, ---- - 1/2) n2 (n2 + 2) (3 n2 + 8) | ||| | | |n1 - 1 |{ 2 | ||| | | |----- |{ ----------------------------------------------------- n2::odd | ||| | | n1 | \ \{ n2 + 1 / ||| | | (-1) (3 n1 + 5) (3 n1 + 10) | ) ------------------------------------------------------------------------------------||| |n - 1 | | / (3 n2 + 13) (3 n2 + 8) (3 n2 + 10) (3 n2 + 5) ||| |----- | |----- ||| | \ | \n2 = 0 /|| | ) |- ---------------------------------------------------------------------------------------------------------------------------||, n + 8/3 | / \ (3 n1 + 4) (3 n1 + 7) /| |----- | \n1 = 0 / } "A191398" memory used=95478.6MB, alloc=2519.5MB, time=696.18 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 3 _Z + 2 _Z + 1 "A191501" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z - 1, index = 1) , RootOf(_Z - _Z - _Z - 1, index = 2) , RootOf(_Z - _Z - _Z - 1, index = 3) } "A191520" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 (n + 3) { ------------------ n::even n { binomial(n, n/2) (n/2 + 2) n::even { n binomial(n, n/2) {2 , { , { } { binomial(n - 1, n/2 - 1/2) (n + 3) n::odd { (2 n + 2) { 2 2 (n + 4) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A191522" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n - 1) { binomial(n, n/2) (n - 2) n::even { -------------------------- n::even { { n (n + 1) binomial(n, n/2) {{ 2 , { } { binomial(n + 1, n/2 + 1/2) (n - 1) { (2 n - 2) { 1/2 ----------------------------------- n::odd { 2 2 (n - 2) { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A191524" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n - 4 n - 1) { binomial(n, n/2) (n - 4) n::even { -------------------------- n::even n { { n (n + 1) binomial(n, n/2) {2 , { 2 , { } { binomial(n + 1, n/2 + 1/2) (n - 4 n - 1) { (2 n - 2) { 1/2 ----------------------------------------- n::odd { 2 2 (n - 4) { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A191526" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { n2 \ \ | | { 12 binomial(n2, ----) (n2 + 1) | | | | { 2 | | | | { ------------------------------ n2::even| | | | { n2 + 2 | | | | { | | | | { n2 | | | | { 2 binomial(n2 + 1, ---- + 1/2) (3 n2 + 7) | | |n - 1 |n1 - 1 { 2 | | |----- |----- { ----------------------------------------- n2::odd | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ { n2 + 3 | | {(-1/2 I 2 ) , (1/2 I 2 ) , (-1/2 I 2 ) | ) (-1) 2 | ) -----------------------------------------------------------| I|, | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / / / { n2 \ \ | | { 4 (3 n2 + 7) | | | | { 1/2 ------------------------------------ n2::even| | | | { n2 | | | | { (n2 + 1) (n2 + 3) binomial(n2, ----) | | | | { 2 | | | | { | | | | { (2 n2 - 2) | | | | { 3 2 (n2 + 1) | | | | { ---------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) | | 1/2 n | \ n1 1/2 | \ { 2 | | (-1/2 I 2 ) | ) (-1) 2 | ) ----------------------------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (1/2 I 2 ) | | \n1 = 0 \n2 = 0 / / "A191527" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 1/2 ------------------------ n::even { binomial(n, n/2) (2 n - 2) n::even { (n + 1) binomial(n, n/2) {{ , { } { binomial(n + 1, n/2 + 1/2) (n - 1) n::odd { (2 n - 2) { 2 (n - 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A191529" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n binomial(n, n/2) { 2 4 (n - 3) { ------------------ n::even { -------------------------- n::even { n - 1 { n (n + 1) binomial(n, n/2) {{ , { } { binomial(n + 1, n/2 + 1/2) (n - 3) { (2 n - 2) { 1/2 ---------------------------------- n::odd { 4 2 { n { ---------------------------------- n::odd { (n - 1) binomial(n - 1, n/2 - 1/2) "A191531" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | |{ --------------------------- n1::even| n - 1 |{ n1 | n - 1 /{ n1 \ ----- |{ (n1 + 1) binomial(n1, ----) | ----- |{ 2 binomial(n1, ----) n1::even| \ |{ 2 | \ |{ 2 | {1, n, ) |{ |, ) |{ |} / |{ (2 n1 - 2) | / |{ n1 | ----- |{ 2 2 | ----- |{ binomial(n1 + 1, ---- + 1/2) n1::odd | n1 = 0 |{ ------------------------------- n1::odd | n1 = 0 \{ 2 / |{ n1 | |{ n1 binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A191585" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 3 3 | | 3 3 | {|3/2 - ------| , |3/2 + ------| , \ 2 / \ 2 / / / / 1/2\(-n2 - 1) \\ |n - 1 |n1 - 1 | 3 3 | || / 1/2\n |----- / 1/2\n1 / 1/2\(-n1 - 1) |----- |3/2 + ------| (2 n2 + 1) binomial(2 n2, n2) (n2 + 6)|| | 3 3 | | \ | 3 3 | | 3 3 | | \ \ 2 / || |3/2 - ------| | ) |3/2 + ------| |3/2 - ------| | ) --------------------------------------------------------------||} \ 2 / | / \ 2 / \ 2 / | / (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A191586" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191605" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2 _Z + 2, index = 1) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 2) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 3) } "A191625" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2 _Z + 2, index = 1) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 2) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 3) } "A191649" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191652" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - 2 _Z + 2, index = 1) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 2) , RootOf(_Z - 2 _Z - 2 _Z + 2, index = 3) } "A191662" n {(n + 1) n!, (n + 1) (-1) n!} "A191684" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A191782" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (3 n + 5) { 4 binomial(n, n/2) (3 n + 8) (n + 1) { 1/2 -------------------------------- n::even { ------------------------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {1, { , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (3 n + 5) { 2 (n + 1) (3 n + 8) { -------------------------------------- n::odd { -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A191786" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z - 1, index = 1) , RootOf(_Z - _Z - _Z - 1, index = 2) , RootOf(_Z - _Z - _Z - 1, index = 3) } "A191787" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n - 8 n - 1) { binomial(n, n/2) (n - 8) n::even { -------------------------- n::even n { { n (n + 1) binomial(n, n/2) {2 , { 2 , { } { binomial(n + 1, n/2 + 1/2) (n - 8 n - 1) { (2 n - 2) { 1/2 ----------------------------------------- n::odd { 2 2 (n - 8) { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A191789" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := 2 _Z - 2 _Z + 1 "A191790" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n1 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n1::even||| |n - 1 | |{ n1 ||| |----- | |{ (n1 + 1) binomial(n1, ----) ||| n n | \ | n1 |{ 2 ||| {1, (-1) , (-1) | ) |-(-1) |{ |||, | / | |{ (2 n1 - 2) ||| |----- | |{ 2 (n1 + 1) ||| |n1 = 0 | |{ ---------------------------------------- n1::odd ||| | | |{ n1 ||| | | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n1 \\\ |n - 1 | |{ 4 binomial(n1, ----) (n1 + 1) ||| |----- | |{ 2 ||| n | \ | n1 |{ ----------------------------- n1::even||| (-1) | ) |-(-1) |{ n1 + 2 |||} | / | |{ ||| |----- | |{ n1 ||| |n1 = 0 | |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd ||| \ \ \{ 2 /// "A191792" memory used=96312.8MB, alloc=2519.5MB, time=701.97 LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191794" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A191796" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { n 2 { binomial(n, n/2) (n - 7 n + 8) { 4 4 (n - 6 n + 1) { 1/2 ------------------------------- n::even { -------------------------- n::even n { n - 1 { n (n + 1) binomial(n, n/2) {2 , { , { } { 2 { (2 n + 2) 2 { binomial(n - 1, n/2 - 1/2) (n - 6 n + 1) { 2 2 (n - 7 n + 8) { ----------------------------------------- n::odd { ------------------------------------------ n::odd { n + 1 { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A191993" n (2 n + 1) binomial(2 n, n) {3 , --------------------------} n + 1 "A192008" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192009" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A192070" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n1 / 1/2 \n1 n - 1 | 5 | n - 1 |5 | ----- (2 n1 + 1) |1/2 - ----| binomial(2 n1, n1) ----- (2 n1 + 1) |---- + 1/2| binomial(2 n1, n1) \ \ 2 / \ \ 2 / {1, ) --------------------------------------------, ) --------------------------------------------} / n1 + 1 / n1 + 1 ----- ----- n1 = 0 n1 = 0 "A192132" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192238" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 5 5 {GAMMA(n + 3/2 - ----), GAMMA(n + 3/2 + ----)} 2 2 "A192239" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 5 5 {GAMMA(n + 3/2 - ----), GAMMA(n + 3/2 + ----)} 2 2 "A192250" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n1 / 1/2 \n1 n - 1 | 5 | n - 1 |5 | ----- (2 n1 + 1) |1/2 - ----| binomial(2 n1, n1) ----- (2 n1 + 1) |---- + 1/2| binomial(2 n1, n1) \ \ 2 / \ \ 2 / {1, ) --------------------------------------------, ) --------------------------------------------} / n1 + 1 / n1 + 1 ----- ----- n1 = 0 n1 = 0 "A192251" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n1 / 1/2 \n1 n - 1 | 5 | n - 1 |5 | ----- (2 n1 + 1) |1/2 - ----| binomial(2 n1, n1) ----- (2 n1 + 1) |---- + 1/2| binomial(2 n1, n1) \ \ 2 / \ \ 2 / {1, ) --------------------------------------------, ) --------------------------------------------} / n1 + 1 / n1 + 1 ----- ----- n1 = 0 n1 = 0 "A192252" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n - 1 n - 1 ----- / 1/2\n1 ----- / 1/2 \n1 \ | 5 | \ |5 | {1, ) (n1 + 1) |1/2 - ----| n1!, ) (n1 + 1) |---- + 1/2| n1!} / \ 2 / / \ 2 / ----- ----- n1 = 0 n1 = 0 "A192253" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n - 1 n - 1 ----- / 1/2\n1 ----- / 1/2 \n1 \ | 5 | \ |5 | {1, ) (n1 + 1) |1/2 - ----| n1!, ) (n1 + 1) |---- + 1/2| n1!} / \ 2 / / \ 2 / ----- ----- n1 = 0 n1 = 0 "A192264" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ (n1 + 2) (n1 + 1) n1! | 2 {(n + n - 1) | ) -------------------------------|, n + n - 1} | / 2 2 | |----- ((n1 + 1) + n1) (n1 + n1 - 1)| \n1 = 0 / "A192364" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192365" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A192368" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A192415" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A192417" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A192440" memory used=97151.7MB, alloc=2519.5MB, time=707.68 memory used=97746.4MB, alloc=2519.5MB, time=711.66 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192441" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192442" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192446" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A192457" n n {(1/2) n! binomial(2 n, n), (n + 1) 2 n!} "A192458" n n {(1/2) n! binomial(2 n, n), (n + 1) 2 n!} "A192459" n n {(1/2) n! binomial(2 n, n), (n + 1) 2 n!} "A192460" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n 1/2 / 1/2\n 1/2 | 5 | 5 | 5 | 5 {|3/2 - ----| GAMMA(n + 1/2 - ----), |3/2 + ----| GAMMA(n + 1/2 + ----)} \ 2 / 2 \ 2 / 2 "A192461" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n 1/2 / 1/2\n 1/2 | 5 | 5 | 5 | 5 {|3/2 - ----| GAMMA(n + 1/2 - ----), |3/2 + ----| GAMMA(n + 1/2 + ----)} \ 2 / 2 \ 2 / 2 "A192462" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n 1/2 / 1/2 \n 1/2 | 5 | 5 |5 | 5 {|1/2 - ----| GAMMA(n + 1/2 - ----), |---- + 1/2| GAMMA(n + 1/2 + ----)} \ 2 / 2 \ 2 / 2 "A192463" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n 1/2 / 1/2 \n 1/2 | 5 | 5 |5 | 5 {|1/2 - ----| GAMMA(n + 1/2 - ----), |---- + 1/2| GAMMA(n + 1/2 + ----)} \ 2 / 2 \ 2 / 2 "A192480" binomial(2 n, n) {n, 1/2 ----------------} n - 1/2 "A192481" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 binomial(2 n, n) {-------------------} n + 1 "A192482" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192744" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ / 1/2\n / 1/2 \n / 1/2\n |----- | |----- / 1/2 \(-n2 - 1) ||| | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | 2 ||| {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| (n2 + 1) n2!|||} \ 2 / \ 2 / \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A192745" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ / 1/2\n / 1/2 \n / 1/2\n |----- | |----- / 1/2 \(-n2 - 1) ||| | 5 | |5 | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | ||| {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| (n2 + 1) n2!|||} \ 2 / \ 2 / \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A192826" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 8 (n + 2) { n { 1/2 ------------------------ n::even n 2 n 2 { 2 2 (n + 1) (n + 2) binomial(n, n/2) n::even { (n + 1) binomial(n, n/2) {2 (n + 2) , 4 (n + 2) , { , { } { (n + 1) 2 { (3 n - 3) { 1/2 2 (n + 2) binomial(n + 1, n/2 + 1/2) n::odd { 2 2 (n + 1) (n + 2) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A192835" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 (2 n + 1) (2 n + 3) binomial(2 n, n) { ---------------------------------------- n::even 3 2 { 2 2 (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) {----------------------------, (2 n + 1) binomial(2 n, n) (n + 1), { , n + 1 { (4 n - 4) 2 { 8 2 (2 n + 1) (2 n - 1) binomial(2 n - 2, n - 1) { ---------------------------------------------------------- n::odd { 3 2 { n binomial(n - 1, n/2 - 1/2) { 2 2 { 8 (2 n + 1) binomial(n, n/2) binomial(2 n, n) n::even { } { 2 { (n + 1) (2 n + 3) binomial(n + 1, n/2 + 1/2) binomial(2 n + 2, n + 1) n::odd "A192856" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192924" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 5 | 1/2 1/2 1/2 1/2 {|1/2 - ----| (5 - 1) (2 n BesselI(n, 5 - 1) + (1 - 5 ) BesselI(n - 1, 5 - 1)), \ 2 / / 1/2\n | 5 | 1/2 1/2 1/2 1/2 |1/2 - ----| (5 - 1) (2 n BesselK(n, 1 - 5 ) + (1 - 5 ) BesselK(n - 1, 1 - 5 )), \ 2 / / 1/2 \n |5 | 1/2 1/2 1/2 1/2 |---- + 1/2| (-1 - 5 ) (2 n BesselI(n, -1 - 5 ) + (5 + 1) BesselI(n - 1, -1 - 5 )), \ 2 / / 1/2 \n |5 | 1/2 1/2 1/2 1/2 |---- + 1/2| (-1 - 5 ) (2 n BesselK(n, 5 + 1) + (5 + 1) BesselK(n - 1, 5 + 1))} \ 2 / "A192925" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 5 | 1/2 1/2 1/2 1/2 {|1/2 - ----| (5 - 1) (2 n BesselI(n, 5 - 1) + (1 - 5 ) BesselI(n - 1, 5 - 1)), \ 2 / / 1/2\n | 5 | 1/2 1/2 1/2 1/2 |1/2 - ----| (5 - 1) (2 n BesselK(n, 1 - 5 ) + (1 - 5 ) BesselK(n - 1, 1 - 5 )), \ 2 / / 1/2 \n |5 | 1/2 1/2 1/2 1/2 |---- + 1/2| (-1 - 5 ) (2 n BesselI(n, -1 - 5 ) + (5 + 1) BesselI(n - 1, -1 - 5 )), \ 2 / / 1/2 \n |5 | 1/2 1/2 1/2 1/2 |---- + 1/2| (-1 - 5 ) (2 n BesselK(n, 5 + 1) + (5 + 1) BesselK(n - 1, 5 + 1))} \ 2 / "A192936" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 5 5 {GAMMA(n + 3/2 - ----), GAMMA(n + 3/2 + ----)} 2 2 "A192939" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 n 5 n 5 {2 GAMMA(n + 5/4 - ----), 2 GAMMA(n + 5/4 + ----)} 4 4 "A192940" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 n 5 n 5 {2 GAMMA(n + 5/4 - ----), 2 GAMMA(n + 5/4 + ----)} 4 4 "A192941" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 5 + 2), GAMMA(n + 5 + 2)} "A192942" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 5 + 2), GAMMA(n + 5 + 2)} "A192945" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 7 7 | {|17/4 - ------| \ 4 / / 1/2 1/2 \ | 343 119 7 1/2 343 119 7 | |(54 n + 72) hypergeom([5/6, - 2/3 - n], [5/3], --- + --------) + 17 (17 + 7 7 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], --- + --------)| \ 27 27 27 27 / / 1/2\ | 119 7 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- 316/9 + --------|/(GAMMA(n + 1) GAMMA(n + 2/3))} \ 9 / "A192946" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192947" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A192948" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A192950" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 5 + 2), GAMMA(n + 5 + 2)} "A192989" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A193215" memory used=98662.8MB, alloc=2519.5MB, time=717.46 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ | ----- | | \ (2 n2 + 1) binomial(2 n2, n2) (5 n2 + 9)| | ) ----------------------------------------| |n - 1 / (n2 + 3) (n2 + 2) (n2 + 1) | |----- ----- | n n | \ n2 = 0 | {1, (-1/2) , (-1/2) | ) -----------------------------------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A193282" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 2 { 4 ((n/2)!) n::even { 4 ((n/2)!) binomial(n, n/2) n::even {{ , { } { (2 n - 2) 2 { 2 2 { 4 2 ((n/2 - 1/2)!) n::odd { ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) n::odd "A193361" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 (n + 1) HermiteH(n, 1/2 I 2 ) I) 2 {---------------------------------------------------------------------------------------------} n (n - 1) "A193364" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n! n! (n - 7 n + 12 n - 1) {---------, ---------------------------------, (n - 1) n n (n - 1) (n - 2) (n - 3) (n - 4) /n - 1 \ |----- n1 2 | 3 2 | \ (-1) (n1 - 13) (n1 + 1) n1 (n1 - 1) (n1 - 2) (n1 - 3) | n! (n - 7 n + 12 n - 1) | ) --------------------------------------------------------------------------| | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) - 7 (n1 + 1) + 12 n1 + 11) (n1 - 7 n1 + 12 n1 - 1)| \n1 = 0 / -------------------------------------------------------------------------------------------------------------} n (n - 1) (n - 2) (n - 3) (n - 4) "A193384" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n 2 2 { 2 (n + 1) GAMMA(n/4 + 3/4) { ----------------------------- irem(n, 4) = 0 { 2 2 { (n - 1) GAMMA(n/4 + 5/4) { { 2 { (n - 1) 2 { 2 binomial(n/2, n/4) irem(n, 4) = 0 { 2 2 GAMMA(n/4 + 1/2) { { ---------------------------- irem(n, 4) = 1 { 2 { 2 { 4 binomial(n/2 - 1/2, n/4 - 1/4) irem(n, 4) = 1 { GAMMA(n/4 + 1) { {{ , { 2 , { n 2 { 8 binomial(n/2 - 1, n/4 - 1/2) irem(n, 4) = 2 { 2 GAMMA(n/4 + 1/4) { { -------------------- irem(n, 4) = 2 { 2 2 { 2 { binomial(n/2 + 1/2, n/4 + 1/4) (n + 1) { GAMMA(n/4 + 3/4) { ---------------------------------------- irem(n, 4) = 3 { { 2 { n 2 { (n - 1) { 2 GAMMA(n/4) { ----------------- irem(n, 4) = 3 { 2 { GAMMA(n/4 + 1/2) { n 2 { n { 4 2 GAMMA(n/4 + 1/4) { 128 4 { ---------------------- irem(n, 4) = 0 { ---------------------- irem(n, 4) = 0 { 2 { 2 2 { GAMMA(n/4 + 3/4) { n binomial(n/2, n/4) { { { (n - 1) 2 { n 2 { 8 2 GAMMA(n/4) { 1024 4 (n + 1) { ---------------------- irem(n, 4) = 1 { ------------------------------------------------- irem(n, 4) = 1 { 2 { 2 2 2 { GAMMA(n/4 + 1/2) { (n - 1) (n + 3) binomial(n/2 + 3/2, n/4 + 3/4) { , { } { n 2 2 { n { 4 2 (n + 1) GAMMA(n/4 + 3/4) { 512 4 { ------------------------------- irem(n, 4) = 2 { -------------------------------------- irem(n, 4) = 2 { 2 2 { 2 2 { (n - 1) GAMMA(n/4 + 5/4) { (n + 2) binomial(n/2 + 1, n/4 + 1/2) { { { (n + 1) 2 { (2 n + 2) { 2 2 GAMMA(n/4 + 1/2) { 64 2 { ---------------------------- irem(n, 4) = 3 { ---------------------------------------- irem(n, 4) = 3 { 2 { 2 2 { GAMMA(n/4 + 1) { (n + 1) binomial(n/2 + 1/2, n/4 + 1/4) "A193385" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | |n - 1 { \ 2 / / n1 \ n1 | |----- { n1 (-1/2) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | | \ { \ 2 / 2 | {n! | ) ----------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |- ---- + 1/2| | | { (-2) |----|! n1::even| |n - 1 { \ 2 / n1 / n1 \ | |n - 1 { \ 2 / | |----- { 2 n1 binomial(n1 - 1, ---- - 1/2) |---- - 1/2|! n1::odd | |----- { | | \ { 2 \ 2 / | | \ { 0 n1::odd | n! | ) -------------------------------------------------------------------------------|, n! | ) ------------------------------------|, | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / / n1 \ | | { 2 |----|! n1::even| |n - 1 { \ 2 / | |----- { | | \ { 0 n1::odd | n! | ) ---------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A193437" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A193463" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n - 1) (n + 1) BesselJ(n, -2) + n BesselJ(n - 1, -2)), (-1) ((n - 1) (n + 1) BesselY(n, -2) + n BesselY(n - 1, -2))} "A193464" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((n + n - 1) BesselJ(n, -2) + (n + 1) BesselJ(n - 1, -2)), (-1) ((n + n - 1) BesselY(n, -2) + (n + 1) BesselY(n - 1, -2))} "A193465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n!, (n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A193618" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ / 1/2 1/2 \ n | 2 2 | n | 2 2 | (-8) |(4 n + 4) LegendreP(2 n + 2, ----) - LegendreP(2 n, ----)| (-8) |(4 n + 4) LegendreQ(2 n + 2, ----) - LegendreQ(2 n, ----)| \ 2 2 / \ 2 2 / {-----------------------------------------------------------------, -----------------------------------------------------------------} (4 n + 3) n (4 n + 3) n "A193619" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ n | 2 2 2 | (-8) |4 (n + 1) (2 n + 1) LegendreP(2 n + 2, ----) + (8 n + 4 n - 1) LegendreP(2 n, ----)| \ 2 2 / {--------------------------------------------------------------------------------------------, (2 n - 1) (4 n + 3) n / 1/2 1/2 \ n | 2 2 2 | (-8) |4 (n + 1) (2 n + 1) LegendreQ(2 n + 2, ----) + (8 n + 4 n - 1) LegendreQ(2 n, ----)| \ 2 2 / --------------------------------------------------------------------------------------------} (2 n - 1) (4 n + 3) n "A193624" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A193638" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A193651" n {1, (2 n + 1) (1/2) n! binomial(2 n, n)} "A193657" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A193658" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ /n - 1 \ |----- |----- n2 || |----- | | \ | \ 2 || | \ | {1, (n - 3) | ) n1! | ) ------------------------------------||, (n - 3) | ) n1!|, n - 3} | / | / (n2 - 2) (n2 - 1) (n2 + 1)! (n2 - 3)|| | / | |----- |----- || |----- | \n1 = 0 \n2 = 0 // \n1 = 0 / "A193659" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- n2 | \ 2 \ 2 | \ 2 | {1, ) n1! (n1 - n1 + 2), ) n1! (n1 - n1 + 2) | ) ---------------------------------------------|, / / | / 2 2 | ----- ----- |----- (n2 + 1)! ((n2 + 1) - n2 + 1) (n2 - n2 + 2)| n1 = 0 n1 = 0 \n2 = 0 / n - 1 /n1 - 1 \ ----- |----- 2 | \ 2 | \ n2 + 3 n2 + 6 | ) n1! (n1 - n1 + 2) | ) ---------------------------------------------|} / | / 2 2 | ----- |----- (n2 + 1)! ((n2 + 1) - n2 + 1) (n2 - n2 + 2)| n1 = 0 \n2 = 0 / "A193665" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) (n + 1) ((n + 1) (n + 2 n - 2) BesselJ(n, -2) + (n + 3 n + 1) BesselJ(n - 1, -2)), n 2 2 (-1) (n + 1) ((n + 1) (n + 2 n - 2) BesselY(n, -2) + (n + 3 n + 1) BesselY(n - 1, -2))} "A193668" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1 + 1 | n! (n - n + 1) | ) -----------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) - n1) (n1 - n1 + 1)| n! (n - n + 1) \n1 = 0 / {---------------, ------------------------------------------------------------------} n n "A193778" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-I 2 ) HermiteH(n, 1/2 I 2 )} "A193913" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n | 3 5 | | 3 5 | {|7/2 - ------| , |7/2 + ------| , \ 2 / \ 2 / /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- / 1/2\(-n2 - 1) ||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 3 5 | ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) |7/2 + ------| (2 LegendreP(n2 + 1, 3) - LegendreP(n2, 3))|||, \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- / 1/2\(-n2 - 1) ||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 3 5 | ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) |7/2 + ------| (2 LegendreQ(n2 + 1, 3) - LegendreQ(n2, 3))|||} \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A193930" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A193931" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A193932" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A193933" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A194019" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / 1/2\n1 \ |n - 1 | 2 | 1/2 1/2 1/2 | |----- |- ----| ((n1 - 7) HermiteH(n1 + 1, 2 ) + 4 2 (n1 + 1) HermiteH(n1, 2 ))| 2 2 | \ \ 2 / | {n! (n + 9 n + 16), n! (n + 9 n + 16) | ) ---------------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 9 n1 + 25) (n1 + 9 n1 + 16) | \n1 = 0 / "A194124" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { 8 binomial(n, n/2) (n + 1) (n + 3) { 1/4 -------------------------------- n::even { ---------------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {n + 3, { , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) (n + 2) { 2 (n + 1) (n + 3) { ------------------------------------ n::odd { 1/2 -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A194460" n (2 n + 1) binomial(2 n, n) (n + 3) {4 , ----------------------------------} n + 1 "A194588" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) ((n + 2) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 5) hypergeom([1/2, -n], [1], 4))} "A194589" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 2) hypergeom([1/2, -n], [1], 4))} "A194590" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (-2 n - 2) n::even { binomial(n, n/2) {(-2) , { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { - ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A194723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 binomial(2 n1, n1)| {9 , 9 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194724" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 3 2 binomial(2 n1, n1)| {16 , 16 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 4 5 binomial(2 n1, n1)| {25 , 25 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 5 6 binomial(2 n1, n1)| {36 , 36 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194727" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 6 7 binomial(2 n1, n1)| {49 , 49 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194728" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-6 n1 - 6) | n n | \ 7 2 binomial(2 n1, n1)| {64 , 64 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194729" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 8 3 binomial(2 n1, n1)| {81 , 81 | ) -----------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A194730" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 9 10 binomial(2 n1, n1)| {100 , 100 | ) ------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A195254" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 (n1 + 1)| n! | ) ------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, ------------------------} n n "A195255" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 3 (n1 + 1)| n! | ) ------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, ------------------------} n n "A195256" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 4 (n1 + 1)| n! | ) ------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, ------------------------} n n "A195257" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 5 (n1 + 1)| n! | ) ------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, ------------------------} n n "A195422" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-2) (n + 1) (n BesselI(n, 4) - 2 BesselI(n - 1, 4)), (-2) (n + 1) (n BesselK(n, -4) - 2 BesselK(n - 1, -4))} "A196148" LREtools/SearchTable: "SearchTable successful" hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) {-----------------------------------------------} n + 2 "A196265" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 2 GAMMA(n/2 + 3/4) n::even { 2 GAMMA(n/2 + 5/4) { { ------------------- n::even {{ (n + 1) , { 2 n + 1 } { 2 GAMMA(n/2 + 5/4) { { ------------------------- n::odd { (n - 1) { 2 n + 1 { 2 GAMMA(n/2 + 3/4) n::odd "A196411" 4 {n , n!} "A196412" 5 {n , n!} "A196413" 6 {n , n!} "A196414" 7 {n , n!} "A196532" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (2 n1 + 3) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A196738" 8 {n , n!} "A196739" 10 {n , n!} "A196864" LREtools/SearchTable: "SearchTable not successful" {} "A196865" LREtools/SearchTable: "SearchTable not successful" {} "A196866" memory used=99512.1MB, alloc=2551.5MB, time=723.53 LREtools/SearchTable: "SearchTable not successful" {} "A196867" LREtools/SearchTable: "SearchTable not successful" {} "A197130" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (n1 + 1) (2 n1 + 3) n1!\| {(n + 1) 2 n!, (n + 1) 2 n! | ) |1/2 -----------------------||} | / \ (n1 + 2) (n1 + 1)! /| |----- | \n1 = 0 / "A197209" LREtools/SearchTable: "SearchTable successful" ((7 n + 16) (3 n + 5) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + 8 (3 n + 7) (n + 1) hypergeom([-n, -n, -n], [1, 1], -1)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A197355" LREtools/SearchTable: "SearchTable successful" n {(-I) HermiteH(n, 4 I)} "A197657" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + hypergeom([-n, -n, -n], [1, 1], -1)} "A198059" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A198808" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A198888" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A198951" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A198953" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A199033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (7 n1 + 22 n1 + 17)| {(-1/4) , (-1/4) | ) -------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) (-1/4) | \n1 = 0 / "A199042" LREtools/SearchTable: "SearchTable successful" n {-(-1 + I) n! hypergeom([-n + 1, 1 + 1/2 I], [2], 1 - I)} "A199094" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n { n { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { ------------------------------------ irem(n, 3) = 0 { ------------------------------------ irem(n, 3) = 0 { 2 { 2 { GAMMA(5/3 + n/3) { GAMMA(n/3 + 4/3) { { { (n - 1) { n { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 9 3 (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) {{ ------------------------------------------ irem(n, 3) = 1, { ---------------------------------------------- irem(n, 3) = 1, { 2 { 2 { GAMMA(n/3 + 4/3) { (n + 2) GAMMA(n/3 + 2) { { { (n + 1) { (n + 1) { 3 (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { -------------------------------------------------- irem(n, 3) = 2 { ------------------------------------------ irem(n, 3) = 2 { 2 { 2 { (n + 2) GAMMA(n/3 + 2) { GAMMA(5/3 + n/3) { 2 n { 9 binomial(n, n/3) binomial(---, n/3) (n + 1) { 3 { --------------------------------------------- irem(n, 3) = 0 { n + 3 { { 2 n { 27 binomial(n - 1, n/3 - 1/3) binomial(--- - 2/3, n/3 - 1/3) (n + 1) n { 3 { ---------------------------------------------------------------------- irem(n, 3) = 1} { 2 { (n + 2) { { 2 n { 27 binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) (n - 1) n { 3 { ---------------------------------------------------------------------- irem(n, 3) = 2 { 2 { (n + 1) "A199126" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=100306.5MB, alloc=2583.5MB, time=729.15 memory used=100869.0MB, alloc=2583.5MB, time=732.78 memory used=101433.2MB, alloc=2583.5MB, time=736.24 memory used=101997.7MB, alloc=2583.5MB, time=739.69 { 4 %2 { 9 %3 { - -------- irem(n, 3) = 0 { - ----------- irem(n, 3) = 0 { 2 { 2 2 { (n + 1) { n (n + 3) { { { %3 { 18 %1 (n + 5) {{ - ----------- irem(n, 3) = 1, { - ------------------------ irem(n, 3) = 1, { 2 2 { 2 { n (n + 3) { (n - 1) (n + 2) (n + 8) { { { 2 %1 (n + 5) { 36 %2 { - ------------------------ irem(n, 3) = 2 { - -------- irem(n, 3) = 2 { 2 { 2 { (n - 1) (n + 2) (n + 8) { (n + 1) { 2 %1 (n + 5) { - ------------------------ irem(n, 3) = 0 { 2 { (n - 1) (n + 2) (n + 8) { { 4 %2 { - -------- irem(n, 3) = 1} { 2 { (n + 1) { { %3 { - ----------- irem(n, 3) = 2 { 2 2 { n (n + 3) 3 2 %1 := (n + 6 n - 3 n + 8) hypergeom([- n/3 - 5/3, - n/3 - 5/3, - n/3 - 5/3], [1, 1], -1) 3 2 + (n + 3 n - 64) hypergeom([- n/3 - 2/3, - n/3 - 2/3, - n/3 - 2/3], [1, 1], -1) %2 := 1/3 (n + 4) n hypergeom([- n/3 - 4/3, - n/3 - 4/3, - n/3 - 4/3], [1, 1], -1) 2 + (1/3 n + 1/3 n + 2) hypergeom([- n/3 - 1/3, - n/3 - 1/3, - n/3 - 1/3], [1, 1], -1) 2 %3 := (n - 2) (n + 6) (n + 3 n + 6) hypergeom([- n/3 - 2, - n/3 - 2, - n/3 - 2], [1, 1], -1) 4 3 2 + (n + 4 n + 9 n + 78 n + 360) hypergeom([- n/3 - 1, - n/3 - 1, - n/3 - 1], [1, 1], -1) "A199127" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n { n { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { ------------------------------------ irem(n, 3) = 0 { ------------------------------------ irem(n, 3) = 0 { 2 { 2 { GAMMA(5/3 + n/3) { GAMMA(n/3 + 4/3) { { { (n - 1) { n { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 9 3 (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) {{ ------------------------------------------ irem(n, 3) = 1, { ---------------------------------------------- irem(n, 3) = 1, { 2 { 2 { GAMMA(n/3 + 4/3) { (n + 2) GAMMA(n/3 + 2) { { { (n + 1) { (n + 1) { 3 (n + 3) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) { 3 3 GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) { -------------------------------------------------- irem(n, 3) = 2 { ------------------------------------------ irem(n, 3) = 2 { 2 { 2 { (n + 2) GAMMA(n/3 + 2) { GAMMA(5/3 + n/3) { 2 n { 9 binomial(n, n/3) binomial(---, n/3) (n + 1) { 3 { --------------------------------------------- irem(n, 3) = 0 { n + 3 { { 2 n { 27 binomial(n - 1, n/3 - 1/3) binomial(--- - 2/3, n/3 - 1/3) (n + 1) n { 3 { ---------------------------------------------------------------------- irem(n, 3) = 1} { 2 { (n + 2) { { 2 n { 27 binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) (n - 1) n { 3 { ---------------------------------------------------------------------- irem(n, 3) = 2 { 2 { (n + 1) "A199128" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {1, (-2) } "A199495" 2 n n! (2 n - 2 n - 1) (-1) n! (2 n - 1) {-------------------, ------------------} (n - 1) n (n - 1) n "A199516" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n 3 { 4 (n + 5) GAMMA(n/4 + 2) GAMMA(n/4 + 3/2) GAMMA(n/4 + 7/4) { ------------------------------------------------------------ irem(n, 4) = 0 { 3 { (n + 2) (n + 3) (n + 4) GAMMA(n/4 + 9/4) { { (2 n - 2) 2 { 16 2 (n + 4) GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/4) GAMMA(n/4 + 7/4) { ------------------------------------------------------------------------- irem(n, 4) = 1 { 3 { (n + 2) (n + 3) GAMMA(n/4 + 2) {{ , { n { 6 4 (n + 3) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/4) { ------------------------------------------------------------- irem(n, 4) = 2 { 3 { (n + 2) GAMMA(n/4 + 7/4) { { n { 4 4 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { ----------------------------------------------------- irem(n, 4) = 3 { 3 { GAMMA(n/4 + 3/2) { 2 { 512 binomial(n/2, n/4) binomial(n, n/2) (n + 1) { ------------------------------------------------ irem(n, 4) = 0 { n + 4 { { 2 { 3072 binomial(n/2 - 1/2, n/4 - 1/4) binomial(n - 1, n/2 - 1/2) (n + 1) n { ------------------------------------------------------------------------- irem(n, 4) = 1 { 2 { (n + 3) , { { 2 { 8192 binomial(n/2 - 1, n/4 - 1/2) binomial(n - 2, n/2 - 1) (n + 1) (n - 1) n { ----------------------------------------------------------------------------- irem(n, 4) = 2 { 3 { (n + 2) { { 2 { 32 binomial(n/2 + 1/2, n/4 + 1/4) binomial(n + 1, n/2 + 1/2) irem(n, 4) = 3 { n { 2048 4 (n + 1) binomial(n, n/2) { -------------------------------- irem(n, 4) = 0 { 3 2 { (n + 2) binomial(n/2, n/4) { { n { 512 4 binomial(n + 3, n/2 + 3/2) { ----------------------------------------------- irem(n, 4) = 1 { 2 { (n + 2) (n + 3) binomial(n/2 + 3/2, n/4 + 3/4) { , { n { 2048 4 binomial(n + 2, n/2 + 1) { --------------------------------------------- irem(n, 4) = 2 { 2 { (n + 2) (n + 4) binomial(n/2 + 1, n/4 + 1/2) { { (2 n + 2) { 768 2 binomial(n + 1, n/2 + 1/2) { ----------------------------------------- irem(n, 4) = 3 { 2 2 { (n + 3) binomial(n/2 + 1/2, n/4 + 1/4) { n { 96 4 (n + 3) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/4) { -------------------------------------------------------------- irem(n, 4) = 0 { 3 { (n + 2) GAMMA(n/4 + 7/4) { { (2 n - 2) { 256 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { --------------------------------------------------------------- irem(n, 4) = 1 { 3 { GAMMA(n/4 + 3/2) { } { n 3 { 16 4 (n + 5) GAMMA(n/4 + 2) GAMMA(n/4 + 3/2) GAMMA(n/4 + 7/4) { --------------------------------------------------------------- irem(n, 4) = 2 { 3 { (n + 2) (n + 3) (n + 4) GAMMA(n/4 + 9/4) { { (2 n + 2) 2 { 16 2 (n + 4) GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/4) GAMMA(n/4 + 7/4) { ------------------------------------------------------------------------- irem(n, 4) = 3 { 3 { (n + 2) (n + 3) GAMMA(n/4 + 2) "A199524" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A199578" LREtools/SearchTable: "SearchTable successful" n {(n + 1) (-1) n! LaguerreL(n + 1, 1)} "A199660" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { 2 binomial(n, n/2) (n/2)! n::even { 0 n::even { {{ , { (-n - 1) , { (- n/2 + 1/2) { 2 (n - 2) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! { 2 binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { --------------------------------------------------------- n::odd { n { (n - 2) (n/2)! { (n/2) { -------------- n::even { 2 2 (n/2)! { n , { --------------- n::even} { { n { (n/2 - 1/2)! n::odd { { 0 n::odd "A199697" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A199825" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A199826" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A199841" memory used=102915.4MB, alloc=2603.5MB, time=744.94 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A199892" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A200028" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A200031" LREtools/SearchTable: "SearchTable successful" (n/2 + 1/2) 1/2 1/2 1/2 (n/2 + 1/2) 1/2 1/2 1/2 5 (-5 LegendreP(n + 1, 5 ) + 5 LegendreP(n, 5 )) 5 (-5 LegendreQ(n + 1, 5 ) + 5 LegendreQ(n, 5 )) {- ------------------------------------------------------------------, - ------------------------------------------------------------------} n n "A200074" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A200075" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200172" n (2 n + 3) (2 n + 1) (-1) binomial(2 n, n) (n + 2) {1, --------------------------------------------------} (n + 4) (n + 3) "A200173" n (2 n + 5) (2 n + 3) (2 n + 1) (-1) binomial(2 n, n) (n + 2) {1, ------------------------------------------------------------} (n + 5) (n + 4) "A200174" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A200312" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 13 | | 13 | |1/2 - -----| binomial(2 n, n) |1/2 + -----| binomial(2 n, n) \ 2 / \ 2 / {-------------------------------, -------------------------------} n + 1 n + 1 "A200375" n n (-1) binomial(2 n, n) 2 binomial(2 n, n) {----------------------, -------------------} n + 1 n + 1 "A200376" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { /5 n2 \ ||| | | | { |---- - 5/2| ||| | | | { \ 2 / ||| | | | { 2 ||| | | | { ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| n n n | \ | n1 | \ { 2 ||| {(-3) , 3 , 3 | ) |1/3 (-1) | ) ----------------------------------------------------------|||, | / | | / (n2 + 1) ||| |----- | |----- (-3) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 \ \\\ | | | { |----| ||| | | | { \ 2 / n2 ||| | | | { 2 2 binomial(n2, ----) ||| | | | { 2 ||| | | | { ---------------------------- n2::even||| |n - 1 | |n1 - 1 { n2 + 2 ||| |----- | |----- { ||| n | \ | n1 | \ { 0 n2::odd ||| 3 | ) |1/3 (-1) | ) ----------------------------------------------|||} | / | | / (n2 + 1) ||| |----- | |----- (-3) ||| \n1 = 0 \ \n2 = 0 /// "A200380" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200613" n (2 n + 1) binomial(2 n, n) (n + 5) {4 , ----------------------------------} n + 1 "A200719" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200731" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200740" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A200753" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200754" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200757" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200770" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { 2 n { |---| { binomial(---, n/3) (4 n + 9) { \ 3 / { 3 { 3 2 GAMMA(n/3 + 7/6) { ---------------------------- irem(n, 3) = 0 { ------------------------- irem(n, 3) = 0 { n + 3 { GAMMA(5/3 + n/3) { { { 2 n { /2 n \ { 3 binomial(--- - 2/3, n/3 - 1/3) (2 n + 1) { |--- - 2/3| {{ 3 , { \ 3 / , { ------------------------------------------ irem(n, 3) = 1 { 2 2 GAMMA(n/3 + 5/6) { n + 2 { ------------------------------- irem(n, 3) = 1 { { GAMMA(n/3 + 4/3) { 2 n { { 2 binomial(--- - 4/3, n/3 - 2/3) (2 n - 1) { /2 n \ { 3 { |--- + 2/3| { ------------------------------------------ irem(n, 3) = 2 { \ 3 / { n + 1 { 2 (4 n + 9) GAMMA(n/3 + 3/2) { --------------------------------------- irem(n, 3) = 2 { (2 n + 3) GAMMA(n/3 + 2) { /2 n\ { |---| { \ 3 / { 2 2 GAMMA(n/3 + 5/6) { ------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 4/3) { { /2 n \ { |--- + 4/3| { \ 3 / } { 2 (4 n + 9) GAMMA(n/3 + 3/2) { --------------------------------------- irem(n, 3) = 1 { (2 n + 3) GAMMA(n/3 + 2) { { /2 n \ { |--- + 2/3| { \ 3 / { 3 2 GAMMA(n/3 + 7/6) { ------------------------------- irem(n, 3) = 2 { GAMMA(5/3 + n/3) "A200850" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A200966" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 %1 := _Z - _Z - 1 "A201164" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| , \ 2 / \ 2 / / / / / 1/2 \(-n2 - 1) \\\ |n - 1 | |n1 - 1 |5 | ||| / 1/2\n |----- | |----- |---- + 1/2| binomial(2 n2, n2) (5 n2 + 4)||| | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 2 / ||| |1/2 - ----| | ) |-2 (-1) (5 - 1) (5 + 1) | ) ---------------------------------------------------|||} \ 2 / | / | | / (n2 + 1) (n2 + 2) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A201202" LREtools/SearchTable: "SearchTable successful" n {(-1) (n LaguerreL(n + 1, 1) + (-n - 1) LaguerreL(n, 1)) n! (n + 1)} "A201203" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 2) LaguerreL(n + 1, -1) + (-n - 1) LaguerreL(n, -1)) n! (n + 1)} "A201204" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 { { 4 binomial(n, n/2) (2 n + 1) binomial(2 n, n) { (4 n - 4) { ------------------- n::even {--------------------------, { 2 , { 2 } (n + 1) (n + 2) { --------------------------------------- n::odd { (n + 2) { 2 2 2 { { n (n + 2) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A201205" 2 binomial(2 n, n) (4 n + 1) binomial(4 n, 2 n) {-----------------, ----------------------------} 2 (n + 1) (2 n + 1) (n + 1) "A201546" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- 2 | 2 | \ (n1 + 1) (2 n1 + 1) (n1!) binomial(2 n1, n1) | (n + 1) (2 n + 1) (n!) binomial(2 n, n) | ) -----------------------------------------------------------|} | / 2 | |----- (n1 + 2) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A201549" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A201631" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=103806.1MB, alloc=2583.5MB, time=750.78 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A201640" LREtools/SearchTable: "SearchTable successful" ((2 n + 3) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + 2 n hypergeom([-n, -n, -n], [1, 1], -1)) (n + 1) {-------------------------------------------------------------------------------------------------------------} 2 (n + 2) "A201645" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A201686" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A201805" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A201806" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A201807" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A201996" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-I) n1! (LegendreP(n1, I) + LegendreP(n1 + 1, I) I) (n1 + 1)|| {(5/2) n!, (5/2) n! | ) |- ---------------------------------------------------------------||, | / | (n1 + 1) || |----- \ n1 (5/2) (n1 + 1)! /| \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ | (-I) n1! (LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I) (n1 + 1)|| (5/2) n! | ) |- ---------------------------------------------------------------||} | / | (n1 + 1) || |----- \ n1 (5/2) (n1 + 1)! /| \n1 = 0 / "A202020" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 / /n1 - 1 / 1/2\n / 1/2\n / 1/2\n |----- | |----- | 2 | | 2 | | 2 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ n2 1/2 (-n2 - 1) {|1 - ----| , |1 + ----| , |1 - ----| | ) |2 (2 + 2 ) (2 - 2 ) | ) (2 4 (2 + 2 ) \ 2 / \ 2 / \ 2 / | / | | / |----- | |----- \n1 = 0 \ \n2 = 0 3 2 2 ((18 n2 + 72 n2 + 80 n2 + 20) hypergeom([-1/2, -n2 - 1], [1], -2) - 3 (6 n2 + 21 n2 + 17) (n2 + 1) hypergeom([-1/2, -n2], [1], -2))/((n2 + 2) \\\ ||| ||| (n2 + 3) (n2 + 4)))|||} ||| ||| /// "A202065" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) binomial(2 n, n) LaguerreL(n, -n - 1/4, -1/4)} "A202329" (2 n + 1) binomial(2 n, n) (9 n + 14) {1, -------------------------------------} (n + 1) (n + 2) "A202364" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {n!} "A202410" LREtools/SearchTable: "SearchTable successful" ((2 n + 2) LaguerreL(n + 1, 1) - 3 n LaguerreL(n, 1)) n! {--------------------------------------------------------} n "A202411" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202454" 2 (2 n + 1) binomial(2 n, n) (169 n + 713 n + 690) 2 {-------------------------------------------------, 3 n + 18 n + 47} (n + 3) (n + 2) (n + 1) "A202482" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- / n1 n1 \| |----- / n1 n1 \| n n | \ | 7 (-1) 63 GAMMA(n1 + 4/3)|| n | \ | 7 (-1) 63 GAMMA(n1 + 5/3)|| {(-1/7) , (-1/7) | ) |- -----------------------------||, (-1/7) | ) |- -----------------------------||} | / \ GAMMA(n1 + 3) /| | / \ GAMMA(n1 + 3) /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A202554" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 (n1 + 2) | | { 1/4 --------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | | { (2 n1 - 2) | | { 2 (n1 + 1) (n1 + 3) | | { ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | n | \ { 2 | {1, 2 , (3 n + 10) | ) ----------------------------------------------------------|, | / (3 n1 + 13) (3 n1 + 10) | |----- | \n1 = 0 / / { n1 \ | { 8 binomial(n1, ----) (n1 + 1) (n1 + 3) | | { 2 | | { -------------------------------------- n1::even| | { n1 + 2 | | { | |n - 1 { n1 | |----- { binomial(n1 + 1, ---- + 1/2) (2 n1 + 4) n1::odd | | \ { 2 | (3 n + 10) | ) ---------------------------------------------------------|, 3 n + 10} | / (3 n1 + 13) (3 n1 + 10) | |----- | \n1 = 0 / "A202736" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n n { { 4 {2 , { 2 binomial(n - 1, n/2 - 1/2) n , { ------------------------ n::even} { ------------------------------ n::odd { (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A202737" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 n n { {3 , (-1) hypergeom([1/2, -n - 1], [1], 4), { 2 n , { 9 binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) (n - 1) n { 3 { --------------------------------------------------------------------- irem(n, 3) = 2 { 2 { (n + 1) { 0 irem(n, 3) = 0 { n { { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { (n - 1) { ---------------------------------- irem(n, 3) = 0 { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 2 { ---------------------------------------- irem(n, 3) = 1, { GAMMA(n/3 + 4/3) } { 2 { { GAMMA(n/3 + 4/3) { 0 irem(n, 3) = 1 { { { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A202743" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202814" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202824" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202825" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A202826" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202827" LREtools/SolveLRE: "Reduced the order of" E^3+(-n-6)*E^2-(n+6)*(n+2)*E+(n+2)*(n+1)^2 "to two: Symmetric square" E^2-2*E-n-1 LREtools/SearchTable: "SearchTable successful" n 1/2 2 {(-1/2) HermiteH(n, 2 I) } "A202830" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 6 ) HermiteH(n, 1/3 I 6 )} "A202831" memory used=104646.0MB, alloc=2615.5MB, time=756.87 LREtools/SolveLRE: "Reduced the order of" E^3+(-5*n-14)*E^2-5*(5*n+14)*(n+2)*E+125*(n+2)*(n+1)^2 "to two: Symmetric square" E^2-2*E-5*n-5 LREtools/SearchTable: "SearchTable successful" n 1/2 2 {(-5/2) HermiteH(n, 1/5 I 10 ) } "A202832" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 10 ) HermiteH(n, 1/5 I 10 )} "A202833" LREtools/SolveLRE: "Reduced the order of" E^3+(-n-11)*E^2-(n+11)*(n+2)*E+(n+2)*(n+1)^2 "to two: Symmetric square" E^2-3*E-n-1 LREtools/SearchTable: "SearchTable successful" n 1/2 2 {(-1/2) HermiteH(n, 3/2 I 2 ) } "A202834" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 3/2 I 2 )} "A202837" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-I 2 ) HermiteH(n, 3/4 I 2 )} "A202840" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A202849" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A202850" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A202856" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A202879" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 2 I 2 )} "A202932" 3 2 (2 n + 1) binomial(2 n, n) (5625 n + 44453 n + 109238 n + 82824) 4 3 2 {------------------------------------------------------------------, 36 n + 432 n + 2916 n + 12120 n + 29317} (n + 4) (n + 3) (n + 2) (n + 1) "A202950" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 {BesselJ(2 n + 1, -2 2 ), BesselY(2 n + 1, -2 2 )} "A203019" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A203147" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- | | \ (n1 + 1) n1! | (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) -------------------|} | / (n1 + 12) (n1 + 1)!| |----- | \n1 = 0 / "A203152" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 { (-n) 2 2 2 { 1/4 ((n/2)!) (n + 2) n::even { 2 4 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) n::even {{ , { } { 2 { (-2 n + 2) 2 2 2 2 { (n/2 + 3/2) ((n/2 + 1/2)!) n::odd { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd "A203153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 2 2 { 2 2 { 4 (n + 1) (n + 3) binomial(n, n/2) ((n/2)!) n::even { 1/8 ((n/2)!) (n + 4) (n + 2) n::even {{ , { { (-2 n - 2) 2 2 2 { 2 2 2 { 2 2 (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 4) n::odd { 1/16 ((n/2 - 1/2)!) (n + 1) (n + 3) n::odd } "A203154" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 2 2 { (-n) 2 2 2 2 { 1/16 ((n/2)!) (n + 2) (n + 4) n::even { 2 4 (n + 1) (n + 3) binomial(n, n/2) ((n/2)!) (n + 5) n::even {{ , { { 2 2 { (-2 n + 2) 2 2 2 2 2 { 1/8 ((n/2 + 1/2)!) (n + 5) (n + 3) n::odd { 2 n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd } "A203159" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / (n1 + 1) n1! \| {(n + 1) 2 n!, (n + 1) 2 n! | ) |1/2 ------------------||} | / \ (n1 + 2) (n1 + 1)!/| |----- | \n1 = 0 / "A203485" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 n::even { n n n n { { 4 {2 , 3 , (-1) hypergeom([1/2, -n - 1], [1], 4), { 2 binomial(n - 1, n/2 - 1/2) n , { ------------------------ n::even} { ------------------------------ n::odd { (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A203558" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 n::even { n n { { 4 {1, (-1) hypergeom([1/2, -n - 1], [1], 4), { 2 binomial(n - 1, n/2 - 1/2) n , { ------------------------ n::even} { ------------------------------ n::odd { (n + 1) binomial(n, n/2) { n + 1 { { 0 n::odd "A203576" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 2 {((n + 3) (2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-2 n - 2 n + 2) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1)) { 0 n::even { { (6 n - 6) { 3 binomial(2 n, n)/(n + 1), { 2 , { binomial(n, n/2) n::even} { ------------------------------ n::odd { { 3 3 { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A203577" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 n::even { binomial(2 n, n) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1) { (6 n - 6) {----------------------------------------------------------------, { 2 , n + 1 { --------------------------------------- n::odd { 3 2 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) { 3 { 4 binomial(n, n/2) { ------------------- n::even { 2 } { (n + 2) { { 0 n::odd "A203604" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 17 | | 17 | {|1/2 - -----| , |1/2 + -----| } \ 2 / \ 2 / "A203611" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A204061" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A204643" n (2 n + 3) (2 n + 1) binomial(2 n, n) 4 3 2 {(-1) , ------------------------------------, n + 10 n + 41 n + 62 n + 27} (n + 1) (n + 2) "A205825" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { (n/2)! binomial(n, n/2) n::even { 2 (n/2)! { { --------- n::even {{ (n/2 + 1/2)! binomial(n + 1, n/2 + 1/2) , { n + 1 } { --------------------------------------- n::odd { { n + 1 { (n - 1) { 2 (n/2 - 1/2)! n::odd "A206178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A206180" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A206259" (2 n + 3) (2 n + 1) binomial(2 n, n) {1, ------------------------------------} (n + 1) (n + 2) "A206300" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 3 n 3 n { { binomial(3 n, ---) binomial(---, n/2) { (2 n - 2) 3 n { 2 2 {{ 2 binomial(--- - 3/2, n/2 - 1/2) , { ------------------------------------- n::even} { 2 { binomial(n, n/2) (3 n - 1) { ----------------------------------------- n::odd { { n { 0 n::odd "A206307" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) (2 n + 3) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- n1 | 2 | \ (-1) | (n + 1) (2 n + 3) (2 n + 1) (n!) binomial(2 n, n) | ) ----------------------------------------------------------------------|} | / 2 | |----- (n1 + 2) (2 n1 + 5) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A206308" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) (2 n + 3) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- n1 | 2 | \ (-1) | (n + 1) (2 n + 3) (2 n + 1) (n!) binomial(2 n, n) | ) ----------------------------------------------------------------------|} | / 2 | |----- (n1 + 2) (2 n1 + 5) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A206531" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- n1 | 2 | \ (-1) | (n + 1) (2 n + 1) (n!) binomial(2 n, n) | ) -----------------------------------------------------------|} | / 2 | |----- (n1 + 2) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A206532" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) (2 n + 1) (n!) binomial(2 n, n), /n - 1 \ |----- n1 | 2 | \ (-1) | (n + 1) (2 n + 1) (n!) binomial(2 n, n) | ) -----------------------------------------------------------|} | / 2 | |----- (n1 + 2) (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A206603" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 (n + 1), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A206604" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {1, 2 (n + 1), { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A206816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 3 {1, ) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A206817" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 2 | {1, (n + 3) | ) (n1 + 2) (n1 + 1) n1!|, n + 3} | / | |----- | \n1 = 0 / "A207318" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) hypergeom([-n1, -n1, -n1], [1, 1], -1)} / ----- n1 = 0 "A207319" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) n1 hypergeom([-n1, -n1, -n1], [1, 1], -1)} / ----- n1 = 0 "A207321" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) hypergeom([1/2, -n1 - 1, -n1 - 1], [1, 1], 4)} / ----- n1 = 0 "A207322" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) hypergeom([1/2, -n1 - 1, -n1 - 1], [1, 1], 4)} / ----- n1 = 0 "A207323" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) (n1 + 1) hypergeom([1/2, -n1 - 1, -n1 - 1], [1, 1], 4)} / ----- n1 = 0 "A207969" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A208034" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A208137" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 3 n 2 3 n 2 { 16 (n + 3) (3 n + 1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ------------------------------------------------------------------ n::even { 2 2 { (n + 1) binomial(n, n/2) {{ , { (-4 n - 4) 2 3 n 2 3 n 2 { 4 2 (n + 1) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) (n + 4) { 2 2 { -------------------------------------------------------------------------------------------- n::odd { 2 { (3 n + 2) binomial(n + 1, n/2 + 1/2) { 3 n 2 { 1/4 binomial(---, n/2) (3 n + 2) (n + 4) n::even { 2 { { 3 n 2 2 2 2 } { binomial(--- - 3/2, n/2 - 1/2) (n + 3) (3 n - 1) (3 n + 1) { 2 { 1/16 -------------------------------------------------------------- n::odd { 2 2 { n (n + 1) "A208184" n 3 n! (1/6) (n!) binomial(2 n, n) binomial(3 n, n) {----, ----------------------------------------------} n n "A208186" n! /625\n {----, |---| GAMMA(n) GAMMA(n + 3/5) GAMMA(n + 1/5) GAMMA(n + 4/5) GAMMA(n + 2/5)} n \24 / "A208275" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) (HermiteH(n + 1, I) + (n + 3) HermiteH(n, I) I) I} "A208355" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 2 binomial(n, n/2) { -------------------------------- n::even { ------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {{ , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) { 2 { ---------------------------- n::odd { ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A208425" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A208426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A208446" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A208473" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 5 ) binomial(2 n, n), (2 + 5 ) binomial(2 n, n)} "A208588" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 1/2 n 1/2 1/2 n 1/2 {(2 - 5 ) LegendreP(n, -5 ), (2 - 5 ) LegendreQ(n, -5 ), (2 + 5 ) LegendreP(n, 5 ), (2 + 5 ) LegendreQ(n, 5 ), { 0 n::even { { (n/2) { (n/2 - 1/2) , { (-1) binomial(n, n/2) n::even} { (-16) { { ---------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A208616" binomial(2 n, n) binomial(2 n, n) binomial(3 n, n) {1, ----------------, ---------------------------------} n + 1 2 (n + 2) (n + 1) "A208632" n (2 n + 1) 2 binomial(2 n, n) (3 n + 1) (3 n + 2) binomial(3 n, n) n {-----------------------------, --------------------------------------} n + 1 (n + 1) (2 n + 3) (2 n + 1) "A208675" LREtools/SearchTable: "SearchTable successful" memory used=105564.6MB, alloc=2615.5MB, time=763.24 2 {(4 (n + 1) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-21 n - 22 n - 6) hypergeom([-n, -n, -n], [1, -2 n], 1)) / 2 binomial(2 n, n) / n } / "A208887" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" / / / { 0 irem(n2, 4) = 0\\\ | | | { ||| | | | { 0 irem(n2, 4) = 1||| | | | { ||| | | | { 0 irem(n2, 4) = 2||| | | | { ||| | | | { / n2 \ ||| | | | { |---- - 3/4| ||| | | | { \ 4 / ||| | | | { 4 (-16) ||| | | | { -------------------------------------------------- irem(n2, 4) = 3||| |n - 1 | |n1 - 1 { n2 n2 ||| |----- | |----- { binomial(---- - 3/2, ---- - 3/4) (n2 + 3) (n2 - 1) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 4 ||| {(2 ) , (-2 ) , (2 ) | ) |1/2 2 (-1) | ) ---------------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { 0 irem(n2, 4) = 0\\\ | | | { ||| | | | { 0 irem(n2, 4) = 1||| | | | { ||| | | | { / n2 \ ||| | | | { |---- - 1/2| ||| | | | { \ 4 / n2 ||| | | | { (-4) GAMMA(---- + 1/4) ||| | | | { 4 ||| | | | { ---------------------------------- irem(n2, 4) = 2||| | | | { n2 ||| | | | { GAMMA(---- + 7/4) ||| |n - 1 | |n1 - 1 { 4 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 irem(n2, 4) = 3||| (2 ) | ) |1/2 2 (-1) | ) -----------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { 0 irem(n2, 4) = 0\\\ | | | { ||| | | | { / n2 \ ||| | | | { |---- - 1/4| ||| | | | { \ 4 / n2 n2 ||| | | | { 4 (-1) binomial(---- - 1/2, ---- - 1/4) ||| | | | { 2 4 ||| | | | { --------------------------------------------------- irem(n2, 4) = 1||| | | | { n2 + 3 ||| | | | { ||| |n - 1 | |n1 - 1 { 0 irem(n2, 4) = 2||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 irem(n2, 4) = 3||| (2 ) | ) |1/2 2 (-1) | ) ----------------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 \ \\\ | | | { |----| ||| | | | { \ 4 / n2 ||| | | | { (-4) GAMMA(---- + 1/4) ||| | | | { 4 ||| | | | { ---------------------------- irem(n2, 4) = 0||| | | | { n2 ||| | | | { GAMMA(---- + 7/4) ||| | | | { 4 ||| | | | { ||| | | | { 0 irem(n2, 4) = 1||| | | | { ||| |n - 1 | |n1 - 1 { 0 irem(n2, 4) = 2||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 irem(n2, 4) = 3||| (2 ) | ) |1/2 2 (-1) | ) -----------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// "A208888" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" /n - 1 /{ 0 n1::even\\ |----- |{ || n n | \ (-n1 - 1) |{ (n1 - 1) / n1 n1 \ || {2 , 2 | ) 2 |{ 2 2 |LegendreP(---- + 1/2, 1/2) - 2 LegendreP(---- - 1/2, 1/2)| ||, | / |{ \ 2 2 / || |----- |{ ----------------------------------------------------------------------- n1::odd || \n1 = 0 \{ n1 + 3 // /n - 1 /{ 0 n1::even\\ |----- |{ || n | \ (-n1 - 1) |{ (n1 - 1) / n1 n1 \ || 2 | ) 2 |{ 2 2 |LegendreQ(---- + 1/2, 1/2) - 2 LegendreQ(---- - 1/2, 1/2)| ||, | / |{ \ 2 2 / || |----- |{ ----------------------------------------------------------------------- n1::odd || \n1 = 0 \{ n1 + 3 // /n - 1 /{ n1 / n1 n1 \ \\ |----- |{ 2 |LegendreP(---- + 1/2, 1/2) - 2 LegendreP(---- - 1/2, 1/2)| || n | \ (-n1 - 1) |{ \ 2 2 / || 2 | ) 2 |{ --------------------------------------------------------------- n1::even||, | / |{ n1 + 3 || |----- |{ || \n1 = 0 \{ 0 n1::odd // /n - 1 /{ n1 / n1 n1 \ \\ |----- |{ 2 |LegendreQ(---- + 1/2, 1/2) - 2 LegendreQ(---- - 1/2, 1/2)| || n | \ (-n1 - 1) |{ \ 2 2 / || 2 | ) 2 |{ --------------------------------------------------------------- n1::even||} | / |{ n1 + 3 || |----- |{ || \n1 = 0 \{ 0 n1::odd // "A208976" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 6 4 { 2 binomial(n, n/2) (3 n + 4) { ------------------------ n::even { ---------------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {1, { , { } { (2 n + 2) { 12 binomial(n - 1, n/2 - 1/2) n { 2 (3 n + 4) { ------------------------------- n::odd { ------------------------------------------ n::odd { n + 1 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A208977" LREtools/SearchTable: "SearchTable not successful" {} "A208983" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 (2 n + 1) binomial(2 n, n/2) { 1/2 ----------------------------------------------------- n::even { 3 n { 8 binomial(2 n, n/2) (n + 1) { (n + 1) (3 n + 1) binomial(n, n/2) binomial(3 n, ---) { ---------------------------- n::even { 2 {{ 3 n + 2 , { } { { (4 n - 4) { 2 binomial(2 n + 2, n/2 + 1/2) n::odd { 2 2 (n + 1) (2 n - 1) binomial(2 n - 2, n/2 - 1/2) { ----------------------------------------------------------------------------- n::odd { 3 n { n (3 n - 2) (3 n + 2) binomial(n - 1, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 "A209083" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {(-1) (n - 1), n + 1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A209200" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([-1/2, -n - 3/4], [5/4], -1) + (-2 n - 3) hypergeom([-1/2, 1/4 - n], [5/4], -1))} "A209245" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 2)|| {(-1) , (-1) | ) |- -------------------------------------------------------||} | / \ n1 + 1/2 /| |----- | \n1 = 0 / "A209256" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n!} "A209288" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / | | | /n - 1 \ |n - 1 |----- | |----- n n | \ binomial(2 n1, n1)| n | \ {(-1/2) , (-1/2) | ) ------------------|, (-1/2) | ) | / (n1 + 1) | | / |----- (-1/2) | |----- \n1 = 0 / \n1 = 0 /n1 - 1 \\ |----- 2 5 4 3 2 || | \ (2 n2 + 1) binomial(2 n2, n2) binomial(4 n2, 2 n2) (10773 n2 + 33975 n2 + 39765 n2 + 21569 n2 + 5414 n2 + 504)|| binomial(2 n1, n1) | ) -------------------------------------------------------------------------------------------------------------------|| | / 3 || |----- (n2 + 1) (3 n2 + 4) (3 n2 + 1) (3 n2 + 2) binomial(2 n2 + 2, n2 + 1) || \n2 = 0 /| -----------------------------------------------------------------------------------------------------------------------------------------------|} (n1 + 1) | (-1/2) | / "A209352" LREtools/SolveLRE: "Reduced the order of" (3*n+4)*(3*n+1)*(21*n^3+73*n^2+74*n+16)*(n+2)^2*(n+3)^4*E^3-3*(3*n+1)*(21*n^3+136*n^2+283*n+184)*(57*n^ 5+481*n^4+1533*n^3+2247*n^2+1434*n+264)*(n+2)^2*E^2-24*(3*n+7)*(21*n^3+73*n^2+74*n+16)*(57*n^5+481*n^4+1533*n^3+2247*n^2+1434*n+264)*(n+1)^2*E+512*(3 *n+7)*(3*n+4)*(21*n^3+136*n^2+283*n+184)*(n+1)^2*n^4 "to two: Symmetric square" 2*(n+2)^2*E^2+(-7*n^2-21*n-16)*E-4*(n+1)^2 LREtools/SearchTable: "SearchTable successful" 2 2 2 ((n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (n - 7 n - 2) hypergeom([-n, -n, -n], [1, 1], -1)) {----------------------------------------------------------------------------------------------------------------} 4 n "A209358" LREtools/SearchTable: "SearchTable successful" n {4 hypergeom([3/4, 5/8 - n], [13/8], -1)} "A210064" n (2 n + 1) binomial(2 n, n) (3 n + 5) {4 (n + 5), ------------------------------------} n + 1 "A210474" LREtools/SearchTable: "SearchTable successful" 2 (32 n - 42 n + 7) hypergeom([-1/2, -n - 1], [1], -8) - (2 n + 1) (16 n - 21) hypergeom([-1/2, -n], [1], -8) {------------------------------------------------------------------------------------------------------------} (n - 1) n "A210486" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! (n - 1) BesselJ(n + 1/2, -1), (-1) n! (n - 1) BesselY(n + 1/2, -1)} "A210496" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| , n - 1} \ 2 / \ 2 / "A210573" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A210670" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- ----- \ binomial(2 n1, n1) \ (4 n1 + 1) binomial(4 n1, 2 n1) (10 n1 + 9) {1, ) ------------------, ) -------------------------------------------} / n1 + 1 / (n1 + 1) (2 n1 + 3) (2 n1 + 1) ----- ----- n1 = 0 n1 = 0 "A210685" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 2 {(-1/2) (27 n + 63 n + 34), /n - 1 \ |----- | n 2 | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) | (-1/2) (27 n + 63 n + 34) | ) -----------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (n1 + 1) (n1 + 2) (-1/2) (27 (n1 + 1) + 63 n1 + 97) (54 n1 + 126 n1 + 68)| \n1 = 0 / "A210736" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ------------------ n::even { n binomial(n, n/2) { binomial(n, n/2) n::even {{ , { } { (2 n + 2) { 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A211276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A211277" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A211278" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ (-1) ((3 n1 + 4) hypergeom([1/2, -n1 - 1], [1], 4) + (-3 n1 - 3) hypergeom([1/2, -n1], [1], 4))| {1, (2 n + 3) | ) -------------------------------------------------------------------------------------------------|, 2 n + 3} | / (2 n1 + 5) (2 n1 + 3) | |----- | \n1 = 0 / "A211279" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211281" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211288" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ n1 (-n1 - 1) | {2 , 2 | ) (-1) 2 hypergeom([1/2, -n1 - 1], [1], 4)|} | / | |----- | \n1 = 0 / "A211289" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {2 } "A211290" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A211292" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A211293" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A211294" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A211295" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A211296" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 8 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" / 1/2\n / 1/2 \n | 5 | |5 | {1, |1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A211297" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A211298" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A211300" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A211301" memory used=106458.4MB, alloc=2615.5MB, time=769.61 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(1/2) } "A211303" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {3 } "A211304" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1 - 3 ) , (1 + 3 ) } "A211305" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| } \ 2 / \ 2 / "A211307" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211308" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211374" n {n! (n + 3), (-1) n! (n - 1)} "A211606" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (2 HermiteH(n + 1, 1/2 I 2 ) - 2 (3 n + 1) HermiteH(n, 1/2 I 2 ) I) n} "A211774" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211789" LREtools/SearchTable: "SearchTable successful" (3 n + 1) hypergeom([2 n + 3, -n - 1], [1], -1) + (-8 n - 4) hypergeom([-n, 2 n + 1], [1], -1) {----------------------------------------------------------------------------------------------} (17 n + 11) n "A211867" LREtools/SearchTable: "SearchTable successful" n {(-1) binomial(2 n, n) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)} "A211880" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (- n/2) {{ , { 2 binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2) { { 2 (n/2 - 1/2)! n::odd { 0 n::odd "A211894" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211895" LREtools/ReduceToOrderTwo: "Checking Symmetric Cube... (can be time consuming...)" LREtools/ReduceToOrderTwo: "Galois group is Sp4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A211896" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A212199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n n | \ -I (-I 7 ) 7 (LegendreP(n1, 1/7 I 7 ) + 7 LegendreP(n1 + 1, 1/7 I 7 ) I)| {(-2) , (-2) | ) --------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- n1 (-2) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n | \ -I (-I 7 ) 7 (LegendreQ(n1, 1/7 I 7 ) + 7 LegendreQ(n1 + 1, 1/7 I 7 ) I)| (-2) | ) --------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- n1 (-2) | \n1 = 0 / "A212205" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" / / / { n2 n2 \\\ | | | { LegendreP(---- + 3/2, 3) - 3 LegendreP(---- + 1/2, 3) ||| | | | { 2 2 ||| | | | { ----------------------------------------------------- n2::even||| | | | { n2 + 1 ||| | | | { ||| | | | { / n2 n2 \ ||| | | | { 2 |LegendreP(---- + 1, 3) - 3 LegendreP(----, 3)| ||| |n - 1 | |n1 - 1 { \ 2 2 / ||| |----- | |----- { ------------------------------------------------- n2::odd ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { n2 ||| {(6 ) , (-6 ) , (6 ) | ) |1/6 6 (-1) | ) -----------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-6 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 n2 \\\ | | | { LegendreQ(---- + 3/2, 3) - 3 LegendreQ(---- + 1/2, 3) ||| | | | { 2 2 ||| | | | { ----------------------------------------------------- n2::even||| | | | { n2 + 1 ||| | | | { ||| | | | { / n2 n2 \ ||| | | | { 2 |LegendreQ(---- + 1, 3) - 3 LegendreQ(----, 3)| ||| |n - 1 | |n1 - 1 { \ 2 2 / ||| |----- | |----- { ------------------------------------------------- n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { n2 ||| (6 ) | ) |1/6 6 (-1) | ) -----------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-6 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 n2 \ \\\ | | | { 4 |LegendreP(---- + 1, 3) - 3 LegendreP(----, 3)| ||| | | | { \ 2 2 / ||| | | | { ------------------------------------------------- n2::even||| | | | { n2 ||| | | | { ||| | | | { / n2 n2 \ ||| | | | { 2 |LegendreP(---- + 3/2, 3) - 3 LegendreP(---- + 1/2, 3)| ||| |n - 1 | |n1 - 1 { \ 2 2 / ||| |----- | |----- { --------------------------------------------------------- n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { n2 + 1 ||| (6 ) | ) |1/6 6 (-1) | ) ---------------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-6 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 n2 \ \\\ | | | { 4 |LegendreQ(---- + 1, 3) - 3 LegendreQ(----, 3)| ||| | | | { \ 2 2 / ||| | | | { ------------------------------------------------- n2::even||| | | | { n2 ||| | | | { ||| | | | { / n2 n2 \ ||| | | | { 2 |LegendreQ(---- + 3/2, 3) - 3 LegendreQ(---- + 1/2, 3)| ||| |n - 1 | |n1 - 1 { \ 2 2 / ||| |----- | |----- { --------------------------------------------------------- n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { n2 + 1 ||| (6 ) | ) |1/6 6 (-1) | ) ---------------------------------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-6 ) ||| \n1 = 0 \ \n2 = 0 /// "A212233" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n {9 , hypergeom([1/2, -2 n - 2], [1], 4)} "A212234" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n hypergeom([1/2, -2 n - 2], [1], 4) + (-12 n - 6) hypergeom([1/2, -2 n], [1], 4) {9 , -------------------------------------------------------------------------------} 4 n + 3 "A212235" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n (26 n + 27) hypergeom([1/2, -2 n - 2], [1], 4) + (-62 n - 24) hypergeom([1/2, -2 n], [1], 4) {9 , --------------------------------------------------------------------------------------------} 4 n + 3 "A212236" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 2 2 n (142 n + 145 n + 2) hypergeom([1/2, -2 n - 2], [1], 4) + (-250 n - 88 n - 6) hypergeom([1/2, -2 n], [1], 4) {9 , -------------------------------------------------------------------------------------------------------------} (4 n + 3) n "A212237" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 3 2 3 2 n (3556 n + 1868 n - 1753 n - 56) hypergeom([1/2, -2 n - 2], [1], 4) + (-5500 n + 1002 n + 832 n + 168) hypergeom([1/2, -2 n], [1], 4) {9 , ----------------------------------------------------------------------------------------------------------------------------------------} (2 n - 1) (4 n + 3) n "A212238" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 4 3 2 {9 , ((120196 n - 56308 n - 122611 n + 56803 n + 2748) hypergeom([1/2, -2 n - 2], [1], 4) 4 3 2 + (-172684 n + 209286 n - 11534 n - 22068 n - 8244) hypergeom([1/2, -2 n], [1], 4))/((4 n + 3) (2 n - 1) (n - 1) n)} "A212239" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 5 4 3 2 {9 , ((2561296 n - 5029992 n - 834620 n + 5119698 n - 1743722 n - 104322) hypergeom([1/2, -2 n - 2], [1], 4) 5 4 3 2 + (-3506080 n + 9611228 n - 6852236 n + 81829 n + 552819 n + 312966) hypergeom([1/2, -2 n], [1], 4))/((4 n + 3) (2 n - 3) (2 n - 1) (n - 1) n)} "A212260" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A212291" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) n1| {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A212303" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 1/2 n binomial(n, n/2) n::even {{ , { } { (2 n - 2) { 1/2 binomial(n + 1, n/2 + 1/2) n::odd { 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A212334" LREtools/SearchTable: "SearchTable successful" 3 {(7 (n + 1) hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) 3 2 / 3 + (-239 n - 357 n - 189 n - 35) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1)) / n } / "A212348" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n / 1/2\n / 1/2\n / 1/2\ / 1/2 \n / 1/2 \n / 1/2\ / 1/2\n | 5 | |5 | | 5 | | 5 | | 48 16 5 | |5 | |5 | | 48 16 5 | | 5 | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| n, |1/2 - ----| |n + -- + -------|, |---- + 1/2| n, |---- + 1/2| |n + -- - -------|, |1/2 - ----| \ 2 / \ 2 / \ 2 / \ 2 / \ 11 55 / \ 2 / \ 2 / \ 11 55 / \ 2 / / / |n - 1 | |----- | 1/2 | \ | n1 1/2 (-n1 - 1) 1/2 n1 2 (55 n + 240 + 16 5 ) | ) |- 2 (-1) (5 - 1) (5 + 1) (11 n1 + 107 n1 + 256) | / | |----- | \n1 = 0 \ / / 1/2 \(-n2 - 1) / 1/2\ \ |n1 - 1 |5 | | 59 16 5 | | |----- |---- + 1/2| (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) (2 n2 + 9) (2 n2 + 11) |n2 + -- + -------| binomial(2 n2, n2)| / | \ \ 2 / \ 11 55 / | / | | ) -------------------------------------------------------------------------------------------------------------------------------| / | | / 2 2 | / \ |----- (n2 + 1) (n2 + 2) (n2 + 3) (n2 + 4) (n2 + 5) (11 (n2 + 1) + 107 n2 + 363) (11 n2 + 107 n2 + 256) | \n2 = 0 / \\ || / 1/2\ \|| | 48 16 5 | 1/2 ||| |n1 + -- + -------| (55 n1 + 295 + 16 5 )|||} \ 11 55 / /|| || // "A212349" memory used=107335.2MB, alloc=2615.5MB, time=775.93 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n 1/2 n 1/2 n n 1/2 n 1/2 {2 , (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , 2 (n + 36/7), (- 1/2 - 1/2 I 3 ) (n + 7 + 2/21 I 3 ), /n - 1 |----- 1/2 n 1/2 1/2 n 1/2 | \ 1/2 n1 (- 1/2 + 1/2 I 3 ) (n + 7 - 2/21 I 3 ), (- 1/2 - 1/2 I 3 ) (2 I 3 + 21 n + 147) | ) (- 1/2 + 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 / |----- | 1/2 (-n1 - 1) 2 | \ | n2 1/2 (-n2 - 1) 1/2 (- 1/2 - 1/2 I 3 ) (147 n1 + 2205 n1 + 8222) | ) |2 4 (3 I - 1) (n2 + 8 + 2/21 I 3 ) | / | |----- | \n2 = 0 \ /n2 - 1 |----- 3 2 | \ (-n3 - 1) (21 n2 + 465 n2 + 3404 n2 + 8234) | ) 2 (2 n3 + 1) (2 n3 + 3) (2 n3 + 5) (2 n3 + 7) (2 n3 + 9) (2 n3 + 11) (2 n3 + 13) (2 n3 + 15) | / |----- \n3 = 0 2 / (2 n3 + 17) (147 n3 + 2499 n3 + 10574) binomial(2 n3, n3) / ((n3 + 1) (n3 + 2) (n3 + 3) (n3 + 4) (n3 + 5) (n3 + 6) (n3 + 7) / \ | 3 2 3 2 | / 2 (21 (n3 + 1) + 465 (n3 + 1) + 3404 n3 + 11638) (21 n3 + 465 n3 + 3404 n3 + 8234))| / ((147 (n2 + 1) + 2205 n2 + 10427) | / | / \\ \ || | 2 || / 1/2 1/2 | (147 n2 + 2205 n2 + 8222))|| / ((n1 + 7 + 2/21 I 3 ) (2 I 3 + 21 n1 + 168))|} || / | || | // / "A212364" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A212383" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A212395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 5) n1| {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A212403" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} n + 1 "A212404" n binomial(2 n, n) n {4 , ------------------} n + 1 "A212405" n binomial(2 n, n) (3 n + 4) {4 , --------------------------} n + 1 "A212406" 2 n binomial(2 n, n) (59 n + 34 n - 31) {4 , ------------------------------------} (n + 1) (2 n - 1) "A212407" 2 n binomial(2 n, n) (181 n + 98 n - 89) {4 , -------------------------------------} (n + 1) (2 n - 1) "A212408" 3 2 n binomial(2 n, n) (955 n - 917 n - 1224 n + 678) {4 , -------------------------------------------------} (n + 1) (2 n - 1) (2 n - 3) "A212418" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 2 n { 0 n::even { (- n/2) n! (6 n - 2 n - 3) (-1) n! (2 n - 3) { { 2 binomial(n, n/2) (n/2)! {-------------------, ------------------, { (n/2 - 1/2) , { -------------------------------- n::even} (n - 1) n (n - 1) n { 2 2 (n/2 - 1/2)! { n - 1 { --------------------------- n::odd { { n - 1 { 0 n::odd "A212419" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 4 { 2 binomial(n, n/2) (5 n - 2) { -------------------------- n::even { ---------------------------- n::even { n (n + 1) binomial(n, n/2) { (n + 2) (n - 1) {n!, { , { } { (2 n + 2) { 16 binomial(n - 1, n/2 - 1/2) { 2 2 (5 n - 2) { ----------------------------- n::odd { -------------------------------------------------- n::odd { n + 1 { (n - 1) (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A212472" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (6 n + 5) (2 n + 1) (6 n + 1) binomial(6 n, 3 n) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) {16 , ---------------------------------------------------------------------------------------------------} (n + 1) (3 n + 1) (3 n + 2) "A212473" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {16 , ((6 n + 5) (7 n + 8) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - 2 (n + 1) (3 n + 1) (3 n + 2) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n) (2 n + 1)/((n + 1) (3 n + 1) (3 n + 2) (5 n + 4)) } "A212474" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 {16 , ((6 n + 5) (6 n + 1) (839 n + 1277 n + 434) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (109 n + 45) (3 n + 2) (3 n + 1) (n + 1) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/((5 n + 4) (n + 1) (3 n + 1) (3 n + 2) )} "A212475" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 2 {16 , ((6 n + 5) (6 n + 1) (21467 n + 32786 n + 11381 n + 78) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 2 - (n + 1) (3 n + 1) (3 n + 2) (2337 n + 913 n + 39) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/((3 n + 2) (3 n + 1) n (n + 1) (5 n + 4))} "A212476" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 4 3 2 {16 , ((6 n + 5) (6 n + 1) (909865 n + 941246 n - 207517 n - 242330 n - 3624) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 3 2 - (n + 1) (3 n + 1) (3 n + 2) (91205 n - 12457 n - 16635 n - 1812) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n + 1) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A212477" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" memory used=108180.9MB, alloc=2615.5MB, time=781.93 n 5 4 3 2 {16 , ((6 n + 5) (6 n + 1) (8874443 n + 351557 n - 11197974 n - 376007 n + 2347087 n + 48774) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 4 3 2 - (3 n + 2) (3 n + 1) (n + 1) (847363 n - 982678 n - 12799 n + 145672 n + 24387) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n - 1) (n + 1) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A212478" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {16 , ((6 n + 5) (6 n + 1) (661193009 n - 964193320 n - 875126553 n + 1220197891 n + 216753778 n - 259146051 n - 5992434) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (3 n + 2) (3 n + 1) (n + 1) 5 4 3 2 (61112089 n - 164030149 n + 108858683 n + 9060331 n - 14898507 n - 2996217) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n - 1) (n + 1) (2 n - 3) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A212650" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | /n - 1 \ |n - 1 { \ 2 / / n1 \ n1 | |----- n1 | |----- { n1 (-1/2) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | | \ (-1) | | \ { \ 2 / 2 | {n! | ) ---------|, n! | ) ----------------------------------------------------------------------------------|, | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / / n1 \ | | { (-2) |----|! n1::even| / / 1/2\n1 1/2 \ |n - 1 { \ 2 / | |n - 1 | 2 | 2 | |----- { | |----- |- ----| HermiteH(n1 + 1, ----)| | \ { 0 n1::odd | | \ \ 2 / 2 | n! | ) ------------------------------------|, n! | ) ---------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A212694" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 5 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A212696" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {(-1) ((8 n + 4) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (3 n + 3) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(5 n + 3), binomial(2 n, n)} "A212730" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n {9 , hypergeom([1/2, -2 n - 2], [1], 4)} "A212731" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n (24 n + 19) hypergeom([1/2, -2 n - 2], [1], 4) + (-12 n - 6) hypergeom([1/2, -2 n], [1], 4) {9 , -------------------------------------------------------------------------------------------} 4 n + 3 "A212732" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n (98 n + 77) hypergeom([1/2, -2 n - 2], [1], 4) + (-62 n - 36) hypergeom([1/2, -2 n], [1], 4) {9 , --------------------------------------------------------------------------------------------} 4 n + 3 "A212733" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 2 2 n (358 n + 275 n - 2) hypergeom([1/2, -2 n - 2], [1], 4) + (-250 n - 152 n + 6) hypergeom([1/2, -2 n], [1], 4) {9 , --------------------------------------------------------------------------------------------------------------} (4 n + 3) n "A212734" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 3 2 3 2 n (7444 n + 1834 n - 2872 n + 65) hypergeom([1/2, -2 n - 2], [1], 4) + (-5500 n - 704 n + 1865 n - 195) hypergeom([1/2, -2 n], [1], 4) {9 , ----------------------------------------------------------------------------------------------------------------------------------------} (2 n - 1) (4 n + 3) n "A212735" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 4 3 2 {9 , ((225172 n - 175234 n - 136498 n + 89545 n - 3777) hypergeom([1/2, -2 n - 2], [1], 4) 4 3 2 + (-172684 n + 148608 n + 85099 n - 67338 n + 11331) hypergeom([1/2, -2 n], [1], 4))/((4 n + 3) (2 n - 1) (n - 1) n)} "A212736" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=108991.3MB, alloc=2647.5MB, time=787.82 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 5 4 3 2 {9 , ((4450864 n - 10257612 n + 2789092 n + 5673195 n - 2794118 n + 172905) hypergeom([1/2, -2 n - 2], [1], 4) 5 4 3 2 + (-3506080 n + 8253224 n - 2710580 n - 4134038 n + 2398791 n - 518715) hypergeom([1/2, -2 n], [1], 4))/((4 n + 3) (2 n - 3) (2 n - 1) (n - 1) n)} "A212915" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A212916" memory used=109728.3MB, alloc=2648.7MB, time=793.35 memory used=110446.8MB, alloc=2647.5MB, time=798.75 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A213090" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 6 _Z + 5 _Z - 4, index = 1) , RootOf(_Z - 6 _Z + 5 _Z - 4, index = 2) , RootOf(_Z - 6 _Z + 5 _Z - 4, index = 3) } "A213119" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} n + 1 "A213120" n binomial(2 n, n) (15 n + 7) {4 , ---------------------------} n + 1 "A213121" n binomial(2 n, n) (7 n + 3) {4 , --------------------------} n + 1 "A213122" 2 n binomial(2 n, n) (315 n - 35 n - 62) {4 , -------------------------------------} (n + 1) (2 n - 1) "A213123" 2 n binomial(2 n, n) (33 n - 5 n - 6) {4 , ----------------------------------} (n + 1) (2 n - 1) "A213124" 3 2 n binomial(2 n, n) (1001 n - 1694 n + 119 n + 254) {4 , --------------------------------------------------} (n + 1) (2 n - 1) (2 n - 3) "A213168" {n, (n + 1) n! n} "A213169" {n + 1, n!} "A213190" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 1/2 {(-3 ) 3 (3 BesselI(n - 1, 2 3 ) - n BesselI(n, 2 3 )), (-3 ) 3 (3 BesselK(n - 1, -2 3 ) - n BesselK(n, -2 3 ))} "A213203" 2 {(n + 1) n!, (n + 1) (n!) (n - 1)} "A213252" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A213282" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A213290" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { binomial(n, n/2) (5 n - 4) { 4 4 (5 n + 1) { -------------------------- n::even { -------------------------- n::even { n - 1 { n (n + 1) binomial(n, n/2) {1, { , { } { 2 binomial(n - 1, n/2 - 1/2) (5 n + 1) { (2 n + 2) { -------------------------------------- n::odd { 2 2 (5 n - 4) { n + 1 { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A213291" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n | \ n1 6 5 4 3 2 {1, (-1) , (2 n + 1) | ) (-1) ((27 n1 + 148 n1 - 440 n1 - 1174 n1 + 779 n1 + 960 n1 - 108) hypergeom([1/2, -n1 - 1], [1], 4) | / |----- \n1 = 0 6 5 4 3 2 + (81 n1 + 5 n1 - 908 n1 - 1157 n1 + 731 n1 + 1932 n1 - 108) hypergeom([1/2, -n1], [1], 4))/((n1 + 3) (n1 + 2) n1 (n1 - 1) (n1 - 2) \ { binomial(n, n/2) (5 n - 4) | { -------------------------- n::even | { n - 1 (2 n1 + 3) (2 n1 + 1))|, 2 n + 1, { , | { 2 binomial(n - 1, n/2 - 1/2) (5 n + 1) | { -------------------------------------- n::odd / { n + 1 { n { 4 4 (5 n + 1) { -------------------------- n::even { n (n + 1) binomial(n, n/2) { } { (2 n + 2) { 2 2 (5 n - 4) { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A213336" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A213403" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable not successful" n 16 binomial(2 n, n) (6 n + 1) (2 n + 1) binomial(6 n, 3 n) {--------------------, --------------------------------------} n + 1 (n + 1) (3 n + 1) (3 n + 2) "A213465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n (6 n + 5) (2 n + 1) (6 n + 1) binomial(6 n, 3 n) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) {16 , ---------------------------------------------------------------------------------------------------} (n + 1) (3 n + 1) (3 n + 2) "A213466" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {16 , ((6 n + 5) (47 n + 40) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - 2 (n + 1) (3 n + 1) (3 n + 2) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n) (2 n + 1)/((n + 1) (3 n + 1) (3 n + 2) (5 n + 4)) } "A213467" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=111346.4MB, alloc=2647.5MB, time=804.91 LREtools/SearchTable: "SearchTable successful" n 2 {16 , ((6 n + 5) (6 n + 1) (2119 n + 2857 n + 898) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (3 n + 2) (109 n + 61) (3 n + 1) (n + 1) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/((5 n + 4) (n + 1) (3 n + 1) (3 n + 2) )} "A213468" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 2 {16 , ((6 n + 5) (6 n + 1) (41947 n + 55978 n + 17293 n - 66) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 2 - (n + 1) (3 n + 1) (3 n + 2) (2337 n + 1361 n - 33) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n + 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A213469" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 4 3 2 {16 , ((6 n + 5) (6 n + 1) (1565243 n + 1277044 n - 410291 n - 314608 n + 4092) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 3 2 - (n + 1) (3 n + 1) (3 n + 2) (91223 n + 8787 n - 28451 n + 2046) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n + 1) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A213470" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {16 , ( 5 4 3 2 (6 n + 5) (6 n + 1) (14117557 n - 2904929 n - 15063954 n + 1090643 n + 2793833 n - 73830) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) 4 3 2 - (n + 1) (3 n + 1) (3 n + 2) (847597 n - 760086 n - 358693 n + 289452 n - 36915) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n - 1) (n + 1) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A213471" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {16 , ((6 n + 5) (6 n + 1) (996758605 n - 1723870076 n - 710549073 n + 1676517959 n + 50838638 n - 295259043 n + 12400830) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (n + 1) (3 n + 1) (3 n + 2) 5 4 3 2 (61133365 n - 146377733 n + 55768535 n + 61378283 n - 34968855 n + 6200415) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/(n (n - 1) (n + 1) (2 n - 3) (2 n - 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A213507" LREtools/SearchTable: "SearchTable successful" {(2 n + 1) n! binomial(2 n, n) (hypergeom([-n - 1], [n + 2], 1) - hypergeom([-n], [n + 1], 1))} "A213527" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 5 {|1/2 - ----| n! hypergeom([-n, 1/2 + ----], [1], 5/2 + ----)} \ 2 / 10 2 "A213528" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 n 2 1/2 {(1 - 2 ) n! hypergeom([-n, 1/2 + ----], [1], 4 + 2 2 )} 4 "A213593" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 n 5 n 5 {(-1) GAMMA(n - 1/2 - ----), (-1) GAMMA(n - 1/2 + ----)} 2 2 "A213684" LREtools/SearchTable: "SearchTable successful" ((20 n + 10) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (-3 n - 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {---------------------------------------------------------------------------------------------------------------------------------------} n + 1 "A213705" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A213720" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) ((n + 3) HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I)} "A213937" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 - 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A214159" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=112195.7MB, alloc=2647.5MB, time=810.62 LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 |----- n1 2 | |----- n n | \ (2 n1 + 1) (-1/4) binomial(2 n1, n1) (32 n1 + 124 n1 + 123)| n | \ n1 {(1/3) , (1/3) | ) --------------------------------------------------------------|, (1/3) | ) (2 n1 + 1) (-1/4) binomial(2 n1, n1) | / (n1 + 1) | | / |----- (n1 + 3) (n1 + 2) (n1 + 1) (1/3) | |----- \n1 = 0 / \n1 = 0 2 (32 n1 + 124 n1 + 123) /n1 - 1 \ |----- 2 | | \ (3 n2 + 5) (3 n2 + 2) (3 n2 + 4) (3 n2 + 1) binomial(2 n2, n2) binomial(3 n2, n2) (140 n2 + 483 n2 + 423) | | ) --------------------------------------------------------------------------------------------------------------------------------------| | / 2 (n2 + 1) 2 2 | |----- (n2 + 3) (n2 + 2) (n2 + 1) (2 n2 + 3) (-1/4) binomial(2 n2 + 2, n2 + 1) (32 (n2 + 1) + 124 n2 + 247) (32 n2 + 124 n2 + 123)| \n2 = 0 / \ | / (n1 + 1) | / ((n1 + 3) (n1 + 2) (n1 + 1) (1/3) )|} / | | / "A214198" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A214200" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A214201" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A214203" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A214204" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A214282" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) { { 2 (-1) binomial(n, n/2) {{ (n/2 - 1/2) , { ---------------------------- n::even, { (-16) { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A214283" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) { { 2 (-1) binomial(n, n/2) {{ (n/2 - 1/2) , { ---------------------------- n::even, { (-16) { n + 2 { ------------------------------------ n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A214358" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {n + 3} "A214372" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A214377" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 2 4 binomial(---, n/2) { 3 n 3 n { 2 {{ (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { ----------------------- n::even} { 2 2 { n + 2 { --------------------------------------------------------------------- n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A214553" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /12500\(n/2 - 1/2) {{ |-----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) GAMMA(n/2 + 1/5) , { \ 27 / { -------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 4/3) GAMMA(n/2 + 2/3) { /12500\(n/2) { |-----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 1/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) { \ 27 / { -------------------------------------------------------------------------------- n::even} { GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/2) { { 0 n::odd "A214615" LREtools/SearchTable: "SearchTable successful" n {I n! hypergeom([-n, 1/2 + 1/2 I], [1], 2)} "A214649" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreP(n, ----) + 5 (n + 1) LegendreP(n + 1, ----)| \ 5 5 / {- ---------------------------------------------------------------------------, n (n - 1) / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 |(3 n - 1) LegendreQ(n, ----) + 5 (n + 1) LegendreQ(n + 1, ----)| \ 5 5 / - ---------------------------------------------------------------------------} n (n - 1) "A214692" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A214801" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A214817" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 3 2 n n 3 2 {(-1) (n + 1) (243 n + 243 n + 90 n - 8), 8 (n + 1) (225 n - 64), (-1) (n + 1) (243 n + 243 n + 90 n - 8) /n - 1 \ |----- / n1 n1 6 5 4 3 2 \| | \ | (-1) 2 (2 n1 + 1) (567 n1 + 5454 n1 + 14217 n1 + 17126 n1 + 8868 n1 + 568 n1 - 480) binomial(2 n1, n1)|| | ) |- ---------------------------------------------------------------------------------------------------------------||} | / | 3 2 3 2 || |----- \ (n1 + 1) (n1 + 2) (243 (n1 + 1) + 243 (n1 + 1) + 90 n1 + 82) (243 n1 + 243 n1 + 90 n1 - 8) /| \n1 = 0 / "A214818" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 7 6 5 4 3 2 {(-1) (n + 1) (6790635 n - 15844815 n + 7608573 n - 7663005 n - 4490640 n + 9589860 n + 6401712 n - 663040), n 4 3 2 n 8 (n + 1) (59535 n + 639090 n - 2679615 n + 2798142 n - 1212416), (-1) (n + 1) /n - 1 |----- 7 6 5 4 3 2 | \ / n1 n1 (6790635 n - 15844815 n + 7608573 n - 7663005 n - 4490640 n + 9589860 n + 6401712 n - 663040) | ) |- (-1) 2 n1 (n1 - 1) (2 n1 + 1) | / \ |----- \n1 = 0 10 9 8 7 6 5 4 3 (280428075 n1 + 1075966335 n1 - 573765120 n1 - 2894733162 n1 - 4475180241 n1 - 3524758209 n1 - 3001291570 n1 + 3394710092 n1 2 / + 6953732056 n1 + 473855584 n1 - 599208960) binomial(2 n1, n1) / ((n1 + 1) (n1 + 2) / 7 6 5 4 3 2 (6790635 (n1 + 1) - 15844815 (n1 + 1) + 7608573 (n1 + 1) - 7663005 (n1 + 1) - 4490640 (n1 + 1) + 9589860 (n1 + 1) + 6401712 n1 + 5738672) \ | 7 6 5 4 3 2 \| (6790635 n1 - 15844815 n1 + 7608573 n1 - 7663005 n1 - 4490640 n1 + 9589860 n1 + 6401712 n1 - 663040))||} /| | / "A214849" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A214875" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A214916" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 { (n - 1) {{ , { , { 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) irem(n, 4) = 1, { 0 irem(n, 4) = 2 { 1/4 %1 irem(n, 4) = 2 { { { { 0 irem(n, 4) = 2 { 1/8 %1 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { { 0 irem(n, 4) = 3 { %1 irem(n, 4) = 0 { { 0 irem(n, 4) = 1 { } { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 n %1 := 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) "A214938" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215002" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215067" memory used=113064.9MB, alloc=2647.5MB, time=816.55 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215096" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | n n n | \ n1 | \ (n2 + 2) (n2 + 1) n2!| | {(-I) , I , (-I) | ) (-1) | ) ---------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A215125" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215176" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 binomial(n, n/2) ((n/2)!) (2 n + 2) n::even {{ 2 , { } { 2 ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { ----------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { n + 1 "A215287" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n 3 n { binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { ----------------------------------------------- n::even { 2 2 { binomial(n, n/2) (n + 3) (n + 1) {{ , { 3 n 3 n { (3 n + 4) (3 n - 2) (3 n + 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { ----------------------------------------------------------------------------------------- n::odd { 2 2 2 { (n + 4) (n + 2) n binomial(n - 1, n/2 - 1/2) { 3 n { 2 binomial(n, n/2) binomial(---, n/2) (3 n + 4) (3 n + 2) { 2 { --------------------------------------------------------- n::even { 2 { (n + 4) (n + 2) } { { 3 n { 2 binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) { 2 { ----------------------------------------------------------- n::odd { n + 3 "A215288" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 (2 n + 1) binomial(2 n, n) { 1/2 ----------------------------------- n::even { 3 2 2 { (n + 1) (n + 3) binomial(n, n/2) {{ , { (4 n - 4) { 4 2 (n + 1) (2 n - 1) (2 n + 1) binomial(2 n - 2, n - 1) { ----------------------------------------------------------------- n::odd { 3 3 2 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { 2 { 128 binomial(n, n/2) binomial(2 n, n) (n + 1) (2 n + 1) { -------------------------------------------------------- n::even { 3 { (n + 4) (n + 2) { } { 2 { 16 binomial(n + 1, n/2 + 1/2) binomial(2 n + 2, n + 1) { ------------------------------------------------------- n::odd { 2 { (n + 3) "A215289" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /5 n\ { |---| { \ 2 / { 5 GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) GAMMA(n/2 + 8/5) { -------------------------------------------------------------------------- n::even { 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) {{ , { /5 n \ { |--- + 5/2| { \ 2 / 13 17 19 21 { 5 (n + 3) (n + 5) (n + 7) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) { 10 10 10 10 { ---------------------------------------------------------------------------------------------------- n::odd { 2 { (5 n + 7) (5 n + 9) (5 n + 11) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) { /5 n\ { |---| { \ 2 / 13 17 19 21 { 5 (n + 3) (n + 5) (n + 7) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) { 10 10 10 10 { ---------------------------------------------------------------------------------------------- n::even { 2 { (5 n + 7) (5 n + 9) (5 n + 11) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) } { { /5 n \ { |--- - 5/2| { \ 2 / { 5 GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) GAMMA(n/2 + 8/5) { -------------------------------------------------------------------------------- n::odd { 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) "A215290" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n 2 3 n { 64 binomial(---, n/2) (3 n + 1) (3 n + 2) binomial(3 n, ---) { 2 2 { 1/8 -------------------------------------------------------------- n::even { 5 4 2 4 { (n + 1) (n + 3) (n + 5) binomial(n, n/2) {{ , { (6 n - 6) 3 n 2 3 n { 3 2 binomial(--- - 3/2, n/2 - 1/2) (n + 1) (3 n - 2) (3 n - 1) (3 n + 1) (3 n + 2) binomial(3 n - 3, --- - 3/2) { 2 2 { ------------------------------------------------------------------------------------------------------------------------- n::odd { 5 5 3 4 { n (n + 2) (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) { 2 3 n 2 3 n { 6144 binomial(n, n/2) binomial(---, n/2) binomial(3 n, ---) (3 n + 2) (3 n + 1) (n + 1) { 2 2 { ----------------------------------------------------------------------------------------- n::even { 5 3 { (n + 2) (n + 4) (n + 6) { } { 2 3 n 2 3 n { 256 binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { -------------------------------------------------------------------------------------------- n::odd { 4 2 { (n + 3) (n + 5) "A215291" memory used=113835.6MB, alloc=2748.1MB, time=823.22 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { { /7 n\ { |---| {{ \ 2 / { 7 GAMMA(n/2 + 5/7) GAMMA(n/2 + 6/7) GAMMA(n/2 + 8/7) GAMMA(n/2 + 9/7) GAMMA(n/2 + 10/7) GAMMA(n/2 + 11/7) { -------------------------------------------------------------------------------------------------------------- , n::even { 2 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) GAMMA(n/2 + 5) /7 n \ |--- + 7/2| \ 2 / 17 19 23 25 27 29 7 (n + 3) (n + 5) (n + 7) (n + 9) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) 14 14 14 14 14 14 -------------------------------------------------------------------------------------------------------------------------------------------- , 2 2 (7 n + 9) (7 n + 11) (7 n + 13) (7 n + 15) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) GAMMA(n/2 + 11/2) { { { , { { n::odd { { { /7 n\ |---| \ 2 / 17 19 23 25 27 29 7 (n + 3) (n + 5) (n + 7) (n + 9) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) 14 14 14 14 14 14 -------------------------------------------------------------------------------------------------------------------------------------- , n::even 2 2 (7 n + 9) (7 n + 11) (7 n + 13) (7 n + 15) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) GAMMA(n/2 + 11/2) /7 n \ |--- - 7/2| \ 2 / } 7 GAMMA(n/2 + 5/7) GAMMA(n/2 + 6/7) GAMMA(n/2 + 8/7) GAMMA(n/2 + 9/7) GAMMA(n/2 + 10/7) GAMMA(n/2 + 11/7) -------------------------------------------------------------------------------------------------------------------- , n::odd 2 2 GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) GAMMA(n/2 + 5) "A215294" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n 3 n { binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { ----------------------------------------------- n::even { 2 2 { binomial(n, n/2) (n + 3) (n + 1) {{ , { 3 n 3 n { (3 n + 2) (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { ------------------------------------------------------------------------------- n::odd { 2 2 2 { (n + 2) n binomial(n - 1, n/2 - 1/2) { 3 n { 2 binomial(n, n/2) binomial(---, n/2) (3 n + 2) { 2 { ----------------------------------------------- n::even { 2 { (n + 2) } { { 3 n { 2 binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) { 2 { ----------------------------------------------------------- n::odd { n + 3 "A215295" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /5 n\ { |---| { \ 2 / { 5 GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) { -------------------------------------------------------------------------- n::even { 2 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) {{ , { /5 n \ { |--- + 5/2| { \ 2 / 11 13 17 19 { 5 (n + 3) (n + 5) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) { 10 10 10 10 { -------------------------------------------------------------------------------------------- n::odd { 2 2 { (5 n + 7) (5 n + 9) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) { /5 n\ { |---| { \ 2 / 11 13 17 19 { 5 (n + 3) (n + 5) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) { 10 10 10 10 { -------------------------------------------------------------------------------------- n::even { 2 2 { (5 n + 7) (5 n + 9) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) } { { /5 n \ { |--- - 5/2| { \ 2 / { 5 GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) { -------------------------------------------------------------------------------- n::odd { 2 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) "A215296" memory used=114794.4MB, alloc=2745.8MB, time=830.28 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { { /7 n\ { |---| {{ \ 2 / { 7 GAMMA(n/2 + 4/7) GAMMA(n/2 + 5/7) GAMMA(n/2 + 6/7) GAMMA(n/2 + 8/7) GAMMA(n/2 + 9/7) GAMMA(n/2 + 10/7) { ------------------------------------------------------------------------------------------------------------- , n::even { 2 2 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) /7 n \ |--- + 7/2| \ 2 / 15 17 19 23 25 27 7 (n + 3) (n + 5) (n + 7) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) , 14 14 14 14 14 14 ------------------------------------------------------------------------------------------------------------------------------------ , n::odd 2 2 2 (7 n + 9) (7 n + 11) (7 n + 13) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) { /7 n\ { |---| { \ 2 / 15 17 19 23 25 27 { 7 (n + 3) (n + 5) (n + 7) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) { 14 14 14 14 14 14 { ------------------------------------------------------------------------------------------------------------------------------ n::even { 2 2 2 { (7 n + 9) (7 n + 11) (7 n + 13) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/2) GAMMA(n/2 + 9/2) } { { /7 n \ { |--- - 7/2| { \ 2 / { 7 GAMMA(n/2 + 4/7) GAMMA(n/2 + 5/7) GAMMA(n/2 + 6/7) GAMMA(n/2 + 8/7) GAMMA(n/2 + 9/7) GAMMA(n/2 + 10/7) { ------------------------------------------------------------------------------------------------------------------- n::odd { 2 2 2 { GAMMA(n/2 + 2) GAMMA(n/2 + 3) GAMMA(n/2 + 4) "A215340" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215341" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A215342" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A215576" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n n {(-I) , I } "A215654" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215661" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215686" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A215715" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A215763" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 - 1) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- 6 5 4 3 2 || | | \ n2! (n2 - 3 n2 - 11 n2 + 39 n2 + 98 n2 - 4 n2 - 24)|| | n1! (n1 - 1) | ) --------------------------------------------------------|| |n - 1 | / 2 2 || |----- |----- (n2 - 1) n2 (n2 + 1)! (n2 + 3) (n2 + 2) || | \ \n2 = 0 /| (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------|} | / (n1 + 3) (n1 + 1)! | |----- | \n1 = 0 / "A215764" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" memory used=115764.2MB, alloc=2743.5MB, time=836.54 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! n1 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ---------------------------|, | / (n1 + 4) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- 6 5 4 3 2 || | | \ n2! (n2 + 3 n2 - 11 n2 - 15 n2 + 134 n2 + 256 n2 + 96)|| | (n1 + 1) n1! n1 | ) -----------------------------------------------------------|| |n - 1 | / 2 || |----- |----- n2 (n2 + 2) (n2 + 1)! (n2 + 4) (n2 + 1) (n2 + 3) || | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, (n + 3) (n + 2) (n + 1) | / (n1 + 4) (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / /n - 1 /n1 - 1 /n2 - 1 |----- |----- |----- | \ | \ 6 5 4 3 2 | \ n! | ) (n1 + 1) n1! n1 | ) n2! (n2 + 3 n2 - 11 n2 - 15 n2 + 134 n2 + 256 n2 + 96) | ) n3! | / | / | / |----- |----- |----- \n1 = 0 \n2 = 0 \n3 = 0 12 11 10 9 8 7 6 5 4 3 2 (n3 - 6 n3 - 35 n3 + 302 n3 + 747 n3 - 5778 n3 - 14569 n3 + 21882 n3 + 39744 n3 - 116944 n3 - 192160 n3 + 19904 n3 + 91392) 2 / 6 5 4 3 2 (n3 + 1) (n3 + 4) / (n3 (n3 - 1) (n3 - 2) (n3 + 1)! ((n3 + 1) + 3 (n3 + 1) - 11 (n3 + 1) - 15 (n3 + 1) + 134 (n3 + 1) + 256 n3 + 352) / \ \ \ | | | 6 5 4 3 2 | / 2 | | (n3 + 3 n3 - 11 n3 - 15 n3 + 134 n3 + 256 n3 + 96))| / (n2 (n2 + 2) (n2 + 1)! (n2 + 4) (n2 + 1) (n2 + 3))|/((n1 + 4) (n1 + 2) (n1 + 1)!)| | / | | | | | / / / } "A215772" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1) (n1 - 2)| {(n + 1) n!, (n + 1) n! | ) ---------------------|} | / n1 (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A215789" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n 3 n { binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { ----------------------------------------------- n::even { 2 2 { binomial(n, n/2) (n + 3) (n + 2) (n + 1) n {{ , { 3 n 3 n { 8 (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { ----------------------------------------------------------------------- n::odd { 2 2 2 { (n + 4) (n + 2) n binomial(n - 1, n/2 - 1/2) { 3 n { 64 binomial(n, n/2) binomial(---, n/2) { 2 { -------------------------------------- n::even { 2 { (n + 4) (n + 2) } { { 3 n { 8 binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) { 2 { ----------------------------------------------------------- n::odd { (n + 3) (n + 2) n "A215931" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | |3/2 - ----| binomial(2 n, n) |3/2 + ----| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A215973" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A216116" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A216119" {(n + 1) (n - 1) (n - 2) n!} "A216234" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable not successful" {} "A216239" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 | 2 2 | \ (n1 + 1) n1 (-1) (6 n1 + 4 n1 - 1) | {n! (3 n - n + 1), n! (3 n - n + 1) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! (3 (n1 + 1) - n1) (3 n1 - n1 + 1)| \n1 = 0 / "A216314" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216316" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([1/3, -n], [1], 9)} "A216317" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([2/3, -n + 1/2], [3/2], 9)} "A216357" LREtools/SearchTable: "SearchTable successful" n {(-4) hypergeom([1/2, 1/4 - n], [5/4], 5)} "A216358" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216359" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A216434" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) } "A216441" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A216443" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (n/2 + 1) 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A216454" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/3) } "A216466" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A216483" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216490" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216541" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | |1/2 - ----| binomial(2 n, n) |---- + 1/2| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A216584" LREtools/SearchTable: "SearchTable successful" n {(-1) binomial(2 n, n) hypergeom([1/2, -n], [1], 4)} "A216604" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216617" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A216636" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216696" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216698" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216778" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n - 1, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A216779" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n - 1, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A216795" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216831" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A216947" LREtools/SearchTable: "SearchTable successful" 3 2 2 (2 n + 15 n + 55 n + 54) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) - 9 (n + 1) (2 n + 11 n + 18) hypergeom([1/2, -n, -n], [1, 1], 4) {-----------------------------------------------------------------------------------------------------------------------------------------} 2 (n + 3) n (n + 2) "A217213" (n - 2) (n - 1) binomial(2 n, n) {--------------------------------} 2 n - 1 "A217238" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2| | \ (n1 + 1) (n1!) | {n! | ) ----------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A217239" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2| | \ (n1 + 1) (n1!) | {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A217275" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-3 I 3 ) (3 LegendreP(n + 1, 1/9 I 3 ) I - 9 LegendreP(n, 1/9 I 3 )) {-------------------------------------------------------------------------------, n + 2 1/2 n 1/2 1/2 1/2 (-3 I 3 ) (3 LegendreQ(n + 1, 1/9 I 3 ) I - 9 LegendreQ(n, 1/9 I 3 )) -------------------------------------------------------------------------------} n + 2 "A217280" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217282" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A217284" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 3 3 | \ 1 | {(n!) , (n!) | ) ------------|} | / 3| |----- ((n1 + 1)!) | \n1 = 0 / "A217312" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - _Z + 3, index = 1) , RootOf(_Z - 2 _Z - _Z + 3, index = 2) , RootOf(_Z - 2 _Z - _Z + 3, index = 3) } "A217323" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" n 2 3 2 (-1) (-3 (13 n + 81 n + 122) (n + 1) hypergeom([1/2, -n], [1], 4) + (41 n + 318 n + 775 n + 594) hypergeom([1/2, -n - 1], [1], 4)) {--------------------------------------------------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 3) (n + 2) { n { 4 (n + 2) { 8 binomial(n, n/2) (n + 1) (n + 3) { 1/4 -------------------------------- n::even { ---------------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) { , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) (n + 2) { 2 (n + 1) (n + 3) { ------------------------------------ n::odd { 1/2 -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A217324" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" n 3 2 {(-1) (-3 (41 n + 441 n + 1534 n + 1728) (n + 1) hypergeom([1/2, -n], [1], 4) 4 3 2 + (121 n + 1482 n + 6479 n + 11934 n + 7776) hypergeom([1/2, -n - 1], [1], 4))/((n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), { n 2 2 { 16 (n + 2) (n + 4) { 1/4 ----------------------------------------------------- n::even { 2 2 2 2 2 { (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) { , { (4 n + 4) 2 { 2 (n + 3) (n + 5) { 1/16 -------------------------------------------------------------- n::odd { 2 2 2 2 { (n + 2) (n + 4) (n + 6) (n + 8) binomial(n + 1, n/2 + 1/2) { 2 2 2 { 256 binomial(n, n/2) (n + 3) (n + 5) (n + 1) { ----------------------------------------------- n::even { 2 2 2 { (n + 2) (n + 4) (n + 6) (n + 8) { } { 2 2 2 2 { 1024 binomial(n - 1, n/2 - 1/2) n (n + 2) (n + 4) { ----------------------------------------------------- n::odd { 2 2 2 2 { (n + 1) (n + 3) (n + 5) (n + 7) "A217325" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 2 2 2 2 { 1024 binomial(n, n/2) (n + 1) (n + 3) (n + 5) { ------------------------------------------------- n::even { 2 2 2 2 { (n + 2) (n + 4) (n + 6) (n + 8) {{ , { 2 2 2 { 256 binomial(n + 1, n/2 + 1/2) (n + 4) (n + 6) (n + 2) { --------------------------------------------------------- n::odd { 2 2 2 { (n + 3) (n + 5) (n + 7) (n + 9) { n 2 2 { 16 (n + 2) (n + 4) (n + 6) { 1/64 ------------------------------------------------------------- n::even { 2 2 2 2 2 { (n + 1) (n + 3) (n + 5) (n + 7) (n + 9) binomial(n, n/2) { } { (4 n - 4) 2 2 2 { 2 (n + 1) (n + 3) (n + 5) { 1/16 ------------------------------------------------------------------ n::odd { 2 2 2 2 2 2 { n (n + 2) (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) "A217333" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217358" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217359" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217361" LREtools/SearchTable: "SearchTable successful" 1/2 n 250 80 1/2 (10/7 I 5 - 16/7) GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], --- - --- I 5 ) 189 189 {---------------------------------------------------------------------------------------} GAMMA(n + 2) "A217362" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217365" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217421" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A217446" n n {(n + 2) (n + 1) n! (12 n + 23), (n + 2) (n + 1) 10 n! (3 n + 5), (n + 2) (n + 1) 100 n! (21 n + 32)} "A217447" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ n1! (2 n1 - 4 n1 - 9)| {1, (n - 3) | ) ----------------------|, n - 3} | / (n1 - 2) (n1 - 3) | |----- | \n1 = 0 / "A217461" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217464" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A217525" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(-1) } "A217526" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) ((5 n + 1) hypergeom([1/2, -n], [1], 4) + (n + 1) hypergeom([1/2, -n - 1], [1], 4)) {(-1) , -----------------------------------------------------------------------------------------, 2 n - 1} (n - 1) n "A217539" memory used=116795.6MB, alloc=2743.5MB, time=843.46 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n binomial(2 n, n) (-1) ((n - 2) hypergeom([1/2, -n - 1], [1], 4) + (5 n - 2) hypergeom([1/2, -n], [1], 4)) {----------------, -----------------------------------------------------------------------------------------} n + 1 n "A217596" LREtools/SearchTable: "SearchTable successful" 2 {(10 (2 n + 1) (7 n - 1) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (-289 n - 55 n + 26) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((3 n - 2) (3 n - 1) (3 n + 1))} "A217615" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217616" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217617" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217661" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217664" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217666" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217675" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A217676" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A217677" memory used=117677.4MB, alloc=2743.5MB, time=849.98 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {(n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A217703" LREtools/SearchTable: "SearchTable not successful" {} "A217711" memory used=118397.3MB, alloc=2743.5MB, time=855.26 2 n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (n + 9 n + 26) {4 (n + 4), --------------------------------------------------------------} (n + 4) (n + 2) (n + 1) "A217766" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A217767" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A218008" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A218045" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 {((2 n + 5) (n + 2) (7 n + 7 n + 1) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) 4 3 2 + (-114 n - 456 n - 594 n - 276 n - 30) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1))/((n + 2) (2 n + 1) (n - 1) n), n 2 binomial(2 n, n) 1/2 -------------------} n - 1/2 "A218073" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 n { ------------------------ n::even { (n + 1) binomial(n, n/2) { 1/2 n binomial(n, n/2) n::even {{ , { } { (2 n - 2) { 1/2 n binomial(n + 1, n/2 + 1/2) n::odd { 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A218186" LREtools/SearchTable: "SearchTable successful" n 2 4 (2 (2 n - 3) (n + 1) hypergeom([-1/2, -n - 1], [1], -1) + (-4 n + 3) hypergeom([-1/2, -n], [1], -1)) {--------------------------------------------------------------------------------------------------------} (n - 1) n "A218225" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A218262" memory used=119235.0MB, alloc=2746.1MB, time=861.12 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" LREtools/SearchTable: "SearchTable successful" 1/2 n {(-1/2 I 2 ) ( 1/2 3 2 1/2 3 2 1/2 2 (5 n + 115 n + 856 n + 2056) (n + 3) HermiteH(n + 1, 1/2 I 2 ) I - 2 (n + 1) (n + 4) (n + 30 n + 281 n + 832) HermiteH(n, 1/2 I 2 )) } "A218263" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) ((n + 3) HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I), { n { 4 (n + 2) { 8 binomial(n, n/2) (n + 1) (n + 3) { 1/4 -------------------------------- n::even { ---------------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) { , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) (n + 2) { 2 (n + 1) (n + 3) { ------------------------------------ n::odd { 1/2 -------------------------------------------- n::odd { n + 3 { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A218264" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 n {(-1/2 I 2 ) (2 (n + 3) HermiteH(n + 1, 1/2 I 2 ) I - (n + 1) (n + 4) HermiteH(n, 1/2 I 2 )), (-1) ( 4 3 2 (121 n + 1482 n + 6479 n + 11934 n + 7776) hypergeom([1/2, -n - 1], [1], 4) 3 2 - 3 (41 n + 441 n + 1534 n + 1728) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A218265" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) ((n + 6) (n + 3) HermiteH(n + 1, 1/2 I 2 ) + 2 I 2 (n + 1) (n + 4) HermiteH(n, 1/2 I 2 )), { 2 2 2 2 { 1024 binomial(n, n/2) (n + 1) (n + 3) (n + 5) { ------------------------------------------------- n::even { 2 2 2 2 { (n + 2) (n + 4) (n + 6) (n + 8) { , { 2 2 2 { 256 binomial(n + 1, n/2 + 1/2) (n + 4) (n + 6) (n + 2) { --------------------------------------------------------- n::odd { 2 2 2 { (n + 3) (n + 5) (n + 7) (n + 9) { n 2 2 { 16 (n + 2) (n + 4) (n + 6) { 1/64 ------------------------------------------------------------- n::even { 2 2 2 2 2 { (n + 1) (n + 3) (n + 5) (n + 7) (n + 9) binomial(n, n/2) { } { (4 n - 4) 2 2 2 { 2 (n + 1) (n + 3) (n + 5) { 1/16 ------------------------------------------------------------------ n::odd { 2 2 2 2 2 2 { n (n + 2) (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) "A218266" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (2 (3 n + 16) (n + 3) HermiteH(n + 1, 1/2 I 2 ) I - 2 (n + 4) (n + 7) (n + 1) HermiteH(n, 1/2 I 2 ))} "A218267" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" memory used=120078.3MB, alloc=2743.5MB, time=866.98 memory used=120460.2MB, alloc=2743.5MB, time=870.33 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SearchTable: "SearchTable successful" memory used=120958.0MB, alloc=2743.5MB, time=874.10 1/2 n 2 1/2 1/2 1/2 { n {(-1/2 I 2 ) ((n + 15 n + 52) (n + 3) HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) (n + 4) (3 n + 19) HermiteH(n, 1/2 I 2 ) I), { - 12 4 (n + 2) { 11 10 9 8 7 6 5 4 3 (n + 4) (n + 6) ((2 n + 205 n + 9408 n + 1450833 n + 72996036 n + 1736752674 n + 23611564960 n + 197998158386 n + 1044952604010 n 2 + 3385428011505 n + 6153949177200 n + 4806604583325) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) - 9 9 8 7 6 5 4 3 2 2 (2 n + 189 n + 7860 n + 321660 n + 9430440 n + 161471214 n + 1601529100 n + 9130702140 n + 27838963350 n + 35220397725) (n + 3) / 3 3 3 3 2 2 hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)) / ((n + 1) (n + 3) (n + 5) (n + 7) (n + 9) (n + 11) (n + 13) (n + 15) / binomial(n, n/2)) , n::even (2 n + 2) 10 9 8 7 6 5 4 3 - 6 2 (n + 3) (n + 5) (n + 7) ((2 n + 207 n + 9516 n + 654372 n + 27070872 n + 590641632 n + 7387156960 n + 55261998720 n 2 + 244940078592 n + 593876219904 n + 607265538048) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) - 9 8 7 6 5 4 3 2 2 (2 n + 187 n + 7592 n + 219100 n + 4567880 n + 60673408 n + 470105568 n + 1923230592 n + 3202882560) (n + 4) / 3 3 3 3 2 hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) / ((n + 2) (n + 4) (n + 6) (n + 8) (n + 10) (n + 12) (n + 14) (n + 16) / { 10 9 8 7 6 5 4 binomial(n + 1, n/2 + 1/2)) , n::odd, { - 12582912 ((2 n + 207 n + 9516 n + 654372 n + 27070872 n + 590641632 n + 7387156960 n { 3 2 + 55261998720 n + 244940078592 n + 593876219904 n + 607265538048) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) - 9 8 7 6 5 4 3 2 2 (2 n + 187 n + 7592 n + 219100 n + 4567880 n + 60673408 n + 470105568 n + 1923230592 n + 3202882560) (n + 4) / 3 3 hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) binomial(n, n/2) (n + 7) (n + 5) (n + 3) (n + 1) / ((n + 2) (n + 4) (n + 6) (n + 8) / 3 3 2 (n + 10) (n + 12) (n + 14) (n + 16)) , n::even 11 10 9 8 7 6 5 4 3 - 25165824 ((2 n + 205 n + 9408 n + 1450833 n + 72996036 n + 1736752674 n + 23611564960 n + 197998158386 n + 1044952604010 n 2 + 3385428011505 n + 6153949177200 n + 4806604583325) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) - 9 9 8 7 6 5 4 3 2 2 (2 n + 189 n + 7860 n + 321660 n + 9430440 n + 161471214 n + 1601529100 n + 9130702140 n + 27838963350 n + 35220397725) (n + 3) / 3 hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)) binomial(n - 1, n/2 - 1/2) n (n + 2) (n + 4) (n + 6) / ((n + 1) (n + 3) (n + 5) / 3 3 3 2 2 (n + 7) (n + 9) (n + 11) (n + 13) (n + 15) ) , n::odd} "A218268" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 2 1/2 2 1/2 {(-1/2 I 2 ) (2 I 2 (n + 13 n + 41) (n + 3) HermiteH(n + 1, 1/2 I 2 ) - (n + 1) (n + 4) (n + 17 n + 68) HermiteH(n, 1/2 I 2 ))} "A218274" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A218321" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A218473" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 2 3 (3 n1 + 2) binomial(3 n1, n1)| {27 , 27 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218474" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-6 n1 - 6) | n n | \ 3 2 (7 n1 + 5) binomial(3 n1, n1)| {64 , 64 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218475" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 4 5 (4 n1 + 3) binomial(3 n1, n1)| {125 , 125 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218476" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 5 6 (9 n1 + 7) binomial(3 n1, n1)| {216 , 216 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218477" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 6 7 (5 n1 + 4) binomial(3 n1, n1)| {343 , 343 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218478" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-9 n1 - 9) | n n | \ 7 2 (11 n1 + 9) binomial(3 n1, n1)| {512 , 512 | ) -----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218479" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-6 n1 - 6) | n n | \ 8 3 (6 n1 + 5) binomial(3 n1, n1)| {729 , 729 | ) ----------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218480" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 9 10 (13 n1 + 11) binomial(3 n1, n1)| {1000 , 1000 | ) -------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A218540" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n (-3 3 ) GAMMA(n - 1/3) (3 3 ) GAMMA(n - 1/3) {-------------------------, ------------------------} GAMMA(n + 1) GAMMA(n + 1) "A218693" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A218768" LREtools/SearchTable: "SearchTable successful" n n {(-1/2) binomial(2 n, n) n! BesselI(n + 1/2, 1), (-1/2) binomial(2 n, n) n! BesselK(n + 1/2, -1)} "A219024" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { (-n) 2 2 { ((n/2)!) (n/2 + 2) n::even { 4 binomial(n, n/2) ((n/2)!) (n + 1) (n + 3) n::even {{ , { } { 2 { (-2 n - 2) 2 2 { 1/4 ((n/2 - 1/2)!) (n + 1) (n + 3) n::odd { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (2 n + 8) n::odd "A219197" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A219312" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(3 - 5 ) , (3 + 5 ) , (3 - 5 ) | ) (3 + 5 ) (3 - 5 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) || | \ (3 + 5 ) ((5 n2 + 3) hypergeom([-1/2, -n2 - 1], [1], -4) + (-5 n2 - 5) hypergeom([-1/2, -n2], [1], -4))|| | ) ------------------------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A219314" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1 - 5 ) , (5 + 1) , (1 - 5 ) | ) (5 + 1) (1 - 5 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) || | \ (-1) (5 + 1) ((n2 + 1) hypergeom([1/2, -n2 - 1], [1], 4) + (3 n2 + 9) hypergeom([1/2, -n2], [1], 4))|| | ) ------------------------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A219534" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A219535" LREtools/SearchTable: "SearchTable successful" 3 2 2 (2 n + 5) (n + 2) (11 n + 6) hypergeom([-n - 1, -2 n - 4], [2], 3) + (-649 n - 2301 n - 2546 n - 840) hypergeom([-n, -2 n - 2], [2], 3) {-------------------------------------------------------------------------------------------------------------------------------------------} (11 n + 17) (2 n + 1) n "A219536" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([2 n + 2, -n + 1], [2], 3)} "A219538" LREtools/SearchTable: "SearchTable successful" {((5 n + 2) (3 n + 5) (2 n + 3) (n + 2) hypergeom([-n - 1, 3 n + 6], [n + 4], -1) 2 - (51 n + 105 n + 50) (2 n + 1) (n + 3) hypergeom([-n, 3 n + 3], [n + 3], -1)) binomial(2 n, n)/(n (n + 1) (n + 2) (n + 3) (22 n + 29))} "A219562" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n (-1) hypergeom([-n, -n, -n, -n], [1, 1, 1], 1) {-----------------------------------------------} binomial(3 n, n) binomial(4 n, n) (4 n + 1) "A219670" memory used=122088.0MB, alloc=2743.5MB, time=880.83 LREtools/ReduceToOrderTwo: "Checking Symmetric Cube... (can be time consuming...)" memory used=122731.5MB, alloc=2746.1MB, time=886.87 memory used=123130.6MB, alloc=2747.3MB, time=892.74 memory used=123474.8MB, alloc=2775.5MB, time=899.17 memory used=123845.8MB, alloc=2924.9MB, time=905.89 memory used=124138.3MB, alloc=2988.9MB, time=913.20 memory used=124650.9MB, alloc=3148.9MB, time=920.60 memory used=125311.0MB, alloc=3180.9MB, time=928.65 memory used=126052.0MB, alloc=3330.2MB, time=941.17 memory used=126611.6MB, alloc=3330.2MB, time=966.92 memory used=127169.3MB, alloc=3330.2MB, time=988.14 memory used=127728.4MB, alloc=3330.2MB, time=1013.69 memory used=128287.3MB, alloc=3330.2MB, time=1039.52 memory used=128844.9MB, alloc=3330.2MB, time=1060.57 memory used=129406.2MB, alloc=3330.2MB, time=1086.02 LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+5)*(7*n^4+56*n^3+166*n^2+216*n+105)*(n+4)^2*(n+3)^2*E^4-(2*n+7)*(2*n+3)*(n+4)*(n+3)*(70*n^6+ 1050*n^5+6406*n^4+20337*n^3+35449*n^2+32244*n+12048)*E^3-3*(2*n+5)*(n+3)*(490*n^8+9800*n^7+84910*n^6+416150*n^5+1261159*n^4+2417840*n^3+2860095*n^2+ 1905600*n+546588)*E^2+27*(2*n+7)*(2*n+3)*(70*n^6+1050*n^5+6406*n^4+20283*n^3+35044*n^2+31221*n+11178)*(n+2)^2*E+729*(2*n+7)*(7*n^4+84*n^3+376*n^2+744 *n+550)*(n+2)^2*(n+1)^3 "to two: Symmetric cube" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" n 2 {-(-1) hypergeom([1/2, -n], [1], 4) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4))} "A219671" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n (-1) {-----} n + 1 "A219672" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n / 1/2 \n | 5 | 1/2 | 5 | 1/2 |5 | 1/2 {|1/2 - ----| LegendreP(n, -2 - 5 ), |1/2 - ----| LegendreQ(n, -2 - 5 ), |---- + 1/2| LegendreP(n, -2 + 5 ), \ 2 / \ 2 / \ 2 / / 1/2 \n |5 | 1/2 |---- + 1/2| LegendreQ(n, -2 + 5 )} \ 2 / "A219673" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n / 1/2\n | 5 | 1/2 | 5 | 1/2 | 5 | 1/2 {|- 1/2 - ----| LegendreP(n, 2 - 5 ), |- 1/2 - ----| LegendreQ(n, 2 - 5 ), |- 1/2 + ----| LegendreP(n, 2 + 5 ), \ 2 / \ 2 / \ 2 / / 1/2\n | 5 | 1/2 |- 1/2 + ----| LegendreQ(n, 2 + 5 )} \ 2 / "A219692" LREtools/SearchTable: "SearchTable not successful" {} "A219779" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A220092" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 n n n {(-I) , I , (2 n + 1) (1/2) n! binomial(2 n, n)} "A220097" LREtools/SearchTable: "SearchTable successful" {(2 (4 n + 7) (4 n + 5) hypergeom([-n - 3, -2 n - 5], [-2 n - 7/2], 1/4) - (5 n + 7) (2 n + 1) hypergeom([-n - 2, -2 n - 3], [-2 n - 3/2], 1/4)) binomial(4 n, 2 n) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (2 n + 1) (2 n + 3) (7 n + 11))} "A220119" LREtools/SearchTable: "SearchTable not successful" {} "A220433" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 3 2 | 1/2 1/2 1/2 {|3/2 - ------| ((3 n + 4) hypergeom([5/6, - 2/3 - n], [5/3], 4 + 2 2 ) + 3 (1 + 2 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], 4 + 2 2 )) \ 2 / 1/2 GAMMA(n - 1/3) GAMMA(n + 1/3) (-3 + 2 2 )/GAMMA(n + 2/3)} "A220449" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-I) GAMMA(n + 2 - I), I GAMMA(n + 2 + I)} "A220452" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {1, (n + 1) (-2) n! LaguerreL(n + 1, -n - 1/2, 1/2)} "A220589" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 3 2 n (-1) (3 (2 n + 11 n + 18) (n + 1) hypergeom([1/2, -n], [1], 4) + (2 n + 23 n + 63 n + 54) hypergeom([1/2, -n - 1], [1], 4)) {(-1) , -------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A220699" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 I 2 HermiteH(n, 1/2 I 2 )) {-------------------------------------------------------------------------------} (n - 1) (n - 2) "A220700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (2 (n + 3) HermiteH(n + 1, 1/2 I 2 ) I - (n + 1) (n + 4) HermiteH(n, 1/2 I 2 ))} "A220878" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(3/2) } "A220898" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A220899" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A220903" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {(n + 1) n!, (n + 1) n! (n + 5 n + 2), (n + 1) n! | ) ---------------------------------------------------|} | / (n1 + 5) (n1 + 4) (n1 + 1) (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A220910" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 3 binomial(2 n1, n1) | {(-4) (8 n - 1), (-4) (8 n - 1) | ) -------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4) (8 n1 + 7) (8 n1 - 1)| \n1 = 0 / "A221058" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n - 2) { ---------------- n::even n n { binomial(n, n/2) { 2 {(-2) , 2 (2 n - 1), { , { 1/2 n binomial(n, n/2) n::even} { (2 n + 2) 2 { { 2 n { binomial(n - 1, n/2 - 1/2) (n - 1) n n::odd { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A221145" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / 1/2\n1 1/2 \ |n - 1 | 2 | 2 | |----- |- ----| HermiteH(n1 + 1, ----)| | \ \ 2 / 2 | {(n + 1) n!, (n + 1) n! | ) ---------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A221701" 2 n (2 n + 1) n binomial(2 n, n) (n + 9 n + 10) {4 , --------------------------------------------} (n + 4) (n + 3) (n + 2) "A221703" 4 3 2 n (2 n + 1) binomial(2 n, n) (3 n + 31 n + 98 n + 136 n + 72) {4 , --------------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A221704" 2 2 n (2 n + 1) binomial(2 n, n) (n + 7 n + 8) (5 n + 17 n + 18) {4 , ------------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A221957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) binomial(2 n1, n1)| {n! | ) -----------------------------|, n!} | / (n1 + 1) (n1 + 1)! | |----- | \n1 = 0 / "A222050" LREtools/SearchTable: "SearchTable successful" binomial(3 n, n) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4) {------------------------------------------------------------} 2 n + 1 "A222051" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) binomial(3 n, n) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)} "A222052" LREtools/SearchTable: "SearchTable successful" memory used=130787.7MB, alloc=3223.5MB, time=1097.71 {binomial(3 n, n) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)} "A222080" LREtools/SearchTable: "SearchTable successful" (-n) (-n) {2 KummerM(n + 1/2, 1, 1) binomial(2 n, n) n!, 2 LaguerreL(n - 1/2, -1) binomial(2 n, n) n!} "A222283" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222353" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A222364" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 7 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 3 } "A222395" memory used=132172.6MB, alloc=3223.5MB, time=1105.86 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222467" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 1/2 {(-2 ) 2 (-n BesselI(n, 2 2 ) + 2 BesselI(n - 1, 2 2 )), (-2 ) 2 (-n BesselK(n, -2 2 ) + 2 BesselK(n - 1, -2 2 ))} "A222468" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 2 1/2 1/2 1/2 {(-2 ) 2 (-(n + n + 2) BesselI(n, 2 2 ) + 2 (n + 1) BesselI(n - 1, 2 2 )), 1/2 n 1/2 2 1/2 1/2 1/2 (-2 ) 2 (-(n + n + 2) BesselK(n, -2 2 ) + 2 (n + 1) BesselK(n - 1, -2 2 ))} "A222469" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 1/2 {(-2 ) 2 (-n BesselJ(n, -2 2 ) - 2 BesselJ(n - 1, -2 2 )), (-2 ) 2 (-n BesselY(n, -2 2 ) - 2 BesselY(n - 1, -2 2 ))} "A222470" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {(-2 ) 2 (-(n + 2) (n - 1) BesselJ(n, -2 2 ) - 2 (n + 1) BesselJ(n - 1, -2 2 )), 1/2 n 1/2 1/2 1/2 1/2 (-2 ) 2 (-(n + 2) (n - 1) BesselY(n, -2 2 ) - 2 (n + 1) BesselY(n - 1, -2 2 ))} "A222472" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 2 1/2 1/2 1/2 {(-3 ) 3 (-(n + n + 3) BesselI(n, 2 3 ) + 3 (n + 1) BesselI(n - 1, 2 3 )), 1/2 n 1/2 2 1/2 1/2 1/2 (-3 ) 3 (-(n + n + 3) BesselK(n, -2 3 ) + 3 (n + 1) BesselK(n - 1, -2 3 ))} "A222559" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {{ (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A222627" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ / (n1 + 1) n1! \| (-1) n! | ) |- ------------------|| | / \ (n1 - 1) (n1 + 1)!/| n |----- | (-1) n! \n1 = 0 / {---------, ----------------------------------------} (n - 1) n (n - 1) n "A222636" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / (n2 + 1) n2! \| || | | (n1 + 1) | ) |- ------------------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 - 1) (n2 + 1)!/| || |----- | |----- | |----- | || n | \ / (n1 + 1) n1! \| n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------|| (-1) n! | ) |- --------------------------------------------|| | / \ (n1 - 1) (n1 + 1)!/| | / \ (n1 - 1) (n1 + 1)! /| n |----- | |----- | (-1) n! \n1 = 0 / \n1 = 0 / {---------, ----------------------------------------, ------------------------------------------------------------------} (n - 1) n (n - 1) n (n - 1) n "A222684" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222741" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222748" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / (n2 + 1) n2! \| || | | (n1 + 1) | ) |- ------------------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 - 1) (n2 + 1)!/| || |----- | |----- | |----- | || n | \ / (n1 + 1) n1! \| n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------|| (-1) n! | ) |- --------------------------------------------|| | / \ (n1 - 1) (n1 + 1)!/| | / \ (n1 - 1) (n1 + 1)! /| n |----- | |----- | (-1) n! \n1 = 0 / \n1 = 0 / {---------, ----------------------------------------, ------------------------------------------------------------------, (n - 1) n (n - 1) n (n - 1) n / / / / /n2 - 1 \ \\ \\ | | | | |----- | || || | | | | | \ / (n3 + 1) n3! \| || || | | | | (n2 + 1) | ) |- ------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 - 1) (n3 + 1)!/| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | (n1 + 1) | ) |- --------------------------------------------|| n1!|| |n - 1 | | / \ (n2 - 1) (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- ----------------------------------------------------------------------|| | / \ (n1 - 1) (n1 + 1)! /| |----- | \n1 = 0 / --------------------------------------------------------------------------------------------} (n - 1) n "A222763" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222788" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222848" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {3 } "A222884" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A222980" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 9 } "A223006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A223023" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / (n2 + 1) n2! \| || | | (n1 + 1) | ) |- ------------------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 - 1) (n2 + 1)!/| || |----- | |----- | |----- | || n | \ / (n1 + 1) n1! \| n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------|| (-1) n! | ) |- --------------------------------------------|| | / \ (n1 - 1) (n1 + 1)!/| | / \ (n1 - 1) (n1 + 1)! /| n |----- | |----- | (-1) n! \n1 = 0 / \n1 = 0 / {---------, ----------------------------------------, ------------------------------------------------------------------, (n - 1) n (n - 1) n (n - 1) n / / / / /n2 - 1 \ \\ \\ | | | | |----- | || || | | | | | \ / (n3 + 1) n3! \| || || | | | | (n2 + 1) | ) |- ------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 - 1) (n3 + 1)!/| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | (n1 + 1) | ) |- --------------------------------------------|| n1!|| |n - 1 | | / \ (n2 - 1) (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- ----------------------------------------------------------------------|| | / \ (n1 - 1) (n1 + 1)! /| |----- | \n1 = 0 / --------------------------------------------------------------------------------------------, (n - 1) n / / / / / / /n3 - 1 \ \\ \\ \\ | | | | | | |----- | || || || | | | | | | | \ / (n4 + 1) n4! \| || || || | | | | | | (n3 + 1) | ) |- ------------------|| n3!|| || || | | | | |n2 - 1 | | / \ (n4 - 1) (n4 + 1)!/| || || || | | | | |----- | |----- | || || || | | | | | \ | \n4 = 0 / || || || | | | | (n2 + 1) | ) |- --------------------------------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 - 1) (n3 + 1)! /| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | (n1 + 1) | ) |- ----------------------------------------------------------------------|| n1!|| |n - 1 | | / \ (n2 - 1) (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------------------------------------------------------------------------------------|| | / \ (n1 - 1) (n1 + 1)! /| |----- | \n1 = 0 / ----------------------------------------------------------------------------------------------------------------------} (n - 1) n "A223098" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A223120" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {9 } "A223626" memory used=133527.2MB, alloc=3223.5MB, time=1113.90 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 2 \ | { binomial(n1, ----) (2 n1 + 5 n1 + 17) (n1 + 1) (n1 + 3) | | { 2 | | { -------------------------------------------------------- n1::even| | { n1 + 2 | | { | | { n1 2 | | { 5 binomial(n1 - 1, ---- - 1/2) n1 (n1 + 2) (n1 + 3 n1 + 8) | |n - 1 { 2 | |----- { ----------------------------------------------------------- n1::odd | n 2 | \ { n1 + 1 | {1, 2 (n - 1), (3 n + 15 n + 38) | ) -----------------------------------------------------------------------------|, | / 2 2 | |----- (3 (n1 + 1) + 15 n1 + 53) (3 n1 + 15 n1 + 38) | \n1 = 0 / / { n1 2 \ | { 4 (n1 + 2) (n1 + 3 n1 + 8) | | { 5/4 ----------------------------- n1::even| | { n1 | | { (n1 + 1) binomial(n1, ----) | | { 2 | | { | | { (2 n1 + 2) 2 | | { 2 (n1 + 3) (2 n1 + 5 n1 + 17) | | { 1/4 ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { (n1 + 2) binomial(n1 + 1, ---- + 1/2) | 2 | \ { 2 | 2 (3 n + 15 n + 38) | ) --------------------------------------------------------------|, 3 n + 15 n + 38} | / 2 2 | |----- (3 (n1 + 1) + 15 n1 + 53) (3 n1 + 15 n1 + 38) | \n1 = 0 / "A223899" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||} | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A223901" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||, | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / / / / / /n2 - 1 \ \\ \\ | | | | |----- | || || | | | | | \ / n3! \| || || | | | | | ) |- ---------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)!/| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- --------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- -------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A223902" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||, | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / / / / / /n2 - 1 \ \\ \\ | | | | |----- | || || | | | | | \ / n3! \| || || | | | | | ) |- ---------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)!/| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- --------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- -------------------------------------------||, | / \ (n1 + 1)! /| |----- | \n1 = 0 / / / / / / / /n3 - 1 \ \\ \\ \\ | | | | | | |----- | || || || | | | | | | | \ / n4! \| || || || | | | | | | | ) |- ---------|| n3!|| || || | | | | |n2 - 1 | | / \ (n4 + 1)!/| || || || | | | | |----- | |----- | || || || | | | | | \ | \n4 = 0 / || || || | | | | | ) |- --------------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)! /| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- -------------------------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A223904" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \ \\ | | |----- | || | | | \ / n2! \| || | | | ) |- ---------|| n1!|| /n - 1 \ |n - 1 | | / \ (n2 + 1)!/| || |----- | |----- | |----- | || n n | \ / n1! \| n | \ | \n2 = 0 / || {(-1) n!, (-1) n! | ) |- ---------||, (-1) n! | ) |- --------------------------||, | / \ (n1 + 1)!/| | / \ (n1 + 1)! /| |----- | |----- | \n1 = 0 / \n1 = 0 / / / / / /n2 - 1 \ \\ \\ | | | | |----- | || || | | | | | \ / n3! \| || || | | | | | ) |- ---------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)!/| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- --------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- -------------------------------------------||, | / \ (n1 + 1)! /| |----- | \n1 = 0 / / / / / / / /n3 - 1 \ \\ \\ \\ | | | | | | |----- | || || || | | | | | | | \ / n4! \| || || || | | | | | | | ) |- ---------|| n3!|| || || | | | | |n2 - 1 | | / \ (n4 + 1)!/| || || || | | | | |----- | |----- | || || || | | | | | \ | \n4 = 0 / || || || | | | | | ) |- --------------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)! /| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- -------------------------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- ------------------------------------------------------------||, | / \ (n1 + 1)! /| |----- | \n1 = 0 / / / / / / / / / /n4 - 1 \ \\ \\ \\ \\ | | | | | | | | |----- | || || || || | | | | | | | | | \ / n5! \| || || || || | | | | | | | | | ) |- ---------|| n4!|| || || || | | | | | | |n3 - 1 | | / \ (n5 + 1)!/| || || || || | | | | | | |----- | |----- | || || || || | | | | | | | \ | \n5 = 0 / || || || || | | | | | | | ) |- --------------------------|| n3!|| || || | | | | |n2 - 1 | | / \ (n4 + 1)! /| || || || | | | | |----- | |----- | || || || | | | | | \ | \n4 = 0 / || || || | | | | | ) |- -------------------------------------------|| n2!|| || | | |n1 - 1 | | / \ (n3 + 1)! /| || || | | |----- | |----- | || || | | | \ | \n3 = 0 / || || | | | ) |- ------------------------------------------------------------|| n1!|| |n - 1 | | / \ (n2 + 1)! /| || |----- | |----- | || n | \ | \n2 = 0 / || (-1) n! | ) |- -----------------------------------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A224071" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n 1/2 n {(3/2 - 1/2 I 3 ) , (3/2 + 1/2 I 3 ) , (3/2 - 1/2 I 3 ) /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || | \ 1/2 n1 1/2 (-n1 - 1) | \ (3/2 + 1/2 I 3 ) (3 LegendreP(n2 + 1, 3) - LegendreP(n2, 3))|| | ) (3/2 + 1/2 I 3 ) (3/2 - 1/2 I 3 ) | ) -----------------------------------------------------------------------||, | / | / n2 + 2 || |----- |----- || \n1 = 0 \n2 = 0 // 1/2 n (3/2 - 1/2 I 3 ) /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || | \ 1/2 n1 1/2 (-n1 - 1) | \ (3/2 + 1/2 I 3 ) (3 LegendreQ(n2 + 1, 3) - LegendreQ(n2, 3))|| | ) (3/2 + 1/2 I 3 ) (3/2 - 1/2 I 3 ) | ) -----------------------------------------------------------------------||} | / | / n2 + 2 || |----- |----- || \n1 = 0 \n2 = 0 // "A224292" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (2 n + 29 n + 41 n + 72) LegendreP(n + 1, 3) + (-18 n - 123 n - 267 n - 216) LegendreP(n, 3) {------------------------------------------------------------------------------------------------, (n + 3) (n + 2) n (n - 1) 3 2 3 2 (2 n + 29 n + 41 n + 72) LegendreQ(n + 1, 3) + (-18 n - 123 n - 267 n - 216) LegendreQ(n, 3) ------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n (n - 1) "A224500" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {1, (n + 1) (-4) n! LaguerreL(n + 1, -n - 1/2, 1/4)} "A224529" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A224747" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 2 _Z - 1, index = 1) , RootOf(_Z - _Z - 2 _Z - 1, index = 2) , RootOf(_Z - _Z - 2 _Z - 1, index = 3) } "A224869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A224884" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 3 n 3 n { { binomial(3 n, ---) binomial(---, n/2) { (2 n - 2) 3 n { 2 2 {{ 2 binomial(--- - 3/2, n/2 - 1/2) , { ------------------------------------- n::even} { 2 { binomial(n, n/2) (3 n - 1) { ----------------------------------------- n::odd { { n { 0 n::odd "A225006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (3 n1 + 1) binomial(3 n1, n1) (5 n1 + 3)| {(1/2) , (1/2) | ) --------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A225034" LREtools/SearchTable: "SearchTable successful" n (-1) ((4 n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-4 n + 1) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------} n "A225050" {binomial(2 n, n), binomial(3 n, n)} "A225096" n 3 n 3 (n + 1) (1/2) (n!) binomial(3 n, n) {(1/6) (n!) binomial(2 n, n) binomial(3 n, n), -------------------------------------} 2 n + 1 "A225439" n n 9 GAMMA(n + 1/3) 9 GAMMA(n + 2/3) {-----------------, -----------------} GAMMA(n + 1) GAMMA(n + 1) "A225612" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1) {1, ) ---------------------------------------------------} / (n1 + 1) (3 n1 + 1) (3 n1 + 2) ----- n1 = 0 "A225615" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /3125\n1 ----- |----| GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) \ \256 / {1, ) ------------------------------------------------------------------------} / GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 5/4) ----- n1 = 0 "A225692" (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {------------------------------------, n + 2 n + 2} (n + 3) (n + 2) (n + 1) "A225887" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- (-n1 - 1) | |----- (-n1 - 1) | n n | \ 6 (3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3))| n | \ 6 (3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3))| {6 , 6 | ) ------------------------------------------------------|, 6 | ) ------------------------------------------------------|} | / n1 + 2 | | / n1 + 2 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A225988" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A225989" memory used=134818.0MB, alloc=3225.8MB, time=1122.72 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A226012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 (-n1) 3 2 \| n n | \ |2 3125 128 (333 n1 + 404 n1 + 147 n1 + 16) GAMMA(n1 + 1/5) GAMMA(n1 + 2/5) GAMMA(n1 + 3/5) GAMMA(n1 + 4/5)|| {(1/2) , (1/2) | ) |-------------------------------------------------------------------------------------------------------------------||} | / \ GAMMA(n1 + 2) GAMMA(n1 + 3/2) GAMMA(n1 + 3/4) GAMMA(n1 + 5/4) /| |----- | \n1 = 0 / "A226013" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(4 n1, n1) (77 n1 + 64 n1 + 11)| {(-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (3 n1 + 1) (n1 + 1) (-1/2) | \n1 = 0 / "A226022" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A226170" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ n1! | (-1) n! {n! | ) ---------|, ---------, n!} | / (n1 + 1)!| (n - 1) n |----- | \n1 = 0 / "A226282" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 4 2 2 { 2 2 { 4 (n + 1) binomial(n, n/2) ((n/2)!) n::even { 1/8 ((n/2)!) (n + 2) n n::even {{ , { } { (-2 n - 2) 2 2 2 { 2 4 { 2 2 n (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) n::odd { 1/16 ((n/2 - 1/2)!) (n + 1) n::odd "A226283" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 4 2 2 { 4 (n - 1) (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { (-2 n - 2) 2 2 2 2 { 2 2 n (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n - 2) n::odd { 2 2 2 { 1/32 ((n/2)!) (n - 2) (n + 2) n n::even { } { 2 2 4 { 1/64 ((n/2 - 1/2)!) (n - 1) (n + 1) n::odd "A226284" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 4 2 2 { 4 (n - 3) (n - 1) (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { (-2 n - 2) 2 2 2 2 2 { 2 2 n (n - 2) (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n - 4) n::odd { 2 2 2 2 { 1/128 ((n/2)!) n (n - 2) (n + 2) (n - 4) n::even { } { 2 2 2 4 { 1/256 ((n/2 - 1/2)!) (n - 3) (n - 1) (n + 1) n::odd "A226285" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 2 4 2 2 { 4 (n - 5) (n - 3) (n - 1) (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { (-2 n - 2) 2 2 2 2 2 2 { 2 2 n (n - 4) (n - 2) (n + 2) binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n - 6) n::odd { 2 2 2 2 2 { 1/512 ((n/2)!) n (n - 4) (n - 2) (n + 2) (n - 6) n::even { } { 2 2 2 2 4 { 1/1024 ((n/2 - 1/2)!) (n - 5) (n - 3) (n - 1) (n + 1) n::odd "A226302" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { n2 \\\ | | | { 2 4 ||| | | | { - --------------------------- n2::even||| | | | { n2 ||| | | | { (n2 + 1) binomial(n2, ----) ||| | | | { 2 ||| | | | { ||| | | | { (2 n2 + 2) ||| | | | { 2 n2 ||| | | | { ---------------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { (n2 + 1) (n2 + 2) binomial(n2 + 1, ---- + 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(2 ) , (-2 ) , (2 ) | ) |1/2 2 (-1) | ) ----------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { 2 binomial(n2, ----) n2 ||| | | | { 2 ||| | | | { ----------------------- n2::even||| | | | { n2 + 2 ||| | | | { ||| | | | { n2 ||| | | | { 4 binomial(n2 - 1, ---- - 1/2) n2 ||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { - --------------------------------- n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { n2 + 1 ||| (2 ) | ) |1/2 2 (-1) | ) -----------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-2 ) ||| \n1 = 0 \ \n2 = 0 /// "A226312" LREtools/SearchTable: "SearchTable successful" {n hypergeom([1/2, -n, -n], [1, 1], 4)} "A226316" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / /3 n1\ \ n - 1 | |----| | ----- | \ 2 / 1/2 1/2 1/2 | \ | 2 (-2 LegendreP(n1 + 1, 2 ) + LegendreP(n1, 2 ))| {1, ) |- -------------------------------------------------------------|, / \ n1 + 2 / ----- n1 = 0 / /3 n1\ \ n - 1 | |----| | ----- | \ 2 / 1/2 1/2 1/2 | \ | 2 (-2 LegendreQ(n1 + 1, 2 ) + LegendreQ(n1, 2 ))| ) |- -------------------------------------------------------------|} / \ n1 + 2 / ----- n1 = 0 "A226434" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 n \ binomial(2 n1, n1) (n1 - 1) (19 n1 + 40 n1 + 24) {1, 4 , 3 n + 1, ) -------------------------------------------------} / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A226466" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 3/2 I 2 ), n!} "A226705" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- /-64\n /-64\n | \ /46656\n1 {|---| , |---| | ) |-----| GAMMA(n1 + 1/3) GAMMA(n1 + 2/3) GAMMA(n1 + 1/2) GAMMA(n1 + 7/6) GAMMA(n1 + 5/6) \243/ \243/ | / \3125 / |----- \n1 = 0 4 3 2 / / (45068 n1 + 103202 n1 + 85633 n1 + 30412 n1 + 3885) / |GAMMA(n1 + 2) GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) / \ \ | /-64\(n1 + 1)\| |---| ||} \243/ /| | / "A226730" {(n + 1) n!, (2 n + 1) n! binomial(2 n, n)} "A226733" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | /-16\n /-16\n | \ (4 n1 + 1) binomial(4 n1, n1) (102 n1 + 122 n1 + 35)| {|---| , |---| | ) -----------------------------------------------------|} \27 / \27 / | / /-16\(n1 + 1) | |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A226751" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) binomial(3 n1, n1) (55 n1 + 34)| {(-8/9) , (-8/9) | ) ------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (-8/9) | \n1 = 0 / "A226761" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | /-81\n /-81\n | \ (4 n1 + 1) binomial(4 n1, n1) (2653 n1 + 3107 n1 + 870)| {|---| , |---| | ) --------------------------------------------------------|} \64 / \64 / | / /-81\(n1 + 1) | |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \64 / / "A226881" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even n { { binomial(n, n/2) n::even {1, 2 , { (2 n - 2) , { } { 2 { 0 n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A226910" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A226974" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A226994" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, LegendreP(n, 3), LegendreQ(n, 3)} "A226995" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | | \ (n1 + 1) (-LegendreP(n1, 3) + LegendreP(n1 + 1, 3))| | \ / (n1 + 1) (LegendreQ(n1, 3) - LegendreQ(n1 + 1, 3))\| {1, (2 n + 1) | ) ---------------------------------------------------|, (2 n + 1) | ) |- --------------------------------------------------||, | / (2 n1 + 3) (2 n1 + 1) | | / \ (2 n1 + 3) (2 n1 + 1) /| |----- | |----- | \n1 = 0 / \n1 = 0 / 2 n + 1} "A226996" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | | \ n1 LegendreP(n1 + 1, 3) + (-n1 - 2) LegendreP(n1, 3)| | \ (-n1 - 2) LegendreQ(n1, 3) + n1 LegendreQ(n1 + 1, 3)| {1, (2 n + 1) | ) ----------------------------------------------------|, (2 n + 1) | ) ----------------------------------------------------|, | / (2 n1 + 3) (2 n1 + 1) | | / (2 n1 + 3) (2 n1 + 1) | |----- | |----- | \n1 = 0 / \n1 = 0 / 2 n + 1} "A227035" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A227037" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227081" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {(2/3) , (2/3) | ) (1/2 3 ((4 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -2) + (-4 n1 - 3) hypergeom([-1/2, -n1], [1], -2)))|} | / | |----- | \n1 = 0 / "A227094" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227096" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227357" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227546" 2 {n + 1, n!} "A227580" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227583" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227584" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 |----- 4 3 2 | |----- n n | \ (2 n1 + 1) binomial(2 n1, n1) (243 n1 + 2430 n1 + 8919 n1 + 14152 n1 + 8136)| n | \ {(-1/2) , (-1/2) | ) -------------------------------------------------------------------------------|, (-1/2) | ) (2 n1 + 1) | / (n1 + 1) | | / |----- (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) | |----- \n1 = 0 / \n1 = 0 /n1 - 1 |----- 4 3 2 | \ binomial(2 n1, n1) (243 n1 + 2430 n1 + 8919 n1 + 14152 n1 + 8136) | ) (4 n2 + 9) (4 n2 + 5) (4 n2 + 1) (2 n2 + 5) (2 n2 + 1) (4 n2 + 11) | / |----- \n2 = 0 2 (4 n2 + 7) (4 n2 + 3) binomial(2 n2, n2) binomial(4 n2, 2 n2) 6 5 4 3 2 / 2 2 2 3 (21546 n2 + 342315 n2 + 2222316 n2 + 7507383 n2 + 13814788 n2 + 12978040 n2 + 4768512) / ((n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) / 4 3 2 (3 n2 + 10) (3 n2 + 7) (3 n2 + 11) (3 n2 + 8) binomial(2 n2 + 2, n2 + 1) (243 (n2 + 1) + 2430 (n2 + 1) + 8919 (n2 + 1) + 14152 n2 + 22288) \ \ | | 4 3 2 | / (n1 + 1) | (243 n2 + 2430 n2 + 8919 n2 + 14152 n2 + 8136))| / ((n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1/2) )|} | / | | | / / "A227845" LREtools/SolveLRE: "Reduced the order of" (n+4)^2*E^4+(-6*n^2-42*n-74)*E^3+(6*n^2+30*n+38)*E-(n+2)^2 "to two: Half integer product u(n/2) * u(n/2+1/2)" (2*n+3)*(n+1)^2*E^2-(n+2)*(12*n^2+18*n+7)*E+(n+2)*(n+1)*(2*n+1) RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A227918" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ (-1) | {(n + 1) (n + 2) n!, (n + 1) (n + 2) n! | ) ---------------------------|, (n + 1) (n + 2) n! | ) ---------------------------|} | / (n1 + 2) (n1 + 3) (n1 + 1)!| | / (n1 + 2) (n1 + 3) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A227937" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A227995" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 1) (3 n1 + 2) /| |----- | \n1 = 0 / "A227996" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /3125\n1 \ |----- |----| GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) | n n | \ \256 / | {(-1) , (-1) | ) --------------------------------------------------------------------------|} | / (n1 + 1)| |----- GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 5/4) (-1) | \n1 = 0 / "A228002" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2\| n n | \ | (-1) (2 n1 + 1) binomial(2 n1, n1) || {(-1) , (-1) | ) |- --------------------------------------||} | / | 2 || |----- \ (n1 + 1) /| \n1 = 0 / "A228178" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 \ (2 n1 + 1) binomial(2 n1, n1) (17 n1 + 47 n1 + 36) {1, ) ---------------------------------------------------} / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A228180" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) (5 n1 - 2) {1, ) -----------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A228197" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n n | \ (-n1 - 1) | {2 , 4 , 2 | ) 2 binomial(2 n1, n1)|} | / | |----- | \n1 = 0 / "A228229" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A228230" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n 2 n 2 | \ 2 | {(n + 1) (1/2) (n!) , (n + 1) (1/2) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A228231" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n { 4 (n + 2) ((n/2 + 5/2) hypergeom([- n/2 - 3/2], [n/2 + 5/2], 1) + (- n/2 - 3/2) hypergeom([- n/2 - 1/2], [n/2 + 3/2], 1)) {{ - -------------------------------------------------------------------------------------------------------------------------- , n::even { 2 { (n + 1) binomial(n, n/2) (2 n - 2) - 2 (n + 1) 3 2 2 / 2 ((1/8 n + 5/4 n + 11/2 n + 6) hypergeom([- n/2 - 1], [n/2 + 2], 1) - 1/8 (n + 6) (n + 2) hypergeom([- n/2], [n/2 + 1], 1)) / (n (n + 2) / binomial(n - 1, n/2 - 1/2)) , n::odd, { { - 8 3 2 2 ((1/8 n + 5/4 n + 11/2 n + 6) hypergeom([- n/2 - 1], [n/2 + 2], 1) - 1/8 (n + 6) (n + 2) hypergeom([- n/2], [n/2 + 1], 1)) binomial(n, n/2) (n + 1)/((n + 2) n) , n::even - 8 ((n/2 + 5/2) hypergeom([- n/2 - 3/2], [n/2 + 5/2], 1) + (- n/2 - 3/2) hypergeom([- n/2 - 1/2], [n/2 + 3/2], 1)) binomial(n + 1, n/2 + 1/2) } (n + 2)/(n + 1) , n::odd "A228248" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A228338" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((4 n + 20 n + 23) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 18 n - 17) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A228341" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((n + 2 n + 2) (n + 1) BesselI(n, 2) + (-n - 3 n - 3) BesselI(n - 1, 2)), n 2 2 (-1) ((n + 2 n + 2) (n + 1) BesselK(n, -2) + (-n - 3 n - 3) BesselK(n - 1, -2))} "A228343" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 1) n1 binomial(2 n1, n1)| {2 , 2 | ) -------------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) | |----- | \n1 = 0 / "A228511" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228513" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ 2 | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A228514" LREtools/SearchTable: "SearchTable successful" {hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) - 3 hypergeom([1/2, -n, -n], [1, 1], 4)} "A228694" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228713" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A228714" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228769" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 3 2 | n | \ binomial(2 n1, n1) (2 n1 - 15 n1 + 47 n1 + 104 n1 - 18)| {1, 4 , (6 n + 7) | ) ----------------------------------------------------------|, 6 n + 7} | / (n1 + 3) (n1 + 2) (n1 + 1) (6 n1 + 13) (6 n1 + 7) | |----- | \n1 = 0 / "A228770" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 3 2 | n | \ binomial(2 n1, n1) (8 n1 - 38 n1 - 9 n1 + 87 n1 + 48)| {1, 4 , (6 n - 5) | ) --------------------------------------------------------|, 6 n - 5} | / (n1 + 3) (n1 + 2) (n1 + 1) (6 n1 + 1) (6 n1 - 5) | |----- | \n1 = 0 / "A228771" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=136419.6MB, alloc=3223.5MB, time=1132.08 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 3 2 n \ binomial(2 n1, n1) (n1 - 17 n1 - 46 n1 - 58) n1 {1, 4 , ) -------------------------------------------------} / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A228907" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228923" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A228959" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A228960" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228966" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228987" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A228994" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ 42 n1 + 48 n1 + 97 | {(n + 1) n!, n! (7 n - 6), n! (7 n - 6) | ) -------------------------------|} | / (n1 + 1)! (7 n1 + 1) (7 n1 - 6)| |----- | \n1 = 0 / "A228995" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | | \ 4 n1 - 6 n1 - 10 n1 - 33 | {(n + 1) n!, n! (2 n - 9), n! (2 n - 9) | ) -------------------------------|} | / (n1 + 1)! (2 n1 - 7) (2 n1 - 9)| |----- | \n1 = 0 / "A228996" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4 3 2 | | \ 285 n1 + 465 n1 + 1110 n1 + 975 n1 + 2029| {(n + 1) n!, n! (19 n - 45), n! (19 n - 45) | ) --------------------------------------------|} | / (n1 + 1)! (19 n1 - 26) (19 n1 - 45) | |----- | \n1 = 0 / "A228997" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 5 4 3 2 | | \ 57 n1 + 69 n1 + 86 n1 - 240 n1 - 314 n1 - 819| {(n + 1) n!, n! (19 n - 72), n! (19 n - 72) | ) -------------------------------------------------|} | / (n1 + 1)! (19 n1 - 53) (19 n1 - 72) | |----- | \n1 = 0 / "A228998" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 6 5 4 3 2 | | \ 40796 n1 + 95424 n1 + 211946 n1 + 201264 n1 + 424676 n1 + 386064 n1 + 867791| {(n + 1) n!, n! (1457 n - 5334), n! (1457 n - 5334) | ) ---------------------------------------------------------------------------------|} | / (n1 + 1)! (1457 n1 - 3877) (1457 n1 - 5334) | |----- | \n1 = 0 / "A228999" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 1) n!, n! (2434 n - 10417), /n - 1 \ |----- 7 6 5 4 3 2 | | \ 29208 n1 + 79452 n1 + 170028 n1 + 106470 n1 + 41152 n1 - 392812 n1 - 447278 n1 - 1092335| n! (2434 n - 10417) | ) ----------------------------------------------------------------------------------------------|} | / (n1 + 1)! (2434 n1 - 7983) (2434 n1 - 10417) | |----- | \n1 = 0 / "A229000" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" {(n + 1) n!, n! (41671 n - 190500), n! (41671 n - 190500) /n - 1 \ |----- 8 7 6 5 4 3 2 | | \ 1875195 n1 + 6429060 n1 + 16250430 n1 + 19996200 n1 + 26232150 n1 + 19942020 n1 + 52987725 n1 + 51168720 n1 + 118829881| | ) ------------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 1)! (41671 n1 - 148829) (41671 n1 - 190500) | |----- | \n1 = 0 / "A229003" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A229042" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A229043" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A229049" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A229053" memory used=137838.7MB, alloc=3225.3MB, time=1140.68 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A229068" memory used=139196.6MB, alloc=3233.8MB, time=1149.08 memory used=140624.2MB, alloc=3273.8MB, time=1157.83 memory used=141951.5MB, alloc=3306.5MB, time=1166.26 memory used=143209.8MB, alloc=3231.6MB, time=1173.87 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A229111" LREtools/SearchTable: "SearchTable not successful" {} "A229224" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A229225" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A229244" n 2 (-1) n! n! (2 n + 4 n + 1) {---------------, -------------------} (n + 1) (n + 2) (n + 1) (n + 2) "A229245" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) ((n + 3) HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I)} "A229246" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A229414" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A229465" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n {1, (n + 1) n!, (-1) n! BesselI(n + 1/2, 1), (-1) n! BesselK(n + 1/2, -1)} "A229482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(3/2) , (4/3) } "A229554" {1, n!} "A229733" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A229734" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A229828" {1, n!} "A230071" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ (-1) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A230122" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A230137" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even n { n binomial(n, n/2) n::even { binomial(n, n/2) {2 n, { , { } { binomial(n + 1, n/2 + 1/2) (n/2 + 1/2) n::odd { (2 n - 2) { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A230341" 2 n! binomial(2 n, n) (n + n + 1) {1, --------------------------------, binomial(2 n, n)} n + 2 "A230557" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n n {(-I) , I , { (n/2) / /3 n \ \ { (-1) |6 (n/2 + 1) hypergeom([1/2, - n/2 - 1], [1], 4) + 6 |--- + 2| hypergeom([1/2, - n/2], [1], 4)| n::even { \ \ 2 / / { { (n/2 + 1/2) / /5 n \ \ , { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { ------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) / /5 n \ \ { 2 (-1) |(n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + |--- + 7/2| hypergeom([1/2, - n/2 - 1/2], [1], 4)| { \ \ 2 / / { ------------------------------------------------------------------------------------------------------------------- n::even { n + 1 } { { (n/2 - 1/2) / /3 n \ \ { (-1) |6 (n/2 + 1) hypergeom([1/2, - n/2 - 1], [1], 4) + 6 |--- + 2| hypergeom([1/2, - n/2], [1], 4)| n::odd { \ \ 2 / / "A230741" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n ((n + 1) BesselJ(n, -2) + 2 BesselJ(n - 1, -2)), (-1) n ((n + 1) BesselY(n, -2) + 2 BesselY(n - 1, -2))} "A231373" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231482" LREtools/SearchTable: "SearchTable successful" {(n + 1) (hypergeom([-1/2, -n - 1], [1], -8) - hypergeom([-1/2, -n], [1], -8))} "A231530" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" {GAMMA(n + 1 - I), GAMMA(n + 1 + I)} "A231531" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" {GAMMA(n + 1 - I), GAMMA(n + 1 + I)} "A231552" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231553" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231554" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231556" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231615" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231616" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231617" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231618" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A231622" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n - 1/2, 1) + BesselI(n + 1/2, 1)), (-1) (BesselK(n - 1/2, -1) + BesselK(n + 1/2, -1))} "A231690" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A232205" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 ) n!, (-2 ) n!} "A232500" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 3) { ------------------------ n::even { 4 binomial(n, n/2) (n - 2) { (n + 1) binomial(n, n/2) { -------------------------- n::even {{ , { n + 2 } { (2 n - 2) { { 4 2 (n - 2) { binomial(n + 1, n/2 + 1/2) (n - 3) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A232605" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A232617" n n {(n + 1) 2 n!, (2 n + 1) (1/2) n! binomial(2 n, n)} "A232665" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A232845" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1! | 2 {(n - 3 n + 1) | ) ---------------------------------------|, n - 3 n + 1} | / 2 2 | |----- ((n1 + 1) - 3 n1 - 2) (n1 - 3 n1 + 1)| \n1 = 0 / "A232969" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A232980" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { |---| { \ 3 / { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) irem(n, 3) = 0 { { /2 n \ { |--- + 4/3| { \ 3 / {{ 3 GAMMA(5/3 + n/3) GAMMA(n/3 + 1) , { -------------------------------------------- irem(n, 3) = 1 { n + 2 { { /2 n \ { |--- + 2/3| { \ 3 / { 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { ---------------------------------------------- irem(n, 3) = 2 { n + 1 { (- n/3) 2 2 n { 3 ((n/3)!) binomial(n, n/3) binomial(---, n/3) irem(n, 3) = 0 { 3 { { (1/3 - n/3) 2 2 n { 3 n ((n/3 - 1/3)!) binomial(n - 1, n/3 - 1/3) binomial(--- - 2/3, n/3 - 1/3) irem(n, 3) = 1, { 3 { { (- n/3 - 1/3) 2 2 n { 3 ((n/3 + 1/3)!) binomial(n + 1, n/3 + 1/3) binomial(--- + 2/3, n/3 + 1/3) irem(n, 3) = 2 { 3 { /2 n\ { |---| { \ 3 / { 3 GAMMA(5/3 + n/3) GAMMA(n/3 + 1) { -------------------------------------- irem(n, 3) = 0 { n + 2 { { /2 n \ { |--- - 2/3| } { \ 3 / { 3 GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) { ---------------------------------------------- irem(n, 3) = 1 { n + 1 { { /2 n \ { |--- - 4/3| { \ 3 / { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) irem(n, 3) = 2 "A232981" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { /4 n\ { |---| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 1/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) irem(n, 5) = 0 { { /4 n \ { |--- + 16/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 6/5) GAMMA(n/5 + 8/5) GAMMA(n/5 + 9/5) { ------------------------------------------------------------------------------- irem(n, 5) = 1 { (n + 1) (n + 3) (n + 4) { { /4 n \ { |--- + 12/5| { \ 5 / {{ 5 GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 7/5) GAMMA(n/5 + 8/5) , { ------------------------------------------------------------------------------- irem(n, 5) = 2 { (n + 2) (n + 3) { { /4 n \ { |--- + 8/5| { \ 5 / { 5 GAMMA(n/5 + 3/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 7/5) { -------------------------------------------------------------------------------- irem(n, 5) = 3 { (n + 1) (n + 2) { { /4 n \ { |--- + 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 2/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) { ------------------------------------------------------------------------------ irem(n, 5) = 4 { n + 1 { /4 n\ { |---| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 2/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) { ------------------------------------------------------------------------ irem(n, 5) = 0 { n + 1 { { /4 n \ { |--- - 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 1/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) irem(n, 5) = 1 { { /4 n \ { |--- + 12/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 6/5) GAMMA(n/5 + 8/5) GAMMA(n/5 + 9/5) , { ------------------------------------------------------------------------------- irem(n, 5) = 2 { (n + 1) (n + 3) (n + 4) { { /4 n \ { |--- + 8/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 7/5) GAMMA(n/5 + 8/5) { ------------------------------------------------------------------------------ irem(n, 5) = 3 { (n + 2) (n + 3) { { /4 n \ { |--- + 4/5| { \ 5 / { 5 GAMMA(n/5 + 3/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 7/5) { -------------------------------------------------------------------------------- irem(n, 5) = 4 { (n + 1) (n + 2) { /4 n\ { |---| { \ 5 / { 5 GAMMA(n/5 + 3/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 7/5) { -------------------------------------------------------------------------- irem(n, 5) = 0 { (n + 1) (n + 2) { { /4 n \ { |--- - 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 2/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) { ------------------------------------------------------------------------------ irem(n, 5) = 1 { n + 1 { { /4 n \ { |--- - 8/5| , { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 1/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) irem(n, 5) = 2 { { /4 n \ { |--- + 8/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 6/5) GAMMA(n/5 + 8/5) GAMMA(n/5 + 9/5) { ------------------------------------------------------------------------------ irem(n, 5) = 3 { (n + 1) (n + 3) (n + 4) { { /4 n \ { |--- + 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 7/5) GAMMA(n/5 + 8/5) { ------------------------------------------------------------------------------ irem(n, 5) = 4 { (n + 2) (n + 3) { /4 n\ { |---| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 7/5) GAMMA(n/5 + 8/5) { ------------------------------------------------------------------------ irem(n, 5) = 0 { (n + 2) (n + 3) { { /4 n \ { |--- - 4/5| { \ 5 / { 5 GAMMA(n/5 + 3/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 7/5) { -------------------------------------------------------------------------------- irem(n, 5) = 1 { (n + 1) (n + 2) { { /4 n \ { |--- - 8/5| , { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 2/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) { ------------------------------------------------------------------------------ irem(n, 5) = 2 { n + 1 { { /4 n \ { |--- - 12/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 1/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) irem(n, 5) = 3 { { /4 n \ { |--- + 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 6/5) GAMMA(n/5 + 8/5) GAMMA(n/5 + 9/5) { ------------------------------------------------------------------------------ irem(n, 5) = 4 { (n + 1) (n + 3) (n + 4) { /4 n\ { |---| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 6/5) GAMMA(n/5 + 8/5) GAMMA(n/5 + 9/5) { ------------------------------------------------------------------------ irem(n, 5) = 0 { (n + 1) (n + 3) (n + 4) { { /4 n \ { |--- - 4/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 7/5) GAMMA(n/5 + 8/5) { ------------------------------------------------------------------------------ irem(n, 5) = 1 { (n + 2) (n + 3) { { /4 n \ { |--- - 8/5| } { \ 5 / { 5 GAMMA(n/5 + 3/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 6/5) GAMMA(n/5 + 7/5) { -------------------------------------------------------------------------------- irem(n, 5) = 2 { (n + 1) (n + 2) { { /4 n \ { |--- - 12/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 2/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 6/5) { ------------------------------------------------------------------------------- irem(n, 5) = 3 { n + 1 { { /4 n \ { |--- - 16/5| { \ 5 / { 5 GAMMA(n/5 + 1) GAMMA(n/5 + 1/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) irem(n, 5) = 4 "A233130" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 binomial(2 n, n) {-------------------} n + 1 "A233347" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A233389" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / |n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 5 5 | | 5 5 | | 5 5 | | \ {|11/2 - ------| , |11/2 + ------| , |11/2 - ------| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / / / 1/2\(-n2 - 1) \\\ | |n1 - 1 | 5 5 | ||| | |----- |11/2 + ------| (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (19 n2 + 27)||| | n1 1/2 (-n1 - 1) 1/2 n1 | \ \ 2 / ||| |-2 (-1) (-11 + 5 5 ) (11 + 5 5 ) | ) ------------------------------------------------------------------------------|||} | | / (n2 + 3) (n2 + 2) (2 n2 + 3) (n2 + 1) (2 n2 + 1) ||| | |----- ||| \ \n2 = 0 /// "A233396" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 binomial(2 n, n) {-------------------} n + 1 "A233449" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (-n1 - 1) | {2 , 2 | ) 2 (n1 + 1) n1!|} | / | |----- | \n1 = 0 / "A233481" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1) | | \ 2 (n1 + 1) binomial(2 n1, n1) n1!| {n! | ) --------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A233744" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=144619.2MB, alloc=3223.5MB, time=1182.61 LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 1) (n2 + 2) n2!)|| /n - 1 \ |n - 1 | / || |----- n1 | |----- |----- || | \ (-1) | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) -----------------------------------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A233895" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A234040" n (2 n + 1) binomial(2 n, n) (2 n + 1) (-1) binomial(2 n, n) {--------------------------, --------------------------------} (n + 1) (n + 2) (n + 1) (n + 2) "A234269" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A234270" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1 - 2 ) , (1 + 2 ) , (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) || | \ (-1) (1 + 2 ) ((2 n2 + 3) hypergeom([1/2, -n2 - 1], [1], 4) + 3 hypergeom([1/2, -n2], [1], 4))|| | ) -----------------------------------------------------------------------------------------------------------||} | / (n2 + 2) (n2 + 3) || |----- || \n2 = 0 // "A234276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n RootOf(%1, index = 7) } 7 5 4 2 %1 := _Z - 4 _Z - 2 _Z + 16 _Z + 16 _Z + 4 "A234290" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (3/2) n1! binomial(2 n1, n1)| {(-2) n!, (-2) n! | ) ------------------------------|} | / (n1 + 1) | |----- (-2) (n1 + 1)! | \n1 = 0 / "A234292" LREtools/SearchTable: "SearchTable successful" n / 27 27 \ {(-6) |(33 n + 22) hypergeom([7/6, - 1/3 - n], [4/3], --) + (48 n + 24) hypergeom([7/6, 2/3 - n], [4/3], --)| GAMMA(n + 2/3)} \ 11 11 / "A234627" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A234839" LREtools/SearchTable: "SearchTable successful" {hypergeom([-2 n, -n], [1], -1)} "A235136" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n {{ , { 2 GAMMA(n/2 + 3/4) n::even} { (n - 1) { { 2 GAMMA(n/2 + 3/4) n::odd { 0 n::odd "A235320" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 n { {(-1) hypergeom([1/2, -n], [1], 4), { n , { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { 1/9 ------------------------------------ irem(n, 3) = 2 { 2 { GAMMA(n/3 + 1) { 0 irem(n, 3) = 0 { { 2 n { (n - 1) { binomial(---, n/3) binomial(n, n/3) irem(n, 3) = 0 { 3 GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { 3 { ------------------------------------------ irem(n, 3) = 1, { } { 2 { 0 irem(n, 3) = 1 { GAMMA(n/3 + 1) { { { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A235347" LREtools/SearchTable: "SearchTable successful" {hypergeom([2 n + 2, -n + 1], [2], -2)} "A235348" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A235349" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A235350" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A235351" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A235352" LREtools/SearchTable: "SearchTable successful" n {(-1) hypergeom([2 n + 2, -n + 1], [2], -2)} "A235360" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A235378" n {1, (n + 1) (-1) n!} "A235391" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A236339" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A236407" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (n + 1) LegendreP(n + 1, 3) + (-7 n - 3) LegendreP(n, 3) (7 n + 3) LegendreQ(n, 3) + (-n - 1) LegendreQ(n + 1, 3) {n, --------------------------------------------------------, - --------------------------------------------------------} n n "A236408" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A236438" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A237580" 2 {1, (n!) binomial(2 n, n), n!} "A237622" (n - 1) n binomial(2 n, n) {1, --------------------------} n + 1 "A237664" {1, binomial(2 n, n) (n - 1)} "A237734" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ | |{ || | |{ / n1 \ || | (-2 n1 - 2) |{ |---- - 1/2| || | 2 |{ \ 2 / n1 || | |{ 4 3 binomial(n1 - 1, ---- - 1/2) (n1 + 2) (n1 + 4) n1 || |n - 1 |{ 2 || |----- |{ ----------------------------------------------------------------- n1::odd || n n | \ \{ (n1 + 1) (n1 + 3) /| {4 (n + 4), 4 (n + 4) | ) --------------------------------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 5) | |----- | \n1 = 0 / / /{ / n1 \ \\ | |{ |----| || | |{ \ 2 / || | (-2 n1 - 2) |{ 48 (n1 + 2) (n1 + 4) || | 2 |{ 1/4 ------------------------------------ n1::even|| | |{ n1 || | |{ binomial(n1, ----) (n1 + 1) (n1 + 3) || |n - 1 |{ 2 || |----- |{ || n | \ \{ 0 n1::odd /| 4 (n + 4) | ) -------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 5) | |----- | \n1 = 0 / (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) binomial(2 n, n) --------------------------------------------------------} (n + 3) (n + 2) (n + 1) "A237740" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 1/2| || {4 , 4 | ) 2 |{ \ 2 / n1 ||, | / |{ 4 3 binomial(n1 - 1, ---- - 1/2) n1 (n1 + 2) || |----- |{ 2 || |n1 = 0 |{ -------------------------------------------------------- n1::odd || \ \{ (n1 + 1) (n1 + 3) // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / || { 0 n::even n | \ (-2 n1 - 2) |{ 48 (n1 + 2) || { 4 | ) 2 |{ 1/2 ------------------------------------ n1::even||, { 4 binomial(n - 1, n/2 - 1/2) n (n + 2) , | / |{ n1 || { -------------------------------------- n::odd |----- |{ binomial(n1, ----) (n1 + 1) (n1 + 3) || { (n + 1) (n + 3) |n1 = 0 |{ 2 || | |{ || \ \{ 0 n1::odd // { n { 4 (n + 2) { 1/2 -------------------------------- n::even} { (n + 1) (n + 3) binomial(n, n/2) { { 0 n::odd "A237884" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { ------------------------ n::even { 4 binomial(n, n/2) n { (n + 1) binomial(n, n/2) { -------------------- n::even {{ , { n + 2 } { (2 n - 2) { { 4 2 { binomial(n + 1, n/2 + 1/2) (n - 1) n::odd { ---------------------------------- n::odd { (n + 2) binomial(n - 1, n/2 - 1/2) "A237987" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(-5/2 I 2 ) HermiteH(n, 1/5), (5/2 I 2 ) HermiteH(n, 1/5)} "A238021" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=146017.8MB, alloc=3223.5MB, time=1190.96 memory used=147198.5MB, alloc=3223.5MB, time=1197.83 memory used=148394.3MB, alloc=3223.5MB, time=1204.51 /n - 1 |----- /{ n n | \ (-2 n1 - 2) |{ {4 , 4 | ) 2 |{ n1 n1 n1 | / |{ -24 2 hypergeom([1/2, - ---- - 3/2, - ---- - 3/2], [1, 1], 1) , n1::even |----- \{ 2 2 \n1 = 0 (n1 + 1) / n1 n1 n1 n1 \ 4 2 |(2 n1 + 8) hypergeom([1/2, - ---- - 2, - ---- - 2], [1, 1], 1) + (-2 n1 - 7) hypergeom([1/2, - ---- - 1, - ---- - 1], [1, 1], 1)| \ 2 2 2 2 / ---------------------------------------------------------------------------------------------------------------------------------------------- , n1 + 4 \ /n - 1 \| |----- /{ || n | \ (-2 n1 - 2) |{ ||, 4 | ) 2 |{ n1::odd|| | / |{ /| |----- \{ / \n1 = 0 n1 / n1 n1 n1 n1 \ 4 2 |(2 n1 + 8) hypergeom([1/2, - ---- - 2, - ---- - 2], [1, 1], 1) + (-2 n1 - 7) hypergeom([1/2, - ---- - 1, - ---- - 1], [1, 1], 1)| \ 2 2 2 2 / ---------------------------------------------------------------------------------------------------------------------------------------- , n1 + 4 n1::even \ \| || (n1 - 1) n1 n1 ||} -24 2 hypergeom([1/2, - ---- - 3/2, - ---- - 3/2], [1, 1], 1) , n1::odd|| 2 2 /| / "A238027" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-2 n1 - 2) |{ |---- - 1/2| || {4 , 4 | ) 2 |{ \ 2 / n1 ||, | / |{ 4 3 binomial(n1 - 1, ---- - 1/2) n1 (n1 + 2) || |----- |{ 2 || |n1 = 0 |{ -------------------------------------------------------- n1::odd || \ \{ (n1 + 1) (n1 + 3) // / /{ / n1 \ \\ |n - 1 |{ |----| || |----- |{ \ 2 / || { 0 n::even n | \ (-2 n1 - 2) |{ 48 (n1 + 2) || { 4 | ) 2 |{ 1/2 ------------------------------------ n1::even||, { 4 binomial(n - 1, n/2 - 1/2) n (n + 2) , | / |{ n1 || { -------------------------------------- n::odd |----- |{ binomial(n1, ----) (n1 + 1) (n1 + 3) || { (n + 1) (n + 3) |n1 = 0 |{ 2 || | |{ || \ \{ 0 n1::odd // { n { 4 (n + 2) { 1/2 -------------------------------- n::even} { (n + 1) (n + 3) binomial(n, n/2) { { 0 n::odd "A238032" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=149606.8MB, alloc=3223.5MB, time=1211.40 memory used=150815.5MB, alloc=3223.5MB, time=1218.05 memory used=152017.6MB, alloc=3223.5MB, time=1224.98 /n - 1 |----- /{ n n | \ (-2 n1 - 2) |{ {4 , 4 | ) 2 |{ n1 n1 n1 | / |{ -24 2 hypergeom([1/2, - ---- - 3/2, - ---- - 3/2], [1, 1], 1) , n1::even |----- \{ 2 2 \n1 = 0 (n1 + 1) / n1 n1 n1 n1 \ 4 2 |(2 n1 + 8) hypergeom([1/2, - ---- - 2, - ---- - 2], [1, 1], 1) + (-2 n1 - 7) hypergeom([1/2, - ---- - 1, - ---- - 1], [1, 1], 1)| \ 2 2 2 2 / ---------------------------------------------------------------------------------------------------------------------------------------------- , n1 + 4 \ /n - 1 \| |----- /{ || n | \ (-2 n1 - 2) |{ ||, 4 | ) 2 |{ n1::odd|| | / |{ /| |----- \{ / \n1 = 0 n1 / n1 n1 n1 n1 \ 4 2 |(2 n1 + 8) hypergeom([1/2, - ---- - 2, - ---- - 2], [1, 1], 1) + (-2 n1 - 7) hypergeom([1/2, - ---- - 1, - ---- - 1], [1, 1], 1)| \ 2 2 2 2 / ---------------------------------------------------------------------------------------------------------------------------------------- , n1 + 4 n1::even \ \| || (n1 - 1) n1 n1 ||} -24 2 hypergeom([1/2, - ---- - 3/2, - ---- - 3/2], [1, 1], 1) , n1::odd|| 2 2 /| / "A238111" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {---------------------------------------, ---------------------------------------} n n "A238112" LREtools/SearchTable: "SearchTable successful" (5 n + 7) LegendreP(n + 1, 3) + (-31 n - 21) LegendreP(n, 3) (5 n + 7) LegendreQ(n + 1, 3) + (-31 n - 21) LegendreQ(n, 3) {------------------------------------------------------------, ------------------------------------------------------------} n (n - 1) n (n - 1) "A238113" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) - 3 LegendreP(n, 3) LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) {---------------------------------------, ---------------------------------------} n n "A238299" LREtools/SearchTable: "SearchTable successful" (3 n + 1) LegendreP(n + 1, 5) + (-3 n - 5) LegendreP(n, 5) (3 n + 1) LegendreQ(n + 1, 5) + (-3 n - 5) LegendreQ(n, 5) {----------------------------------------------------------, ----------------------------------------------------------} n (n + 2) n (n + 2) "A238300" LREtools/SearchTable: "SearchTable successful" 2 2 (27 n + 32 n + 1) LegendreP(n + 1, 5) + (-3 n - 4 n - 5) LegendreP(n, 5) {--------------------------------------------------------------------------, (n + 4) (n + 3) n 2 2 (27 n + 32 n + 1) LegendreQ(n + 1, 5) + (-3 n - 4 n - 5) LegendreQ(n, 5) --------------------------------------------------------------------------} (n + 4) (n + 3) n "A238438" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A238452" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 binomial(n, n/2) n { 4 4 { -------------------- n::even { -------------------------------- n::even { n + 2 { (n + 1) (n + 3) binomial(n, n/2) {{ , { } { 16 binomial(n - 1, n/2 - 1/2) n { (2 n + 2) { ------------------------------- n::odd { 2 n { (n + 1) (n + 3) { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A238578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ -I (2 I) ((n1 - 1) LegendreP(n1, I) - (n1 + 1) LegendreP(n1 + 1, I) I)| {(-1) , (-1) | ) ------------------------------------------------------------------------|, | / n1 | |----- | \n1 = 0 / /n - 1 \ |----- n1 | n | \ -I (2 I) ((n1 - 1) LegendreQ(n1, I) - (n1 + 1) LegendreQ(n1 + 1, I) I)| (-1) | ) ------------------------------------------------------------------------|} | / n1 | |----- | \n1 = 0 / "A238755" LREtools/SearchTable: "SearchTable successful" 2 3 2 (n + 1) (3 (2 n + 1) (4 n - 2 n - 3) LegendreP(n + 1, 3) + (-4 n + 4 n + 12 n + 27) LegendreP(n, 3)) {-------------------------------------------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) 2 3 2 (n + 1) (3 (2 n + 1) (4 n - 2 n - 3) LegendreQ(n + 1, 3) + (-4 n + 4 n + 12 n + 27) LegendreQ(n, 3)) -------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) "A238803" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-1/2 I 2 ) (HermiteH(n + 1, 2 I) - 2 (n + 2) HermiteH(n, 2 I) I) {-----------------------------------------------------------------------------} n "A238879" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 binomial(n, n/2) (n + 1) { 4 (n + 2) { --------------------------- n::even { 1/2 -------------------------------- n::even { (n + 2) (n + 4) { (n + 3) (n + 5) binomial(n, n/2) {{ , { } { 4 binomial(n + 1, n/2 + 1/2) (n + 1) (n + 2) { (2 n - 2) { -------------------------------------------- n::odd { 2 2 (n + 1) { (n + 5) (n + 3) { -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A238977" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 2 1/2 (-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 (n - 3 n + 4) HermiteH(n, 1/2 I 2 ) I) {--------------------------------------------------------------------------------------------} (n - 1) (n - 2) "A238978" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 3 2 1/2 (-1/2 I 2 ) (4 HermiteH(n + 1, 1/2 I 2 ) + 2 (n - 5 n + 4 n - 4) HermiteH(n, 1/2 I 2 ) I) {-----------------------------------------------------------------------------------------------------} n (n - 2) (n - 3) "A238979" LREtools/SearchTable: "SearchTable successful" 1/2 n 3 2 1/2 1/2 4 3 2 1/2 {(-1/2 I 2 ) ((5 n - 41 n + 66 n + 12) HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 2) (n - 12 n + 49 n - 60 n - 6) HermiteH(n, 1/2 I 2 ) I)/(n (n - 1) (n - 2) (n - 3) (n - 4))} "A239106" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A239107" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A239108" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A239112" LREtools/SearchTable: "SearchTable not successful" {} "A239116" LREtools/SearchTable: "SearchTable successful" 1/2 n 4 3 2 1/2 {(-1/2 I 2 ) ((16 n - 228 n + 1208 n - 2356 n + 912) HermiteH(n + 1, 1/2 I 2 ) 1/2 6 5 4 3 2 1/2 + 2 (n - 15 n + 71 n - 17 n - 1068 n + 2836 n - 912) HermiteH(n, 1/2 I 2 ) I)/(n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5))} "A239117" LREtools/SearchTable: "SearchTable successful" 1/2 n 5 4 3 2 1/2 {(-1/2 I 2 ) ((35 n - 686 n + 5207 n - 20584 n + 40608 n - 24880) HermiteH(n + 1, 1/2 I 2 ) 1/2 7 6 5 4 3 2 1/2 + 2 (n - 21 n + 175 n - 707 n + 1236 n + 5244 n - 30208 n + 24880) HermiteH(n, 1/2 I 2 ) I)/(n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6))} "A239118" LREtools/SearchTable: "SearchTable successful" 1/2 n 6 5 4 3 2 1/2 {(-1/2 I 2 ) ((64 n - 1576 n + 14744 n - 63192 n + 145384 n - 267120 n + 276720) HermiteH(n + 1, 1/2 I 2 ) 1/2 8 7 6 5 4 3 2 1/2 + 2 (n - 28 n + 374 n - 3052 n + 16361 n - 72080 n + 201176 n - 76080 n - 276720) HermiteH(n, 1/2 I 2 ) I)/(n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7))} "A239119" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 7 6 5 4 3 2 1/2 {(-1/2 I 2 ) (-2 (105 n - 3087 n + 34593 n - 162193 n - 55302 n + 3142444 n - 8062696 n + 3813936) HermiteH(n + 1, 1/2 I 2 ) I 9 8 7 6 5 4 3 2 1/2 1/2 + (2 n - 72 n + 1428 n - 17220 n + 108270 n - 238076 n + 329580 n - 5763832 n + 18376592 n - 7627872) HermiteH(n, 1/2 I 2 )) 2 I/(n (n - 8) (n - 7) (n - 6) (n - 5) (n - 4) (n - 3) (n - 2) (n - 1))} "A239201" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, (2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)} "A239204" LREtools/SearchTable: "SearchTable successful" (7 n + 8) LegendreP(n + 1, 3) - LegendreP(n, 3) n (7 n + 8) LegendreQ(n + 1, 3) - LegendreQ(n, 3) n {-------------------------------------------------, -------------------------------------------------} (n + 2) (n + 3) (n + 2) (n + 3) "A239226" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) 2 ((n + 3) (2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-2 n - 2 n + 2) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1))/( n + 1)} "A239295" LREtools/SearchTable: "SearchTable not successful" {} "A239299" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A239368" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A239425" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) n {--------------------------, (2 n + 1) (-1) binomial(2 n, n) (n + 1) (n + 2) (2 hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) - 3 hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4))/((n + 2) (5 n + 3))} "A239453" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 2 binomial(n, n/2) (n + 1) n { 1/2 ------------------------ n::even { - ---------------------------- n::even n n { (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {(-2) , 2 , { , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 (n + 1) { ------------------------------------ n::odd { -1/2 ------------------------------------------ n::odd { n + 3 { (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A239466" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A239488" LREtools/SearchTable: "SearchTable successful" 5 LegendreP(n + 1, 5) - LegendreP(n, 5) 5 LegendreQ(n + 1, 5) - LegendreQ(n, 5) {---------------------------------------, ---------------------------------------} n + 2 n + 2 "A239840" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) n! HermiteH(n, 1/2 I 2 )} "A240172" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A240558" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 { n { ------------------------ n::even { 16 2 binomial(n, n/2) { (n + 1) binomial(n, n/2) { ---------------------- n::even {{ , { n + 2 } { (3 n - 3) { { 8 2 { (n + 1) { ------------------------------------ n::odd { 2 2 binomial(n + 1, n/2 + 1/2) n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A240586" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 (2 n + 5) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) + (2 n - 10 n - 18) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) {------------------------------------------------------------------------------------------------------------------------------------------, 2 n + 1 n 2 binomial(2 n, n) n ---------------------} n + 1 "A240609" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) / ----- n1 = 0 2 2 4 3 2 ((n1 + 4 n1 + 7) (n1 - 5 n1 - 12) hypergeom([1/2, -n1 - 1], [1], 4) + (3 n1 + 39 n1 + 117 n1 + 69 n1 - 84) hypergeom([1/2, -n1], [1], 4))/( (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2))} "A240688" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ n1 | {(-1) , (-1) | ) (-(2 I) (-LegendreP(n1, I) + LegendreP(n1 + 1, I) I))|, | / | |----- | \n1 = 0 / /n - 1 \ |----- | n | \ n1 | (-1) | ) (-(2 I) (-LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I))|} | / | |----- | \n1 = 0 / "A240721" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (3 n1 + 4) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A240722" LREtools/SearchTable: "SearchTable successful" (2 n + 5) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) - 2 (5 n + 9) (n + 1) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) {-------------------------------------------------------------------------------------------------------------------------------------------} n + 2 "A240879" n {4 (n + 2), (2 n + 1) binomial(2 n, n)} "A240880" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 1/2 3 3 | 1/2 n 1/2 | 1/2 3 3 | (-4 3 ) 3 |2 3 LegendreP(n + 1, ----) - 3 LegendreP(n, ----)| (-4 3 ) 3 |-2 3 LegendreQ(n + 1, ----) + 3 LegendreQ(n, ----)| \ 2 2 / \ 2 2 / {----------------------------------------------------------------------, - -----------------------------------------------------------------------} n n "A240881" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A241023" LREtools/SearchTable: "SearchTable successful" LegendreP(n + 1, 3) (n + 1) + (-7 n - 3) LegendreP(n, 3) (-n - 1) LegendreQ(n + 1, 3) + (7 n + 3) LegendreQ(n, 3) {--------------------------------------------------------, - --------------------------------------------------------} n n "A241193" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (7 n1 + 9)|| {(-1) , (-1) | ) |- -----------------------------------------------------------------------------||} | / | 2 || |----- \ (2 n1 + 3) (n1 + 1) /| \n1 = 0 / "A241478" n {4 n, binomial(2 n, n)} "A241524" n {4 n, (2 n + 1) binomial(2 n, n)} "A241530" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { --------------------------- n::even { 2 { 2 { 4 binomial(n, n/2) n::even { n (n + 1) binomial(n, n/2) { {{ , { 2 } { (4 n - 4) { (n + 1) binomial(n + 1, n/2 + 1/2) { 4 2 { ----------------------------------- n::odd { ------------------------------ n::odd { n { 2 2 { n binomial(n - 1, n/2 - 1/2) "A241543" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { binomial(n, n/2) { ------------------ n::even { ---------------- n::even { n binomial(n, n/2) {{ n - 1 , { } { { (2 n + 2) { binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ------------------------------------------ n::odd { (n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) "A242091" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 n1!| {(n + 1) 2 n!, (n + 1) 2 n! | ) --------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A242172" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { 1/2 ------------------------ n::even { binomial(n, n/2) (2 n + 2) n::even { (n + 1) binomial(n, n/2) {{ , { } { (n + 2) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A242227" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n + 1/2, -1), (-1) BesselY(n + 1/2, -1)} "A242429" n (2 n + 1) binomial(2 n, n) {2 , --------------------------, n + 3} n + 1 "A242499" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242500" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242501" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242502" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242503" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242504" memory used=153423.4MB, alloc=3223.5MB, time=1234.24 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242505" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242506" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242507" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242508" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242511" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 { 2 2 { 2 4 (n - 1) (n - 2 n + 15) { binomial(n, n/2) n (n - 2 n + 6) (n + 8) { -------------------------------- n::even { 1/2 ------------------------------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n + 2) 2 2 { 2 2 { 2 n (n + 8) (n - 2 n + 6) { binomial(n - 1, n/2 - 1/2) n (n - 1) (n - 2 n + 15) { -------------------------------------------------- n::odd { ----------------------------------------------------- n::odd { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) "A242512" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 2 { n 2 { binomial(n, n/2) n (n - n + 30 n - 40) { 4 n (n - 1) (n + 2 n + 33) { ---------------------------------------- n::even { ---------------------------------------- n::even { (n + 2) (n + 4) { (n + 1) (n + 3) (n + 5) binomial(n, n/2) {{ , { } { 2 2 { (2 n + 2) 3 2 { 2 binomial(n - 1, n/2 - 1/2) n (n - 1) (n + 2 n + 33) { 2 n (n - n + 30 n - 40) { ------------------------------------------------------- n::odd { 1/2 -------------------------------------------------- n::odd { (n + 5) (n + 3) (n + 1) { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) "A242513" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 2 { 2 2 { 2 4 (n - 1) (n - 2 n + 15) { binomial(n, n/2) n (n - 2 n + 6) (n + 8) { -------------------------------- n::even { 1/2 ------------------------------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 4) {{ , { } { (2 n + 2) 2 2 { 2 2 { 2 n (n + 8) (n - 2 n + 6) { binomial(n - 1, n/2 - 1/2) n (n - 1) (n - 2 n + 15) { -------------------------------------------------- n::odd { ----------------------------------------------------- n::odd { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) "A242514" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { |---| { \ 3 / 3 2 { 2 (n + 1) (n + 10) (2 n + 3 n + 135 n - 136) GAMMA(n/3 + 11/6) { --------------------------------------------------------------------- irem(n, 3) = 0 { (2 n + 5) GAMMA(n/3 + 13/3) { { /2 n \ { |--- - 2/3| {{ \ 3 / 2 , { 2 n (2 n + 3) (n + 6 n + 81) GAMMA(n/3 + 3/2) { --------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 4) { { /2 n \ { |--- + 2/3| { \ 3 / 3 2 { 2 (n + 2) (n + 11) (2 n + 9 n + 147 n + 4) GAMMA(n/3 + 13/6) { ------------------------------------------------------------------------- irem(n, 3) = 2 { (2 n + 7) GAMMA(n/3 + 14/3) { /2 n\ { |---| { \ 3 / 3 2 { 2 (n + 2) (n + 11) (2 n + 9 n + 147 n + 4) GAMMA(n/3 + 13/6) { ------------------------------------------------------------------- irem(n, 3) = 0 { (2 n + 7) GAMMA(n/3 + 14/3) { { /2 n \ { |--- - 2/3| { \ 3 / 3 2 , { 2 (n + 1) (n + 10) (2 n + 3 n + 135 n - 136) GAMMA(n/3 + 11/6) { --------------------------------------------------------------------------- irem(n, 3) = 1 { (2 n + 5) GAMMA(n/3 + 13/3) { { /2 n \ { |--- - 4/3| { \ 3 / 2 { 2 n (2 n + 3) (n + 6 n + 81) GAMMA(n/3 + 3/2) { --------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 4) { 2 n 2 2 { binomial(---, n/3) n (2 n + 3) (n + 6 n + 81) { 3 { 1/9 ----------------------------------------------- irem(n, 3) = 0 { (n + 3) (n + 6) (n + 9) { { 2 n 3 2 { binomial(--- + 4/3, n/3 + 2/3) (n + 2) (2 n + 9 n + 147 n + 4) { 3 } { 1/9 ---------------------------------------------------------------- irem(n, 3) = 1 { (n + 5) (n + 8) { { 2 n 3 2 { binomial(--- + 2/3, n/3 + 1/3) (n + 1) (2 n + 3 n + 135 n - 136) { 3 { 1/9 ------------------------------------------------------------------ irem(n, 3) = 2 { (n + 7) (n + 4) "A242522" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242566" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A242569" {(n + 1) n!, n + 1} "A242586" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, (2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)} "A242651" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n + 1 - I), (-1) GAMMA(n + 1 + I)} "A242652" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n + 1 - I), (-1) GAMMA(n + 1 + I)} "A242781" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ (2 n2 + 1) binomial(2 n2, n2) (7 n2 + 9)| | {(-1/3 I 3 ) , (1/3 I 3 ) , (-1/3 I 3 ) | ) (-1) 3 | ) ----------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (n2 + 3) (n2 + 1) (1/3 I 3 ) | | \n1 = 0 \n2 = 0 / / "A242798" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / | 1/2 n 1/2 n 1/2 n | {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | | | \ n - 1 /n1 - 1 \\ ----- |----- 1/2 (-n2 - 1) 3 2 || \ 1/2 n1 1/2 (-n1 - 1) | \ (1/2 + 1/2 I 3 ) binomial(3 n2, n2) (91 n2 + 288 n2 + 233 n2 + 48)|| ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) -------------------------------------------------------------------------------|| / | / (n2 + 1) (2 n2 + 3) (2 n2 + 1) || ----- |----- || n1 = 0 \n2 = 0 // } "A242818" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(-1/2 I 2 ) HermiteH(n, I), (1/2 I 2 ) HermiteH(n, I)} "A242986" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 binomial(n, n/2) (n + 1) { 4 (n + 1) { -------------------------- n::even { 1/2 ------------------------ n::even { n + 6 { (n + 5) binomial(n, n/2) {{ , { } { 2 { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 2 (n + 1) { ------------------------------------- n::odd { ------------------------------------ n::odd { n + 5 { n (n + 6) binomial(n - 1, n/2 - 1/2) "A243007" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(323*n^2+1292*n+1211)*E^2+(2*n+3)*(323*n^2+1292*n+1211)*E-(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-18*n-27)*E+n+1 LREtools/SearchTable: "SearchTable successful" 2 2 {LegendreP(n, 9) , LegendreQ(n, 9) , LegendreP(n, 9) LegendreQ(n, 9)} "A243014" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ n1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A243019" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ /5 n1\ \\ | |{ |----| || | |{ \ 2 / || | |{ 2 || | |{ --------------------------- n1::even|| |n - 1 |{ n1 || |----- |{ (n1 + 1) binomial(n1, ----) || n n | \ (-n1 - 1) |{ 2 || {3 , 3 | ) 3 |{ ||, | / |{ /5 n1 \ || |----- |{ |---- - 5/2| || |n1 = 0 |{ \ 2 / || | |{ 4 2 || | |{ - ------------------------------- n1::odd || | |{ n1 || | |{ n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ / n1 \ \\ | |{ |----| || |n - 1 |{ \ 2 / n1 || |----- |{ -4 2 binomial(n1, ----) n1::even|| n | \ (-n1 - 1) |{ 2 || 3 | ) 3 |{ ||} | / |{ / n1 \ || |----- |{ |---- + 1/2| || |n1 = 0 |{ \ 2 / n1 || | |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd || \ \{ 2 // "A243034" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ {(1/4 - 1/4 I 7 ) , (1/4 + 1/4 I 7 ) , (1/4 - 1/4 I 7 ) | ) (1/4 + 1/4 I 7 ) (1/4 - 1/4 I 7 ) | ) | / | / |----- |----- \n1 = 0 \n2 = 0 1/2 (-n2 - 1) (1/4 + 1/4 I 7 ) ((2 n2 + 5) (n2 + 2) (13 n2 + 8) hypergeom([-n2 - 1, -n2 - 2, -n2 - 3/2], [1, 3/2], -1) \\ || 3 2 || 1/2 n + (122 n2 + 366 n2 + 334 n2 + 72) hypergeom([-n2, -n2 - 1, -n2 - 1/2], [1, 3/2], -1))/((n2 + 2) (2 n2 + 1))||, (1/4 - 1/4 I 7 ) || || // /n - 1 /n1 - 1 \\ |----- |----- / n2 1/2 (-n2 - 1) \|| | \ 1/2 n1 1/2 (-n1 - 1) | \ |4 8 (1 + 7 I) (11 n2 + 6) binomial(2 n2, n2)||| | ) (1/4 + 1/4 I 7 ) (1/4 - 1/4 I 7 ) | ) |----------------------------------------------------------|||} | / | / \ n2 + 1 /|| |----- |----- || \n1 = 0 \n2 = 0 // "A243101" LREtools/SearchTable: "SearchTable successful" {n LegendreP(n + 1, 3) + (-n - 2) LegendreP(n, 3), n LegendreQ(n + 1, 3) + (-n - 2) LegendreQ(n, 3)} "A243107" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A243116" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A243156" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A243157" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A243204" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([n, -n], [-n + 1/2], 1/4), binomial(2 n, n)} "A243255" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A243412" memory used=154816.1MB, alloc=3223.5MB, time=1243.71 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A243413" memory used=155988.8MB, alloc=3223.5MB, time=1252.25 memory used=157068.3MB, alloc=3223.5MB, time=1259.65 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A243474" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(-1) ( 4 3 2 3 2 (18 n + 241 n + 1124 n + 2167 n + 1458) hypergeom([1/2, -n - 1], [1], 4) - 3 (6 n + 71 n + 265 n + 314) (n + 1) hypergeom([1/2, -n], [1], 4) )/((n + 5) (n + 4) (n + 3) (n + 2))} "A243569" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 8) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 8) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, (n + 7) | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 1) n5! ||||| | | | | (n4 + 1) n4! | ) ------------------||||| | | | |n3 - 1 | / (n5 + 8) (n5 + 1)!||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------|, (n + 7) (n + 6) (n + 5) | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / / /n5 - 1 \\\\\\ | | | | | |----- |||||| | | | | | | \ (n6 + 1) n6! |||||| | | | | | (n5 + 1) n5! | ) ------------------|||||| | | | | |n4 - 1 | / (n6 + 8) (n6 + 1)!|||||| | | | | |----- |----- |||||| | | | | | \ \n6 = 0 /||||| | | | | (n4 + 1) n4! | ) ----------------------------------------||||| | | | |n3 - 1 | / (n5 + 8) (n5 + 1)! ||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) --------------------------------------------------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ------------------------------------------------------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) --------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / | | | | | | | | | | | | | | | | | | | |n - 1 |----- | \ (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) (n1 + 1) n1! | / |----- \n1 = 0 / / / / / /n6 - 1 \\\\\\ | | | | | |----- |||||| | | | | | | \ (n7 + 1) n7! |||||| | | | | | (n6 + 1) n6! | ) ------------------|||||| | | | | |n5 - 1 | / (n7 + 8) (n7 + 1)!|||||| | | | | |----- |----- |||||| | | | | | \ \n7 = 0 /||||| | | | | (n5 + 1) n5! | ) ----------------------------------------||||| | | | |n4 - 1 | / (n6 + 8) (n6 + 1)! ||||| | | | |----- |----- ||||| | | | | \ \n6 = 0 /|||| | | | (n4 + 1) n4! | ) --------------------------------------------------------------|||| | | |n3 - 1 | / (n5 + 8) (n5 + 1)! |||| | | |----- |----- |||| | | | \ \n5 = 0 /||| | | (n3 + 1) n3! | ) ------------------------------------------------------------------------------------||| | |n2 - 1 | / (n4 + 8) (n4 + 1)! ||| | |----- |----- ||| | | \ \n4 = 0 /|| | (n2 + 1) n2! | ) ----------------------------------------------------------------------------------------------------------|| |n1 - 1 | / (n3 + 8) (n3 + 1)! || |----- |----- || | \ \n3 = 0 /| | ) --------------------------------------------------------------------------------------------------------------------------------|/( | / (n2 + 8) (n2 + 1)! | |----- | \n2 = 0 / \ | | | | | | | | | | | | | | | | | | | | | | (n1 + 8) (n1 + 1)!)|} | | / "A243585" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([1/2, -n], [n + 1], -4)} "A243626" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ LegendreP(n1 + 1, 5) - 5 LegendreP(n1, 5)| n | \ LegendreQ(n1 + 1, 5) - 5 LegendreQ(n1, 5)| {(-2) , (-2) | ) -----------------------------------------|, (-2) | ) -----------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- n1 (-2) | |----- n1 (-2) | \n1 = 0 / \n1 = 0 / "A243632" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A243642" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A243644" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {binomial(2 n, n)} "A243659" LREtools/SearchTable: "SearchTable successful" 2 (394 n + 472 n + 134) hypergeom([-n - 1, 3 n + 4], [1], -1) - 3 (3 n + 1) (3 n + 2) hypergeom([-n, 3 n + 1], [1], -1) {----------------------------------------------------------------------------------------------------------------------} 2 (238 n + 289 n + 83) (3 n + 4) "A243760" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {----------------} n + 1 "A243764" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {1/2 ----------------} n - 1/2 "A243814" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 0 n::even { (n/2) { { 2 (-16) {{ (n/2 - 1/2) , { -------------------------- n::even} { 2 (-1) binomial(n - 1, n/2 - 1/2) { (n + 1) n binomial(n, n/2) { -------------------------------------------- n::odd { { n + 1 { 0 n::odd "A243816" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 0 n::even { (n/2) { { (-1) binomial(n, n/2) {{ (n/2 - 1/2) , { -------------------------- n::even} { 2 (-16) { n - 1 { ------------------------------------ n::odd { { binomial(n - 1, n/2 - 1/2) n (n - 1) { 0 n::odd "A243863" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A243945" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(19*n^2+76*n+71)*E^2+(2*n+3)*(19*n^2+76*n+71)*E-(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-10*n-15)*E+5*n+5 LREtools/SearchTable: "SearchTable successful" 1/2 2 1/2 2 1/2 1/2 {LegendreP(n, 5 ) , LegendreQ(n, 5 ) , LegendreP(n, 5 ) LegendreQ(n, 5 )} "A243946" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 {LegendreP(2 n, 5 ), LegendreQ(2 n, 5 )} "A243947" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 1/2 1/2 (2 n + 1) LegendreP(2 n, 5 ) + (2 n + 2) LegendreP(2 n + 2, 5 ) (2 n + 1) LegendreQ(2 n, 5 ) + (2 n + 2) LegendreQ(2 n + 2, 5 ) {-------------------------------------------------------------------, -------------------------------------------------------------------} 4 n + 3 4 n + 3 "A243948" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A243953" LREtools/SearchTable: "SearchTable successful" n n {(-2) ((2 n - 1) BesselI(n - 1/2, 1) + BesselI(n + 1/2, 1)), (-2) ((2 n - 1) BesselK(n - 1/2, -1) + BesselK(n + 1/2, -1))} "A244038" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { n 3 n { 3 n 3 n { 4 binomial(---, n/2) n::even {{ (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { 2 } { 2 2 { { --------------------------------------------------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A244039" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 1/2 4 binomial(---, n/2) n::even { 2 { {{ (2 n - 2) 3 n , { 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { --------------------------------------------------- n::odd { n { 3 n 3 n { 2 binomial(---, n/2) binomial(3 n, ---) { 2 2 { --------------------------------------- n::even { binomial(n, n/2) { } { 3 n 3 n { (n + 1) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { ------------------------------------------------------------------- n::odd { (3 n + 2) binomial(n + 1, n/2 + 1/2) "A244062" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A244174" {1, binomial(2 n, n)} "A244235" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A244236" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) n} "A244279" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A244280" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A244430" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 5 {|1/2 - ----| n! hypergeom([-n + 1, 1 + ----], [2], 5/2 + ----)} \ 2 / 5 2 "A244432" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 n 2 1/2 {(1 - 2 ) n! hypergeom([-n + 1, 1 + ----], [2], 4 + 2 2 )} 4 "A244451" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 3 5 {|3/2 - ----| n! hypergeom([-n + 1, 1 + ----], [2], - 5/2 - ------)} \ 2 / 5 2 "A244469" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 1/2 4 binomial(---, n/2) n::even { 2 { {{ (2 n - 2) 3 n , { 3 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - ----------------------------------------------------- n::odd { n { 3 n 3 n { 6 binomial(---, n/2) binomial(3 n, ---) { 2 2 { - --------------------------------------- n::even { binomial(n, n/2) { } { 3 n 3 n { (n + 1) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { ------------------------------------------------------------------- n::odd { (3 n + 2) binomial(n + 1, n/2 + 1/2) "A244531" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 memory used=158486.4MB, alloc=3223.5MB, time=1268.11 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n (-1) (n hypergeom([1/2, -n - 1], [1], 4) + (3 n + 12) hypergeom([1/2, -n], [1], 4)) (n + 1) {--------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A244594" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A244627" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A244650" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, ( 4 3 2 3 2 (156 n + 621 n + 526 n - 99 n - 40) hypergeom([-1/2, -n - 1], [1], -4) - 5 (31 n + 106 n + 41 n - 64) (n + 1) hypergeom([-1/2, -n], [1], -4) )/((n + 5) (n + 4) (n + 3) (n + 2))} "A244651" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 (30 n - 10 n - 10) hypergeom([-1/2, -n - 1], [1], -8) - 9 (3 n - 4) (n + 1) hypergeom([-1/2, -n], [1], -8) {-----------------------------------------------------------------------------------------------------------, 2 n - 13} (n + 3) (n + 2) "A244652" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 3 2 2 \| 2 | \ | 4 ((3 n1 + 38 n1 + 97 n1 + 190) LegendreP(n1, 2) + (-34 n1 + 22 n1 - 68) LegendreP(n1 + 1, 2))|| {1, (n - 29 n + 64) | ) |- ---------------------------------------------------------------------------------------------------||, | / | 2 2 || |----- \ (n1 + 2) (n1 + 3) ((n1 + 1) - 29 n1 + 35) (n1 - 29 n1 + 64) /| \n1 = 0 / /n - 1 \ |----- / n1 3 2 2 \| 2 | \ | 4 ((3 n1 + 38 n1 + 97 n1 + 190) LegendreQ(n1, 2) + (-34 n1 + 22 n1 - 68) LegendreQ(n1 + 1, 2))|| 2 (n - 29 n + 64) | ) |- ---------------------------------------------------------------------------------------------------||, n - 29 n + 64 | / | 2 2 || |----- \ (n1 + 2) (n1 + 3) ((n1 + 1) - 29 n1 + 35) (n1 - 29 n1 + 64) /| \n1 = 0 / } "A244653" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1, 25 n - 37} "A244884" LREtools/SearchTable: "SearchTable successful" n (-1) (hypergeom([1/2, -n - 1], [1], 4) + (2 n + 1) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------} n (n + 2) "A244886" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - 2 _Z - 3 _Z - 1, index = 1) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 2) , RootOf(_Z - 2 _Z - 3 _Z - 1, index = 3) } "A244973" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A245001" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 n { 1/2 ------------------------ n::even (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) { (n + 1) binomial(n, n/2) {-------------------------------------------------------------------------, { , n + 2 { (2 n - 2) { 2 (n + 1) { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) { 4 binomial(n, n/2) (n + 1) { -------------------------- n::even { n + 2 } { { 2 binomial(n + 1, n/2 + 1/2) n::odd "A245002" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { n { 4 (3 n + 5) n { -------------------------------- n::even (-1) ((5 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) { (n + 1) (n + 3) binomial(n, n/2) {--------------------------------------------------------------------------------------------, { , (n + 3) (n + 2) { (2 n - 2) { 2 2 (3 n + 2) { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) { n { 4 16 { 2 binomial(n, n/2) (3 n + 2) { ----------------------------------- n::even { ---------------------------- n::even { 2 2 2 { n + 2 { (n + 1) (n + 3) binomial(n, n/2) { , { , { binomial(n + 1, n/2 + 1/2) (3 n + 5) { (4 n + 4) { ------------------------------------ n::odd { 2 { n + 3 { ---------------------------------------------------- n::odd { 2 2 { (n + 1) (n + 2) (n + 4) binomial(n + 1, n/2 + 1/2) { 2 { 16 binomial(n, n/2) (n + 1) { ---------------------------- n::even { 2 { (n + 4) (n + 2) { } { 2 2 { 64 binomial(n - 1, n/2 - 1/2) n { --------------------------------- n::odd { 2 2 { (n + 1) (n + 3) "A245067" LREtools/SearchTable: "SearchTable successful" ((2 n + 7) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-18 n - 45) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) {---------------------------------------------------------------------------------------------------------------------------} 2 (n + 2) "A245088" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A245112" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /12500\(n/2 - 1/2) {{ |-----| GAMMA(n/2 + 1/10) GAMMA(n/2 + 3/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + 7/10) , { \ 27 / { ------------------------------------------------------------------------------------------ n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 7/6) GAMMA(n/2 + 5/6) { /12500\(n/2) { |-----| GAMMA(n/2 + 1/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + 7/10) GAMMA(n/2 + 3/10) { \ 27 / { ------------------------------------------------------------------------------------ n::even} { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 7/6) GAMMA(n/2 + 5/6) { { 0 n::odd "A245114" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { {{ (n/3 - 2/3) 13 , { (84375/4) GAMMA(n/3 + 7/15) GAMMA(n/3 + --) GAMMA(n/3 + 4/15) GAMMA(n/3 + 1/15) { 15 { ------------------------------------------------------------------------------------------ irem(n, 3) = 2 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 7/6) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) 13 { (84375/4) GAMMA(n/3 + 4/15) GAMMA(n/3 + 7/15) GAMMA(n/3 + 1/15) GAMMA(n/3 + --) { 15 , { ------------------------------------------------------------------------------------------ irem(n, 3) = 1 { GAMMA(n/3 + 7/6) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) { { 0 irem(n, 3) = 2 { (n/3) 13 { (84375/4) GAMMA(n/3 + --) GAMMA(n/3 + 4/15) GAMMA(n/3 + 7/15) GAMMA(n/3 + 1/15) { 15 { ------------------------------------------------------------------------------------ irem(n, 3) = 0 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) GAMMA(n/3 + 7/6) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A245176" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ 1/2 n1 1/2 {1, ) (-1/2 I 2 ) HermiteH(n1 + 1, 1/2 I 2 )} / ----- n1 = 0 "A245233" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" 2 binomial(2 n, n) (n - 4 n - 6) {-------------------------------} (n + 3) (n + 2) (n + 1) "A245329" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A245455" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((n + 1) (5 n - 7) hypergeom([1/2, -n - 1], [1], 4) + (-5 n - 7) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------------------} n (n - 1) "A245551" LREtools/SearchTable: "SearchTable successful" n {(-1) ((4 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 15) hypergeom([1/2, -n], [1], 4)) (n + 1)} "A245560" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even n { n binomial(n, n/2) n::even { binomial(n, n/2) {2 (n + 2), { , { } { binomial(n + 1, n/2 + 1/2) (n/2 + 1/2) n::odd { (2 n - 2) { 2 2 { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A245734" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A245735" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A245769" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | | \ LegendreP(n1 + 1, 3) + LegendreP(n1, 3)| | \ LegendreQ(n1 + 1, 3) + LegendreQ(n1, 3)| {(2 n + 1) | ) ---------------------------------------|, (2 n + 1) | ) ---------------------------------------|, 2 n + 1} | / (2 n1 + 3) (2 n1 + 1) | | / (2 n1 + 3) (2 n1 + 1) | |----- | |----- | \n1 = 0 / \n1 = 0 / "A245923" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(7 - 4 3 ) , (7 + 4 3 ) , LegendreP(n, 7), LegendreQ(n, 7)} "A245924" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(7 - 4 3 ) , (7 + 4 3 ) , LegendreP(n + 1, 7), LegendreQ(n + 1, 7)} "A245925" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3+(2*n+5)*(13*n^2+52*n+49)*E^2-(2*n+3)*(13*n^2+52*n+49)*E-(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" 1/2 2 1/2 2 1/2 1/2 {LegendreP(n, 3 I) , LegendreQ(n, 3 I) , LegendreP(n, 3 I) LegendreQ(n, 3 I)} "A245926" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 1/2 n 1/2 {(-1) LegendreP(2 n, 3 I), (-1) LegendreQ(2 n, 3 I)} "A245927" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 1/2 1/2 (-1) ((2 n + 1) LegendreP(2 n, 3 I) + (2 n + 2) LegendreP(2 n + 2, 3 I)) {-------------------------------------------------------------------------------, 4 n + 3 n 1/2 1/2 (-1) ((2 n + 1) LegendreQ(2 n, 3 I) + (2 n + 2) LegendreQ(2 n + 2, 3 I)) -------------------------------------------------------------------------------} 4 n + 3 "A245929" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n n {(-1) binomial(2 n, n) LegendreP(2 n, 3), (-1) binomial(2 n, n) LegendreQ(2 n, 3)} "A245930" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A246056" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A246062" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ / 1/2 1/2 \ n | 2 2 | n | 2 2 | (-8 I) |(4 n + 4) LegendreP(2 n + 2, ----) - LegendreP(2 n, ----)| (-8 I) |(4 n + 4) LegendreQ(2 n + 2, ----) - LegendreQ(2 n, ----)| \ 2 2 / \ 2 2 / {-------------------------------------------------------------------, -------------------------------------------------------------------, n (4 n + 3) n (4 n + 3) / 1/2 1/2 \ / 1/2 1/2 \ n | 2 2 | n | 2 2 | (8 I) |(4 n + 4) LegendreP(2 n + 2, ----) - LegendreP(2 n, ----)| (8 I) |(4 n + 4) LegendreQ(2 n + 2, ----) - LegendreQ(2 n, ----)| \ 2 2 / \ 2 2 / ------------------------------------------------------------------, ------------------------------------------------------------------} n (4 n + 3) n (4 n + 3) "A246065" LREtools/SearchTable: "SearchTable successful" 2 2 {4 (n + 1) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) - 9 (2 n + 1) hypergeom([1/2, -n, -n], [1, 1], 4)} "A246138" LREtools/SearchTable: "SearchTable successful" {(4 n + 5) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-36 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4)} "A246423" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A246437" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) hypergeom([1/2, -n], [1], 4)} "A246459" LREtools/SearchTable: "SearchTable successful" {(4 n + 3) hypergeom([1/2, -n, -n], [1, 1], 4)} "A246460" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) / ----- n1 = 0 ((12 n1 + 19) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) + hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1)) (2 n1 + 3) -------------------------------------------------------------------------------------------------------------------------------------------} 3 (n1 + 2) "A246461" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 2 | \ 4 3 2 {(2 n + 2 n + 1) | ) ((12 n1 + 56 n1 + 90 n1 + 62 n1 + 19) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) | / |----- \n1 = 0 4 3 2 / 2 2 + (-12 n1 - 40 n1 - 42 n1 - 10 n1 + 1) hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1)) / ((2 (n1 + 1) + 2 n1 + 3) (2 n1 + 2 n1 + 1)) / \ | | 2 |, 2 n + 2 n + 1} | | / "A246462" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ {1, ) ((4 n1 + 7) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) + (4 n1 + 1) hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1))} / ----- n1 = 0 "A246467" memory used=159850.3MB, alloc=3223.5MB, time=1277.33 LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(31*n^2+124*n+115)*E^2+5*(2*n+3)*(31*n^2+124*n+115)*E-125*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-5)*E+5*n+5 LREtools/SearchTable: "SearchTable successful" 2 {((2 n + 2) hypergeom([-1/2, -n - 1], [1], -4) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4)) } "A246472" n n {2 , 4 , binomial(2 n, n)} "A246511" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A246512" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 2 |----- 4 n + 8 n + 5 2 | \ 3 2 {--------------, (4 n + 8 n + 5) | ) ((51 n1 + 186 n1 + 232 n1 + 103) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) n + 1 | / |----- \n1 = 0 2 / 2 2 - (n1 + 1) (3 n1 + 9 n1 + 11) hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1)) (2 n1 + 3) / ((n1 + 2) (4 (n1 + 1) + 8 n1 + 13) / \ | 2 | (4 n1 + 8 n1 + 5))|/(n + 1)} | | / "A246513" LREtools/SearchTable: "SearchTable successful" {(2 n - 3) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-18 n + 9) hypergeom([1/2, -n, -n], [1, 1], 4)} "A246542" LREtools/SearchTable: "SearchTable successful" ((12 n + 6) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) - (10 n + 9) (n + 1) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------------------} (n + 1) (n - 1) n "A246555" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A246563" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A246567" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 2 |----- 2 n + 4 n + 1 2 | \ {--------------, (2 n + 4 n + 1) | ) n + 1 | / |----- \n1 = 0 ((17 n1 + 29) hypergeom([n1 + 2, n1 + 2, -n1 - 1, -n1 - 1], [1, 1, 1], 1) + (-n1 - 1) hypergeom([-n1, -n1, n1 + 1, n1 + 1], [1, 1, 1], 1)) \ | 2 2 2 | (2 n1 + 3)/((n1 + 2) (2 (n1 + 1) + 4 n1 + 5) (2 n1 + 4 n1 + 1))|/(n + 1)} | | / "A246604" binomial(2 n, n) {n, ----------------} n + 1 "A246606" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! ((4 n + 3) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) n! ((4 n + 3) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A246607" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A246652" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A246653" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A246662" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A246761" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A246800" n n (2 n + 1) (-1) binomial(2 n, n) {4 , --------------------------------} n + 1 "A246840" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A246860" 2 2 (2 n + 1) binomial(2 n, n) (2 n + 1) binomial(2 n, n) {----------------------------, --------------------------} 2 2 (n + 1) (n + 2) (n + 1) (n + 2) "A246875" LREtools/SearchTable: "SearchTable successful" (2 n - 1) hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + (2 n + 5) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1) {-----------------------------------------------------------------------------------------------------------------------------} 2 2 (n + 2) n "A246876" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-4*(2*n+5)*(13*n^2+52*n+48)*E^2+48*(2*n+3)*(13*n^2+52*n+48)*E-1728*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-4*n-4)*E+3*n+3 LREtools/SearchTable: "SearchTable successful" n 2 {4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 3) hypergeom([-1/2, -n], [1], -2)) } "A246883" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A246884" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A246906" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(79*n^2+316*n+291)*E^2+21*(2*n+3)*(79*n^2+316*n+291)*E-9261*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-10*n-11)*E+21*n+21 LREtools/SearchTable: "SearchTable successful" n 2 {9 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -4/3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -4/3)) } "A246923" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(91*n^2+364*n+339)*E^2+9*(2*n+3)*(91*n^2+364*n+339)*E-729*(2*n+5)*(n+1)^2 "to two: Symmetric square" (3*n+6)*E^2+(-10*n-7)*E+3*n+3 LREtools/SearchTable: "SearchTable successful" 2 {((2 n + 2) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8)) } "A247029" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 {{ , { (n/3 - 2/3) { (-6912) GAMMA(n/3 - 1/12) GAMMA(n/3 + 5/12) GAMMA(1/6 + n/3) { ----------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) 4 n { (-729) binomial(--- - 4/3, n/3 - 1/3) { 3 , { ------------------------------------------------ irem(n, 3) = 1 { n { { 0 irem(n, 3) = 2 { (n/3) { (-6912) GAMMA(n/3 + 5/12) GAMMA(n/3 - 1/12) GAMMA(1/6 + n/3) { ----------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A247102" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n n | \ -I (-I 3 ) 3 ((-n1 - 1) LegendreP(n1, 3 I) + 3 (3 n1 + 5) LegendreP(n1 + 1, 3 I) I)| {(-1) , (-1) | ) ---------------------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- (n1 + 2) (-1) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n | \ (-I 3 ) 3 ((n1 + 1) LegendreQ(n1, 3 I) - 3 (3 n1 + 5) LegendreQ(n1 + 1, 3 I) I) I| (-1) | ) -------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-1) | \n1 = 0 / "A247109" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 7 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A247162" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A247170" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247171" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A247172" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247173" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A247195" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247249" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, - 3/4 - n, -1/4)} "A247287" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, n, ) (-1) hypergeom([1/2, -n1 - 1], [1], 4)} / ----- n1 = 0 "A247289" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 1} "A247291" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A247293" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A247295" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A247296" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247300" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247333" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A247365" memory used=161153.2MB, alloc=3223.5MB, time=1286.10 LREtools/SolveLRE: "Reduced the order of" (16*n^6+192*n^5+957*n^4+2535*n^3+3765*n^2+2977*n+981)*E^4+(n+2)*(64*n^8+1312*n^7+11636*n^6+58278*n^5+ 180210*n^4+352281*n^3+425309*n^2+290241*n+85865)*E^3+(-256*n^10-6400*n^9-71552*n^8-471040*n^7-2021935*n^6-5913027*n^5-11931475*n^4-16405247*n^3-\ 14713832*n^2-7777416*n-1841370)*E^2+(-64*n^9-1376*n^8-12948*n^7-69906*n^6-238388*n^5-531993*n^4-776322*n^3-713829*n^2-374966*n-85590)*E+16*n^6+288*n^ 5+2157*n^4+8603*n^3+19272*n^2+22996*n+11423 "to two: Symmetric product" (2*n-1)*E^2-2*(4*n^2+1)*n*E+2*n+1 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" {BesselI(2 n + 1, 2) BesselI(n, 2), BesselI(2 n + 1, 2) BesselK(n, -2), BesselK(2 n + 1, -2) BesselI(n, 2), BesselK(2 n + 1, -2) BesselK(n, -2)} "A247374" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { 1/2 ------------------------ n::even n { binomial(n, n/2) (2 n + 2) n::even { (n + 1) binomial(n, n/2) {1, 2 , { , { } { (n + 2) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A247386" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(n - 1) n (-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A247471" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A247499" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1 ((n + 2) LaguerreL(n + 1, -1) + (-n - 1) LaguerreL(n, -1)) n! {---------------, -------------------------------------------------------------} (n + 2) (n + 1) n + 2 "A247591" LREtools/SearchTable: "SearchTable successful" 4 3 2 {((4 n + 41 n + 161 n + 298 n + 198) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) 3 2 / 2 3 - 9 (n + 1) (4 n + 33 n + 92 n + 90) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) / ((n + 3) (n + 2) (n + 1))} / "A247623" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 ((n + 3) LegendreP(n/2 + 3/2, 3) + (-7 n - 13) LegendreP(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::even { n + 1 {{ , { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::even { n + 1 { , { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::even { n { , { 2 ((n + 3) LegendreP(n/2 + 3/2, 3) + (-7 n - 13) LegendreP(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::odd { n + 1 { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::even { n { } { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::odd { n + 1 "A247630" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 ((n + 2) LegendreP(n/2 + 1, 3) + (-5 n - 6) LegendreP(n/2, 3)) { ---------------------------------------------------------------- n::even { n {{ , { 2 ((n + 3) (2 n + 1) LegendreP(n/2 + 3/2, 3) + %1 LegendreP(n/2 + 1/2, 3)) { - -------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { 2 ((n + 2) LegendreQ(n/2 + 1, 3) + (-5 n - 6) LegendreQ(n/2, 3)) { ---------------------------------------------------------------- n::even { n { , { 2 ((n + 3) (2 n + 1) LegendreQ(n/2 + 3/2, 3) + %1 LegendreQ(n/2 + 1/2, 3)) { - -------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { (n + 3) (2 n + 1) LegendreP(n/2 + 3/2, 3) + %1 LegendreP(n/2 + 1/2, 3) { - ---------------------------------------------------------------------- n::even { (n - 1) (n + 1) { , { (n + 2) LegendreP(n/2 + 1, 3) + (-5 n - 6) LegendreP(n/2, 3) { ------------------------------------------------------------ n::odd { n { (n + 3) (2 n + 1) LegendreQ(n/2 + 3/2, 3) + %1 LegendreQ(n/2 + 1/2, 3) { - ---------------------------------------------------------------------- n::even { (n - 1) (n + 1) { } { (n + 2) LegendreQ(n/2 + 1, 3) + (-5 n - 6) LegendreQ(n/2, 3) { ------------------------------------------------------------ n::odd { n 2 %1 := -12 n - 29 n - 11 "A247984" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) n { { (-16) {2 , { (n/2 - 1/2) , { ------------------------ n::even} { 2 (-1) binomial(n - 1, n/2 - 1/2) n { binomial(n, n/2) (n + 1) { ---------------------------------------------- n::odd { { n + 1 { 0 n::odd "A248100" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A248167" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(163*n^2+652*n+603)*E^2+33*(2*n+3)*(163*n^2+652*n+603)*E-35937*(2*n+5)*(n+1 )^2 "to two: Symmetric square" (n+2)*E^2+(-14*n-13)*E+33*n+33 LREtools/SearchTable: "SearchTable successful" n 2 {9 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -8/3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8/3)) } "A248168" LREtools/SearchTable: "SearchTable successful" n {3 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -8/3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8/3))} "A248574" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 4 { ------------------ n::even n { n binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n + 2) { binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A248586" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A248658" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A248812" n n! (n + 1) (-1) n! (n - 1) {----------, ----------------} n n "A248876" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A249015" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A249059" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (-(n + 1) (n + 4) HermiteH(n, 1/2 I 2 ) + 2 (n + 3) HermiteH(n + 1, 1/2 I 2 ) I)} "A249060" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (- n/2) {{ , { 2 (n + 1) (n + 3) binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2) { { 1/4 2 (n/2 - 1/2)! (n + 1) (n + 3) n::odd { 0 n::odd "A249062" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A249075" LREtools/SearchTable: "SearchTable successful" n {(-I) (HermiteH(n + 1, 1/2 I) + 2 I (n + 1) HermiteH(n, 1/2 I)) I} "A249101" LREtools/SearchTable: "SearchTable successful" n {(-I) ((-2 n - 1) HermiteH(n - 1/2, 1/2 I) + HermiteH(n + 1/2, 1/2 I) I), n n (n + 1) (-I) (-(-1) (2 n + 1) HermiteH(n - 1/2, -1/2 I) + (-1) HermiteH(n + 1/2, -1/2 I) I)} "A249131" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (- n/2) { 2 (n + 1) KummerM(n/2 + 3/2, 1, 1/2) binomial(n, n/2) (n/2)! n::even {{ , { (- n/2 - 1/2) { 2 (n + 2) (KummerM(n/2 + 2, 1, 1/2) - KummerM(n/2 + 1, 1, 1/2)) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd { (- n/2) { 2 (n + 1) LaguerreL(n/2 + 1/2, -1/2) binomial(n, n/2) (n/2)! n::even { , { (- n/2 - 1/2) { 2 (n + 2) (LaguerreL(n/2 + 1, -1/2) - LaguerreL(n/2, -1/2)) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd { (n/2) { 1/2 2 (n/2)! (n + 2) (KummerM(n/2 + 2, 1, 1/2) - KummerM(n/2 + 1, 1, 1/2)) n::even { , { (n/2 - 1/2) { 1/2 2 (n/2 - 1/2)! (n + 1) KummerM(n/2 + 3/2, 1, 1/2) n::odd { (n/2) { 1/2 2 (n/2)! (n + 2) (LaguerreL(n/2 + 1, -1/2) - LaguerreL(n/2, -1/2)) n::even { } { (n/2 - 1/2) { 1/2 2 (n/2 - 1/2)! (n + 1) LaguerreL(n/2 + 1/2, -1/2) n::odd "A249308" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 8 { n { ------------------ n::even { 2 binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { (n - 1) { (3 n + 3) { 2 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A249332" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" {hypergeom([-2 n, -2 n, -2 n, -2 n], [1, 1, 1], 1)} "A249349" n {1, (2 n + 1) (1/2) n! binomial(2 n, n)} "A249512" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { / n2 \ ||| | | | { |---- - 1/2| ||| | | | { \ 2 / n2 ||| | | | { 2 (-1) binomial(n2 - 1, ---- - 1/2) ||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { ----------------------------------------------- n2::odd ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { n2 + 1 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) -----------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 \ \\\ | | | { |----| ||| | | | { \ 2 / ||| | | | { 2 (-16) ||| | | | { ------------------------------ n2::even||| | | | { n2 ||| | | | { (n2 + 1) n2 binomial(n2, ----) ||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (5 ) | ) |1/5 5 (-1) | ) ------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A249513" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { 0 n2::even\\\ | | | { ||| | | | { / n2 \ ||| | | | { |---- - 1/2| ||| | | | { \ 2 / ||| | | | { (-16) (3 n2 + 8) ||| | | | { ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { (n2 + 2) n2 binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) ----------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { / n2 \ \\\ | | | { |----| ||| | | | { \ 2 / n2 ||| | | | { (-1) binomial(n2, ----) (3 n2 + 8) ||| | | | { 2 ||| | | | { ---------------------------------------- n2::even||| |n - 1 | |n1 - 1 { n2 + 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (5 ) | ) |1/5 5 (-1) | ) ----------------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A249519" n {4 binomial(2 n, n), binomial(4 n, 2 n)} "A249520" n (2 n - 1) binomial(4 n, 2 n) 4 binomial(2 n, n) {----------------------------, 1/2 -------------------} (4 n - 1) (4 n - 3) n - 1/2 "A249608" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A249769" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 { (n/2 + 1) (n/2)! n::even { 2 (n + 1) (n + 2 n + 3) binomial(n, n/2) (n/2)! n::even {{ , { } { 2 { (-n + 1) { (n/2 + 1/2)! (1/2 n + n + 3/2) n::odd { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A249785" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" { 0 n::even { {{ (n/2 - 1/2) 2 3 n , { (2/3) ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 { (- n/2) 2 3 n 3 n { 24 ((n/2)!) binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------------------- n::even} { 3 n - 1 { { 0 n::odd "A249786" RightFactors: "Division check FAILED" RightFactors: "Division check FAILED" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" { {{ 0 , irem(n, 6) = 0 { 0 , irem(n, 6) = 1 (n/2 - 1) 2 2 6 3 binomial(n/2 - 1, n/6 - 1/3) binomial(n/3 - 2/3, n/6 - 1/3) ((n/6 - 1/3)!) (-8 n + 8) binomial(n - 2, n/2 - 1) 2 n binomial(--- - 4/3, n/6 - 1/3) , irem(n, 6) = 2 3 0 , irem(n, 6) = 3 0 , irem(n, 6) = 4 (n/2 + 1/2) 2 2 6 3 binomial(n/2 + 1/2, 1/6 + n/6) binomial(n/3 + 1/3, 1/6 + n/6) ((1/6 + n/6)!) binomial(n + 1, n/2 + 1/2) { 2 n { binomial(--- + 2/3, 1/6 + n/6)/((n + 1) (2 n - 1)) , irem(n, 6) = 5, { 3 { 0 , irem(n, 6) = 0 { 0 , irem(n, 6) = 1 (n/6 - 1/3) 2 6 2 n 6912 binomial(n/2 - 1, n/6 - 1/3) ((n/6 - 1/3)!) binomial(n/3 - 2/3, n/6 - 1/3) binomial(--- - 4/3, n/6 - 1/3) n 3 1/12 ----------------------------------------------------------------------------------------------------------------------------- , n + 1 irem(n, 6) = 2 0 , irem(n, 6) = 3 0 , irem(n, 6) = 4 (- 5/6 + n/6) 2 6 2 n -2/3 6912 binomial(n/2 - 5/2, - 5/6 + n/6) ((- 5/6 + n/6)!) binomial(n/3 - 5/3, - 5/6 + n/6) binomial(--- - 10/3, - 5/6 + n/6) 3 , (2 n - 7) (n - 3) (n - 1) , irem(n, 6) = 5 { 0 irem(n, 6) = 0 { { (n/3 - 1/3) 2 11 { 3456 GAMMA(n/6 + 2/3) GAMMA(n/6 + 1) GAMMA(n/6 + 4/3) GAMMA(n/6 + 5/12) GAMMA(n/6 + --) { 12 { --------------------------------------------------------------------------------------------------- irem(n, 6) = 1 { (n + 1) (n + 2) (2 n - 1) { , { 0 irem(n, 6) = 2 { { 0 irem(n, 6) = 3 { { (n/3 - 4/3) { -8 3456 %1 GAMMA(n/6 - 1/12) GAMMA(n/6 + 1/2) GAMMA(n/6 + 5/6) GAMMA(n/6 + 5/12) irem(n, 6) = 4 { { 0 irem(n, 6) = 5 { 0 irem(n, 6) = 0 { { (n/3 - 1/3) { -8 3456 %1 GAMMA(n/6 - 1/12) GAMMA(n/6 + 1/2) GAMMA(n/6 + 5/6) GAMMA(n/6 + 5/12) irem(n, 6) = 1 { { 0 irem(n, 6) = 2 { { 0 irem(n, 6) = 3, { { (n/3 + 2/3) 2 11 { 3456 GAMMA(n/6 + 2/3) GAMMA(n/6 + 1) GAMMA(n/6 + 4/3) GAMMA(n/6 + 5/12) GAMMA(n/6 + --) { 12 { --------------------------------------------------------------------------------------------------- irem(n, 6) = 4 { (n + 1) (n + 2) (2 n - 1) { { 0 irem(n, 6) = 5 { (n/3) 2 11 { 3456 GAMMA(n/6 + 2/3) GAMMA(n/6 + 1) GAMMA(n/6 + 4/3) GAMMA(n/6 + 5/12) GAMMA(n/6 + --) { 12 { --------------------------------------------------------------------------------------------- irem(n, 6) = 0 { (n + 1) (n + 2) (2 n - 1) { { 0 irem(n, 6) = 1 { , { 0 irem(n, 6) = 2 { { (n/3 - 1) { -8 3456 %1 GAMMA(n/6 - 1/12) GAMMA(n/6 + 1/2) GAMMA(n/6 + 5/6) GAMMA(n/6 + 5/12) irem(n, 6) = 3 { { 0 irem(n, 6) = 4 { { 0 irem(n, 6) = 5 { (n/3) { -8 3456 %1 GAMMA(n/6 - 1/12) GAMMA(n/6 + 1/2) GAMMA(n/6 + 5/6) GAMMA(n/6 + 5/12) irem(n, 6) = 0 { { 0 irem(n, 6) = 1 { { 0 irem(n, 6) = 2 { { (n/3 + 1) 2 11 } { 3456 GAMMA(n/6 + 2/3) GAMMA(n/6 + 1) GAMMA(n/6 + 4/3) GAMMA(n/6 + 5/12) GAMMA(n/6 + --) { 12 { ------------------------------------------------------------------------------------------------- irem(n, 6) = 3 { (n + 1) (n + 2) (2 n - 1) { { 0 irem(n, 6) = 4 { { 0 irem(n, 6) = 5 2 %1 := GAMMA(1/6 + n/6) "A249789" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" {{ { 0 , n::even (2 n - 2) 2 ((2 n + 4) hypergeom([-1/4, - n/2 - 1/2], [3/2], -1) + (-2 n - 5) hypergeom([-1/4, - n/2 + 1/2], [3/2], -1)) GAMMA(n/2 + 1) , { GAMMA(n/2 - 1/4) , n::odd { n 4 ((2 n + 4) hypergeom([-1/4, - n/2 - 1/2], [3/2], -1) + (-2 n - 5) hypergeom([-1/4, - n/2 + 1/2], [3/2], -1)) GAMMA(n/2 + 1) GAMMA(n/2 - 1/4) , n::even } 0 , n::odd "A249792" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A249891" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A249908" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 5 1/2 5 | 1/2 n 1/2 | 5 1/2 5 | {(-5 ) 5 |LegendreP(n, ----) + 5 LegendreP(n + 1, ----)|, (-5 ) 5 |LegendreQ(n, ----) + 5 LegendreQ(n + 1, ----)|} \ 5 5 / \ 5 5 / "A249924" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 | 12 6 | 24 6 |27/5 - -------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], 64/5 + -------) \ 5 / 5 {---------------------------------------------------------------------------------} GAMMA(n + 2) "A249925" LREtools/SearchTable: "SearchTable successful" n n (-4 I) (-2 LegendreP(n, 1/2 I) + LegendreP(n + 1, 1/2 I) I) (-4 I) (-2 LegendreQ(n, 1/2 I) + LegendreQ(n + 1, 1/2 I) I) {------------------------------------------------------------, ------------------------------------------------------------} n + 2 n + 2 "A249945" n {3 , n!} "A249946" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A249975" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { {{ 4 ((n + 2) hypergeom([n + 3, - n/2 - 1], [1], -1) + (3 n + 3) hypergeom([n + 1, - n/2], [1], -1)) { - ------------------------------------------------------------------------------------------------- , n::even { 17 n + 22 2 { 4 ((n + 2) (n + 3) hypergeom([n + 4, - n/2 - 3/2], [1], -1) + (-14 n - 105/2 n - 93/2) hypergeom([n + 2, - n/2 - 1/2], [1], -1)) , { --------------------------------------------------------------------------------------------------------------------------------- , n::odd { (n + 1) (17 n + 39) { 2 2 ((n + 2) (n + 3) hypergeom([n + 4, - n/2 - 3/2], [1], -1) + (-14 n - 105/2 n - 93/2) hypergeom([n + 2, - n/2 - 1/2], [1], -1)) --------------------------------------------------------------------------------------------------------------------------------- , n::even (n + 1) (17 n + 39) 2 ((n + 2) hypergeom([n + 3, - n/2 - 1], [1], -1) + (3 n + 3) hypergeom([n + 1, - n/2], [1], -1)) } - ------------------------------------------------------------------------------------------------- , n::odd 17 n + 22 "A249976" memory used=162506.9MB, alloc=3223.5MB, time=1295.20 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A250307" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n 1/2 n {(11/4 - 1/4 I 7 ) , (11/4 + 1/4 I 7 ) , (11/4 - 1/4 I 7 ) /n - 1 /n1 - 1 \\ |----- |----- / n2 1/2 (-n2 - 1) \|| | \ 1/2 n1 1/2 (-n1 - 1) | \ |4 8 (11 + 7 I) (2 LegendreP(n2 + 1, 2) - LegendreP(n2, 2))||| | ) (11/4 + 1/4 I 7 ) (11/4 - 1/4 I 7 ) | ) |------------------------------------------------------------------------|||, | / | / \ n2 + 2 /|| |----- |----- || \n1 = 0 \n2 = 0 // 1/2 n (11/4 - 1/4 I 7 ) /n - 1 /n1 - 1 \\ |----- |----- / n2 1/2 (-n2 - 1) \|| | \ 1/2 n1 1/2 (-n1 - 1) | \ |4 8 (11 + 7 I) (2 LegendreQ(n2 + 1, 2) - LegendreQ(n2, 2))||| | ) (11/4 + 1/4 I 7 ) (11/4 - 1/4 I 7 ) | ) |------------------------------------------------------------------------|||} | / | / \ n2 + 2 /|| |----- |----- || \n1 = 0 \n2 = 0 // "A250885" LREtools/SearchTable: "SearchTable successful" n (-1/4) ((2 n + 1) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -2) + (-18 n - 6) hypergeom([-3 n, -n], [-2 n], -2)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------} (7 n + 2) (n + 1) "A250886" LREtools/SearchTable: "SearchTable successful" n (-1/9) ((2 n + 1) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -2) + (-18 n - 6) hypergeom([-3 n, -n], [-2 n], -2)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------} (7 n + 2) (n + 1) "A250887" LREtools/SearchTable: "SearchTable successful" /-1\n |--| ((2 n + 1) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -3) + (-30 n - 10) hypergeom([-3 n, -n], [-2 n], -3)) binomial(2 n, n) \16/ {---------------------------------------------------------------------------------------------------------------------------------} (26 n + 9) (n + 1) "A250888" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |162 42 21 | 343 63 21 |--- - --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], --- + --------) \25 25 / 50 50 {---------------------------------------------------------------------------------} GAMMA(n + 2) "A250889" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |56 38 19 | 13718 1064 19 |-- - --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], ----- + ----------) \25 25 / 6075 6075 {------------------------------------------------------------------------------------} GAMMA(n + 2) "A250890" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |324 78 39 | 4394 468 39 |--- - --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], ---- + ---------) \49 49 / 1225 1225 {-----------------------------------------------------------------------------------} GAMMA(n + 2) "A250918" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A251568" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([-n - 1], [n + 1], -1) + (-2 n - 2) hypergeom([-n], [n], -1)) binomial(2 n, n) n! {------------------------------------------------------------------------------------------------------} n + 2 "A251573" LREtools/SearchTable: "SearchTable successful" 3 {((3 n + 1) (3 n - 1) n hypergeom([1/2, -n - 1], [-1/2, -3 n - 1], 3) + (-9 n + 8 n - 5) hypergeom([1/2, -n], [-1/2, -3 n + 2], 3)) binomial(3 n, n) / 2 n! / ((3 n - 1) (9 n + 6 n - 5))} / "A251598" memory used=163377.3MB, alloc=3223.5MB, time=1300.98 memory used=163827.8MB, alloc=3223.5MB, time=1304.80 memory used=164388.0MB, alloc=3223.5MB, time=1308.95 memory used=164892.3MB, alloc=3223.5MB, time=1313.11 LREtools/SearchTable: "SearchTable successful" 2 17 16 15 14 13 12 11 10 {binomial(2 n, n) ((2 n + 1) (128 n + 11584 n + 487648 n + 12684048 n + 228362984 n + 3020987356 n + 30461268890 n + 240048137211 n 9 8 7 6 5 4 3 + 1504226764727 n + 7568008835454 n + 30589446011056 n + 98398072318139 n + 247016567637587 n + 469640303496368 n + 647589573625956 n 2 15 14 + 606082859306352 n + 342059828356224 n + 87403252082688) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1) - 16 (64 n + 5568 n 13 12 11 10 9 8 7 6 + 224080 n + 5535376 n + 93902956 n + 1159351364 n + 10773323735 n + 76876368759 n + 425794797866 n + 1834878904658 n 5 4 3 2 3 + 6109927850495 n + 15433115239759 n + 28560865654884 n + 36426196064340 n + 28507194963360 n + 10277284686336) (n + 1) / 2 2 3 3 3 4 4 4 2 hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)) / ((n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) / )} "A251663" LREtools/SearchTable: "SearchTable successful" binomial(3 n, n) n! hypergeom([1/2, -n], [-1/2, -3 n + 2], 3) {-------------------------------------------------------------} 3 n - 1 "A252284" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A252354" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 2 %1 := _Z - 4 _Z + 2 _Z + 6 _Z - 4 _Z + 1 "A252355" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-2 n - 2, -n - 1], [1], -1) + (6 n + 3) hypergeom([-2 n, -n], [1], -1) {-----------------------------------------------------------------------------------------} 7 n + 4 "A252729" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=165501.7MB, alloc=3223.5MB, time=1317.90 { (n/3) { (n/3) { 3 (n/3)! irem(n, 3) = 0 { 3 GAMMA(n/3 + 4/3) { { ----------------------- irem(n, 3) = 0 { (n/3 + 2/3) { n + 1 { 3 (n/3 + 2/3)! { {{ ------------------------- irem(n, 3) = 1, { (n/3 - 1/3) , { n + 2 { 3 GAMMA(n/3 + 1) irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 + 1/3) { 3 (n/3 + 1/3)! { 3 GAMMA(5/3 + n/3) { ------------------------- irem(n, 3) = 2 { ----------------------------- irem(n, 3) = 2 { n + 1 { n + 2 { (n/3) { 3 GAMMA(5/3 + n/3) { ----------------------- irem(n, 3) = 0 { n + 2 { { (n/3 - 1/3) } { 3 GAMMA(n/3 + 4/3) { ----------------------------- irem(n, 3) = 1 { n + 1 { { (n/3 - 2/3) { 3 GAMMA(n/3 + 1) irem(n, 3) = 2 "A253093" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-7 n - 3) hypergeom([1/2, -n, -n], [1, 1], 4) {----------------------------------------------------------------------------------------------------} 2 (n - 1) n "A253094" LREtools/SearchTable: "SearchTable successful" 2 3 2 (n + 1) (2 n + 6 n - 5) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-20 n - 42 n + 17 n + 15) hypergeom([1/2, -n, -n], [1, 1], 4) {---------------------------------------------------------------------------------------------------------------------------------------} 2 2 (n - 3) (n - 2) (n - 1) (n + 2) n "A253095" LREtools/SearchTable: "SearchTable successful" 3 2 {((2 n + 1) (34 n + 193 n + 353 n + 212) hypergeom([1/2, -n - 1, -n - 1, -n - 1], [1, 1, -n - 1/2], 1) 3 / 2 3 - 16 (5 n + 13) (n + 1) hypergeom([1/2, -n, -n, -n], [1, 1, -n + 1/2], 1)) binomial(2 n, n) / ((n + 4) (n + 3) (n + 2) (n + 1))} / "A253192" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A253390" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 2 { binomial(n, n/2) (25 n + 159 n - 46 n - 144) { 1/2 ---------------------------------------------- n::even n n n { (n + 2) (n - 1) {(-2) , (-1) (2 n + 7), 2 (6 n + 127), 12 n + 7, { , { 3 2 { binomial(n - 1, n/2 - 1/2) (25 n + 213 n + 287 n + 3) { ------------------------------------------------------- n::odd { (n + 1) (n + 3) { n 3 2 { 4 4 (25 n + 213 n + 287 n + 3) { ---------------------------------- n::even { n (n + 1) (n + 3) binomial(n, n/2) { } { (2 n + 2) 3 2 { 2 2 (25 n + 159 n - 46 n - 144) { -------------------------------------------------- n::odd { (n - 1) (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A253502" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { 2 binomial(n, n/2) (n + 1) (13 n + 130 n + 272) { ------------------------------------------------ n::even n 2 { (n + 2) (n + 4) {2 (n + 34), 11 n + 50 n + 103, { , { 2 { binomial(n + 1, n/2 + 1/2) (n + 2) (13 n + 152 n + 403) { -------------------------------------------------------- n::odd { (n + 5) (n + 3) { n 2 { 4 (n + 2) (13 n + 152 n + 403) { 1/2 ---------------------------------------- n::even { (n + 1) (n + 3) (n + 5) binomial(n, n/2) { } { (2 n - 2) 2 { 2 (n + 1) (13 n + 130 n + 272) { -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) "A253665" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 { ---------------- n::even { n { binomial(n, n/2) { 8 2 binomial(n, n/2) n::even {{ , { } { (3 n - 3) { (n + 1) { 8 2 { 2 (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A253831" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(2 _Z - 8 _Z + 4 _Z + 9, index = 1) , RootOf(2 _Z - 8 _Z + 4 _Z + 9, index = 2) , RootOf(2 _Z - 8 _Z + 4 _Z + 9, index = 3) } "A253918" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A254129" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A254314" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A254316" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A254747" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A254795" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { n 2 { 4 GAMMA(n/2 + 5/4) n::even { 2 4 GAMMA(n/2 + 7/4) (n + 1) { { ------------------------------ n::even {{ (2 n + 2) 2 , { 2 } { 2 2 GAMMA(n/2 + 7/4) (n + 1) { (2 n + 3) { -------------------------------------- n::odd { { 2 { (2 n - 2) 2 { (2 n + 3) { 2 GAMMA(n/2 + 5/4) n::odd "A254796" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { n 2 { 4 GAMMA(n/2 + 5/4) n::even { 2 4 GAMMA(n/2 + 7/4) (n + 1) { { ------------------------------ n::even {{ (2 n + 2) 2 , { 2 } { 2 2 GAMMA(n/2 + 7/4) (n + 1) { (2 n + 3) { -------------------------------------- n::odd { { 2 { (2 n - 2) 2 { (2 n + 3) { 2 GAMMA(n/2 + 5/4) n::odd "A254865" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) { (27/4) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) (2 n + 5) { ----------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 11/6) (n + 2) { { (n/3 - 1/3) {{ 3 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) , { ----------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 3/2) { { (n/3 + 1/3) { (27/4) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (2 n + 7) { ----------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 13/6) (n + 2) (n + 3) { (n/3) { (27/4) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (2 n + 7) { ----------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 13/6) (n + 2) (n + 3) { { (n/3 - 1/3) { (27/4) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) (2 n + 5) , { ----------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 11/6) (n + 2) { { (n/3 - 2/3) { 3 (27/4) GAMMA(n/3 + 4/3) GAMMA(n/3 + 2/3) { ----------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 3/2) { 9 binomial(n, n/3) (n/3)! (n + 1) { --------------------------------- irem(n, 3) = 0 { 2 n + 3 { { 3 (n/3 + 2/3)! binomial(n + 2, n/3 + 2/3) } { ----------------------------------------- irem(n, 3) = 1 { n + 2 { { 3 (n/3 + 1/3)! binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 "A255139" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| , n!} \ 2 / \ 2 / "A255197" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A255673" LREtools/SearchTable: "SearchTable successful" binomial(3 n, n) hypergeom([- n/2, - n/2 + 1/2], [2 n + 2], 4) {--------------------------------------------------------------} 2 n + 1 "A255688" LREtools/SearchTable: "SearchTable successful" n n 2 ((n + 1) LegendreQ(n + 1, 2) + (-5 n - 2) LegendreQ(n, 2)) 2 ((5 n + 2) LegendreP(n, 2) + (-n - 1) LegendreP(n + 1, 2)) {-------------------------------------------------------------, - -------------------------------------------------------------} n n "A255806" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -3) + (-n - 4) LaguerreL(n, -3)) n! {-------------------------------------------------------------} n "A255807" LREtools/ReduceToOrderTwo: "Checking Symmetric Cube... (can be time consuming...)" LREtools/ReduceToOrderTwo: "Galois group is Sp4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A255819" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A255839" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A256031" n {n! (3 n + 4), (-1) n! n} "A256092" n n n 8 GAMMA(n + 1/4) 8 GAMMA(n + 3/4) {2 binomial(2 n, n), -----------------, -----------------} GAMMA(n + 1) GAMMA(n + 1) "A256093" n n n 8 GAMMA(n - 3/4) 8 GAMMA(n - 1/4) 2 binomial(2 n, n) {-----------------, -----------------, 1/2 -------------------} GAMMA(n + 1) GAMMA(n + 1) n - 1/2 "A256169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 3)} "A256456" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n + 3) { 2 binomial(n, n/2) n { -------------------------------- n::even { -------------------- n::even (2 n + 1) binomial(2 n, n) n { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {----------------------------, { , { , (n + 3) (n + 2) (n + 1) { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (2 n + 3) { 2 2 { ------------------------------------ n::odd { ---------------------------------- n::odd { n + 3 { (n + 2) binomial(n - 1, n/2 - 1/2) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 { { { 0 irem(n, 4) = 2, { (n/2 - 1) , { { 2 GAMMA(n/4 + 1/4) { n { --------------------------- irem(n, 4) = 2 { 2 { GAMMA(n/4 + 7/4) { 1/2 ---------------------------------------------- irem(n, 4) = 3 { { (n - 1) (n + 3) binomial(n/2 - 3/2, n/4 - 3/4) { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 0 { (n/2) { { 2 GAMMA(n/4 + 1/4) { 4 binomial(n/2 - 1/2, n/4 - 1/4) { ----------------------- irem(n, 4) = 0 { -------------------------------- irem(n, 4) = 1 { GAMMA(n/4 + 7/4) { n + 3 , { } { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 2 { 0 irem(n, 4) = 3 { { 0 irem(n, 4) = 3 "A256467" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) (n - 2) LaguerreL(n + 1, 1) - n (n - 4) LaguerreL(n, 1)) n! {--------------------------------------------------------------------------} n "A256506" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) (4 (n + 1) BesselI(n + 1/2, 1) + (-2 n - 3) BesselI(n - 1/2, 1)), (-1) (4 (n + 1) BesselK(n + 1/2, -1) + (-2 n - 3) BesselK(n - 1/2, -1))} "A256644" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 4 { 2 binomial(n, n/2) (5 n - 2) { -------------------------- n::even { ---------------------------- n::even { n (n + 1) binomial(n, n/2) { (n + 2) (n - 1) {1, { , { } { (2 n + 2) { 16 binomial(n - 1, n/2 - 1/2) { 2 2 (5 n - 2) { ----------------------------- n::odd { -------------------------------------------------- n::odd { n + 1 { (n - 1) (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A256710" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 ) BesselJ(n - 1/2, -2 ), (-2 ) BesselY(n - 1/2, -2 )} "A256880" n (n + 1) (-1) n! (n + 1) n! (2 n + 3) {----------------, --------------------} n + 2 n + 2 "A256881" n (-1) n! n! (2 n + 3) {--------, ------------} n + 2 n + 2 "A256938" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A256939" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A256943" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n (-1) ((5 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------, (n + 2) (n + 3) { n { 4 { 2 binomial(n, n/2) { -------------------------------- n::even { - ------------------ n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 { , { } { (2 n - 2) { 2 binomial(n + 1, n/2 + 1/2) { 2 { ---------------------------- n::odd { - ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A257072" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(23/6 - 1/6 I 11 ) , (23/6 + 1/6 I 11 ) , (23/6 - 1/6 I 11 ) | ) (23/6 + 1/6 I 11 ) (23/6 - 1/6 I 11 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) || | \ (23/6 + 1/6 I 11 ) ((8 n2 + 3) hypergeom([-1/2, -n2 - 1], [1], -8) + (-8 n2 - 7) hypergeom([-1/2, -n2], [1], -8))|| | ) ----------------------------------------------------------------------------------------------------------------------------||} | / n2 + 2 || |----- || \n2 = 0 // "A257104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 5 _Z + 7 _Z - 2 _Z + 1 "A257178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A257290" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A257300" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 3 _Z + 3 _Z - _Z + 1 "A257363" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 3 2 n 3 2 n 3 2 n {RootOf(3 _Z - 18 _Z + 21 _Z + 19, index = 1) , RootOf(3 _Z - 18 _Z + 21 _Z + 19, index = 2) , RootOf(3 _Z - 18 _Z + 21 _Z + 19, index = 3) } "A257386" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n RootOf(%1, index = 7) } 7 6 5 4 3 2 %1 := _Z - 6 _Z + 9 _Z + 6 _Z - 19 _Z + 4 _Z + 5 _Z + 1 "A257388" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A257389" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A257390" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A257515" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A257516" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A257517" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A257519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 8" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n n RootOf(%1, index = 7) , RootOf(%1, index = 8) } 8 7 6 5 4 3 2 %1 := _Z - 7 _Z + 16 _Z - 11 _Z - 4 _Z + 5 _Z - _Z + _Z + 1 "A257520" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {1, (-1) hypergeom([1/2, -n], [1], 4)} "A257546" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {{ , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A257561" memory used=166828.7MB, alloc=3223.5MB, time=1326.73 memory used=167993.6MB, alloc=3223.5MB, time=1340.05 memory used=168830.9MB, alloc=3223.5MB, time=1347.39 memory used=169695.6MB, alloc=3223.5MB, time=1354.58 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" 1/2 n 1/2 n {(4 - 3 2 ) , (4 + 3 2 ) } "A257596" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1 - 2 ) , (1 + 2 ) , (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) 2 2 || | \ (-1) (1 + 2 ) ((2 n2 + 2 n2 - 3) hypergeom([1/2, -n2 - 1], [1], 4) + (-6 n2 - 24 n2 - 27) hypergeom([1/2, -n2], [1], 4))|| | ) ---------------------------------------------------------------------------------------------------------------------------------------|| | / (n2 + 2) (n2 + 3) (n2 + 4) || |----- || \n2 = 0 // } "A257838" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 1) binomial(2 n2, n2) n2|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) ----------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A257859" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n + 1/2, -1), (-1) BesselY(n + 1/2, -1)} "A257995" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A258143" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 8 binomial(n, n/2) (n + 1) n { (n + 1) binomial(n, n/2) { -------------------------- n::even {1, 2 , { , { n + 2 } { (2 n - 2) { { 2 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A258144" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (n/2) { 2 (-16) { 2 (-1) binomial(n, n/2) n { ------------------------ n::even { ------------------------------ n::even { binomial(n, n/2) (n + 1) { n + 2 {1, { , { } { (n/2 + 1/2) { (n/2 - 1/2) { (-16) n { 4 (-1) binomial(n - 1, n/2 - 1/2) n { ------------------------------------------ n::odd { ---------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) (n + 2) (n + 1) { n + 1 "A258172" LREtools/SearchTable: "SearchTable successful" (-n) 2 (LaguerreL(n + 1, -2) - 3 LaguerreL(n, -2)) binomial(2 n, n) n! {---------------------------------------------------------------------} n "A258213" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { (-n) 2 2 2 { ((n/2)!) (n + 2) n::even { 4 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) n::even {{ , { } { 2 { (-2 n + 2) 2 2 2 { (n/2 + 3/2) ((n/2 + 1/2)!) n::odd { 2 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 2) n::odd "A258390" n (2 n + 3) (2 n + 1) binomial(2 n, n) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) {------------------------------------, ---------------------------------------} (n + 3) (n + 2) (n + 1) (n + 3) (n + 2) (n + 1) "A258391" n n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (2 n + 5) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) (2 n + 5) (2 n + 3) (2 n + 1) 3 binomial(2 n, n) {----------------------------------------------, -------------------------------------------------, ------------------------------------------------- (n + 4) (n + 3) (n + 2) (n + 1) (n + 4) (n + 3) (n + 2) (n + 1) (n + 4) (n + 3) (n + 2) (n + 1) } "A258392" n (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) binomial(2 n, n) (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) 2 binomial(2 n, n) {--------------------------------------------------------, -----------------------------------------------------------, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n n (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) 3 binomial(2 n, n) (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) 4 binomial(2 n, n) -----------------------------------------------------------, -----------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A258393" n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) {------------------------------------------------------------------, ---------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 3 binomial(2 n, n) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 4 binomial(2 n, n) ---------------------------------------------------------------------, ---------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 5 binomial(2 n, n) ---------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A258394" (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {-----------------------------------------------------------------------------, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) --------------------------------------------------------------------------------, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 3 binomial(2 n, n) --------------------------------------------------------------------------------, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 4 binomial(2 n, n) --------------------------------------------------------------------------------, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 5 binomial(2 n, n) --------------------------------------------------------------------------------, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 6 binomial(2 n, n) --------------------------------------------------------------------------------} (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A258395" (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {----------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) -------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 3 binomial(2 n, n) -------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 4 binomial(2 n, n) -------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 5 binomial(2 n, n) -------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 6 binomial(2 n, n) -------------------------------------------------------------------------------------------, (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) n (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 7 binomial(2 n, n) -------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 8) (n + 7) (n + 6) (n + 3) (n + 2) (n + 1) "A258396" (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {---------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 2 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 3 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 4 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 5 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 6 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 7 binomial(2 n, n) ------------------------------------------------------------------------------------------------------, (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n (2 n + 15) (2 n + 13) (2 n + 11) (2 n + 9) (2 n + 7) (2 n + 5) (2 n + 3) (2 n + 1) 8 binomial(2 n, n) ------------------------------------------------------------------------------------------------------} (n + 6) (n + 7) (n + 8) (n + 9) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) "A258416" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" (2 n + 3) (2 n + 1) binomial(2 n, n) {------------------------------------} (n + 3) (n + 2) (n + 1) "A258431" n {4 , n binomial(2 n, n)} "A258490" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {1, 9 , 9 | ) --------------------------------------------------------------------|, ----------------------------------------------} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) | (n + 3) (n + 2) (n + 1) |----- | \n1 = 0 / "A258491" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n n | \ 2 3 (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (2 n1 + 7) binomial(2 n1, n1)| {1, 9 , 16 , 9 | ) -------------------------------------------------------------------------------|, | / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / /n - 1 \ |----- n1 (-4 n1 - 4) | n | \ 3 2 (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (2 n1 + 7) binomial(2 n1, n1)| 16 | ) -------------------------------------------------------------------------------|, | / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) binomial(2 n, n) --------------------------------------------------------} (n + 4) (n + 3) (n + 2) (n + 1) "A258664" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n + 3) BesselI(n, 2) + (n - 4) BesselI(n - 1, 2)), (-1) ((n + 3) BesselK(n, -2) + (n - 4) BesselK(n - 1, -2))} "A258723" memory used=170959.6MB, alloc=3223.5MB, time=1363.30 LREtools/SearchTable: "SearchTable successful" 1/2 1/2 1/2 n 3 1/2 n 3 {(4 3 ) LegendreP(n, ----), (4 3 ) LegendreQ(n, ----)} 2 2 "A258902" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |21 15 5 | 125 35 5 {|-- - -------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], --- + -------)} \19 19 / 38 38 "A258916" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n - 1) BesselI(n, 1) - BesselI(n - 1, 1)), (-1) ((2 n - 1) BesselK(n, -1) - BesselK(n - 1, -1))} "A258973" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A259056" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { (n/2 + 1) (-8) (n/2)! n::even} { (n/2 - 1/2) { { (n + 2) n (-2) (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { 0 n::odd "A259104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ n1 + 3 | {n! (n + 3 n + 1), n! (n + 3 n + 1) | ) -------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 3 n1 + 4) (n1 + 3 n1 + 1)| \n1 = 0 / "A259206" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {1, (-1) } "A259239" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A259554" LREtools/SearchTable: "SearchTable successful" (5 n + 3) hypergeom([2 n + 3, -n - 1], [1], -1) + (-2 n - 1) hypergeom([-n, 2 n + 1], [1], -1) {----------------------------------------------------------------------------------------------} 17 n + 11 "A259757" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A259775" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 12 binomial(n, n/2) (n + 1) (n + 2) { 4 (n + 2) (3 n + 7) { ----------------------------------- n::even { 1/2 ---------------------------------------- n::even { (n + 6) (n + 4) { (n + 1) (n + 3) (n + 5) binomial(n, n/2) {{ , { } { 2 binomial(n + 1, n/2 + 1/2) (n + 2) (3 n + 7) { (2 n - 2) { ---------------------------------------------- n::odd { 3 2 (n + 1) (n + 2) { (n + 5) (n + 3) { -------------------------------------------- n::odd { n (n + 4) (n + 6) binomial(n - 1, n/2 - 1/2) "A259834" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (2 n1 - 3) n1 (n1 + 1) | n! (2 n - 1) | ) -------------------------------| | / (n1 + 1)! (2 n1 + 1) (2 n1 - 1)| |----- | n! (2 n - 1) \n1 = 0 / {------------, -----------------------------------------------------} (n - 1) n (n - 1) n "A259845" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A259859" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) ((n + 2 n - 1) BesselJ(n, -2) + (n + 2) BesselJ(n - 1, -2)), (-1) ((n + 2 n - 1) BesselY(n, -2) + (n + 2) BesselY(n - 1, -2))} "A259877" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (- n/2) { (n/2) { 2 (6 n + 6) binomial(n, n/2) (n/2)! n::even { 1/2 2 (n/2)! (n + 1) (n + 2) n::even {{ , { } { (- n/2 - 1/2) { (n/2 - 1/2) { 2 (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd { 2 (n/2 - 1/2)! (3 n + 3) n::odd "A259900" LREtools/SearchTable: "SearchTable successful" n 2 2 (-2) (2 (4 n - 9 n + 7) (n + 1) LaguerreL(n + 1, - 5/4 - n, 1/4) + (2 n - 5 n + 7) LaguerreL(n, - 1/4 - n, 1/4)) n! {----------------------------------------------------------------------------------------------------------------------} n (n - 1) "A259901" LREtools/SearchTable: "SearchTable successful" n 2 2 (-2) (2 (4 n - 9 n + 7) (n + 1) LaguerreL(n + 1, - 5/4 - n, 1/4) + (2 n - 5 n + 7) LaguerreL(n, - 1/4 - n, 1/4)) n! {----------------------------------------------------------------------------------------------------------------------} n (n - 1) "A259902" LREtools/SearchTable: "SearchTable successful" n 2 2 (-2) ((4 n - 3 n - 2) (n + 1) LaguerreL(n + 1, - 5/4 - n, 1/4) + (n - n - 1) LaguerreL(n, - 1/4 - n, 1/4)) n! {----------------------------------------------------------------------------------------------------------------} n (n - 1) "A259903" LREtools/SearchTable: "SearchTable successful" n 2 2 (-2) ((4 n - 3 n - 2) (n + 1) LaguerreL(n + 1, - 5/4 - n, 1/4) + (n - n - 1) LaguerreL(n, - 1/4 - n, 1/4)) n! {----------------------------------------------------------------------------------------------------------------} n (n - 1) "A259904" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 1) BesselI(n, 1) - BesselI(n - 1, 1)), (-1) ((2 n + 1) BesselK(n, -1) - BesselK(n - 1, -1))} "A259905" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 1) BesselI(n, 1) - BesselI(n - 1, 1)), (-1) ((2 n + 1) BesselK(n, -1) - BesselK(n - 1, -1))} "A259906" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n - 1) BesselI(n, 1) - BesselI(n - 1, 1)), (-1) ((2 n - 1) BesselK(n, -1) - BesselK(n - 1, -1))} "A259914" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n + 2) (13 n + 60 n + 71) { 4 binomial(n, n/2) (13 n + 24) (n + 1) (n + 2) { ------------------------------------------------ n::even { ---------------------------------------------- n::even { (n + 1) (n + 3) (n + 5) (n + 7) binomial(n, n/2) { (n + 4) (n + 6) (n + 8) {{ , { { (2 n + 2) { 2 { 2 (n + 2) (13 n + 24) { 8 binomial(n - 1, n/2 - 1/2) n (n + 2) (13 n + 60 n + 71) { 1/2 -------------------------------------------------- n::odd { ---------------------------------------------------------- n::odd { (n + 4) (n + 6) (n + 8) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) (n + 5) (n + 7) } "A260076" 3 6 {1, (n!) , (n!) } "A260153" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { n { -------------------------- n::even { 4 GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) { 2 2 { ---------------------------------- n::even { (n + 1) binomial(n, n/2) { GAMMA(n/2 + 3/2) GAMMA(n/2 + 7/6) {{ , { , { (4 n - 4) { (2 n - 2) { 4 2 { 4 2 GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) { ------------------------------ n::odd { ---------------------------------------------- n::odd { 2 2 { GAMMA(n/2 + 1) GAMMA(n/2 + 2/3) { n binomial(n - 1, n/2 - 1/2) { n { 4 4 GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) { 2 { -------------------------------------- n::even { 4 binomial(n, n/2) n::even { GAMMA(n/2 + 1) GAMMA(n/2 + 2/3) { , { } { 2 { (2 n + 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) { ------------------------------------------ n::odd { GAMMA(n/2 + 3/2) GAMMA(n/2 + 7/6) "A260154" 2 n n binomial(2 n, n) 16 GAMMA(n + 1/2) GAMMA(n + 5/6) 16 GAMMA(n + 1/2) GAMMA(n + 7/6) {-----------------, ---------------------------------, ---------------------------------} 2 GAMMA(n + 2) GAMMA(n + 5/3) GAMMA(n + 2) GAMMA(n + 4/3) (n + 1) "A260155" 2 n 2 n 2 (2 n + 1) binomial(2 n, n) 16 GAMMA(n + 1/2) GAMMA(n + 5/6) (12 n + 30 n + 5) 16 GAMMA(n + 1/2) GAMMA(n + 7/6) (24 n + 60 n + 29) {---------------------------, ----------------------------------------------------, -----------------------------------------------------} 2 GAMMA(n + 3) GAMMA(n + 8/3) GAMMA(n + 3) GAMMA(n + 7/3) (n + 2) (n + 1) "A260346" n 3 2 n (n - 1) (n - 2) (n - 3) binomial(2 n, n) {4 (4 n - 30 n + 101 n + 249), ------------------------------------------} (2 n - 3) (2 n - 1) "A260478" 4 {1, (n!) } "A260667" LREtools/SolveLRE: "Reduced the order of" (n+5)*(1100*n^6+13420*n^5+67529*n^4+179205*n^3+264247*n^2+205123*n+65484)*(n+3)^2*(n+4)^3*E^3-(134200*n ^10+3514940*n^9+41024148*n^8+280825406*n^7+1247906397*n^6+3759190146*n^5+7769520497*n^4+10871738858*n^3+9850091004*n^2+5214671088*n+1224343776)*(n+3) ^2*E^2-(134200*n^10+3517140*n^9+41128208*n^8+282424322*n^7+1260428091*n^6+3817331810*n^5+7939416936*n^4+11187863016*n^3+10213623325*n^2+5450040008*n+ 1289902112)*(n+2)^2*E+(1100*n^6+20020*n^5+151129*n^4+605521*n^3+1357736*n^2+1615048*n+796108)*(n+2)^3*(n+1)^3 "to two: Symmetric square" (n+2)^2* E^2+(-11*n^2-33*n-25)*E-(n+1)^2 LREtools/SearchTable: "SearchTable successful" 2 2 2 {- binomial(2 n, n) (4 (n + 11) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) - 2 (2 n + 1) (11 n + 16) (n + 1) hypergeom([-n, -n, -n], [1, -2 n], 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) 3 2 / 2 - (n + 1) hypergeom([-n, -n, -n], [1, -2 n], 1) ) / ((n + 2) (n + 1) )} / "A260668" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {(n - 1) n} "A260697" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 2 {(n - 3) (n + 3 n + 8)} "A260769" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" {{ n { -8 2 binomial(n, n/2) hypergeom([n + 1, - n/2], [n/2 + 1], -1) , n::even (n + 1) 2 2 8 2 (6 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-17 n - 68 n - 66) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) { { , { binomial(n + 1, n/2 + 1/2)/((n + 1) (5 n + 12)) , n::odd { { n 2 2 8 8 (6 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-17 n - 68 n - 66) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) ------------------------------------------------------------------------------------------------------------------------------------------ , 2 (n + 1) (5 n + 12) binomial(n, n/2) n::even (3 n - 3) 8 2 hypergeom([n + 1, - n/2], [n/2 + 1], -1) - ----------------------------------------------------- , n::odd} n binomial(n - 1, n/2 - 1/2) "A260770" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n 2 {{ - 16 2 (2 (11 n + 13) (n + 3) hypergeom([n + 5, - n/2 - 2], [n/2 + 3], -1) { 3 2 / 2 + (-123/2 n - 883/2 n - 981 n - 643) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) binomial(n, n/2) / ((n + 2) (5 n + 17)) , n::even / (n - 1) 32 2 n (n + 2) ((2 n + 4) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-4 n - 9) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) { / 2 { binomial(n - 1, n/2 - 1/2) / ((n + 1) (5 n + 12)) , n::odd, { / { { n 8 8 (n + 2) ((2 n + 4) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-4 n - 9) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) --------------------------------------------------------------------------------------------------------------------------------------- , n::even 2 (n + 1) (5 n + 12) binomial(n, n/2) (3 n + 3) 2 - 4 2 (2 (11 n + 13) (n + 3) hypergeom([n + 5, - n/2 - 2], [n/2 + 3], -1) 3 2 / 2 + (-123/2 n - 883/2 n - 981 n - 643) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) / ((n + 1) (n + 2) (5 n + 17) binomial(n + 1, n/2 + 1/2) / ) , n::odd} "A260771" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n 2 {{ - 16 2 (4 (n + 3) hypergeom([n + 5, - n/2 - 2], [n/2 + 3], -1) - 1/2 (3 n + 10) (7 n + 19) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) { / 2 binomial(n, n/2) / ((n + 2) (5 n + 17)) , n::even / (n - 1) 32 2 n ((2 n + 4) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-4 n - 9) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) { / 2 { binomial(n - 1, n/2 - 1/2) / ((n + 1) (5 n + 12)) , n::odd, { / { { n 16 8 ((2 n + 4) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) + (-4 n - 9) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) -------------------------------------------------------------------------------------------------------------------------------- , n::even 2 (n + 1) (5 n + 12) binomial(n, n/2) (3 n + 3) 2 8 2 (4 (n + 3) hypergeom([n + 5, - n/2 - 2], [n/2 + 3], -1) - 1/2 (3 n + 10) (7 n + 19) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) - ----------------------------------------------------------------------------------------------------------------------------------------------- 2 (n + 1) (n + 2) (5 n + 17) binomial(n + 1, n/2 + 1/2) , n::odd} "A260772" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n {{ 16 2 ((-4 n - 5) hypergeom([n + 1, - n/2], [n/2 + 1], -1) + (2 n + 2) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) binomial(n, n/2) { - ------------------------------------------------------------------------------------------------------------------------------------- , n::even { n (5 n + 7) (n + 1) - 8 2 2 (4 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) - 1/2 (3 n + 7) (7 n + 12) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) { { , { binomial(n + 1, n/2 + 1/2)/(n (n + 1) (5 n + 12)) , n::odd { { n 2 8 8 (4 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) - 1/2 (3 n + 7) (7 n + 12) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) - ---------------------------------------------------------------------------------------------------------------------------------------------- 2 n (n + 1) (5 n + 12) binomial(n, n/2) , n::even (3 n - 3) 16 2 ((-4 n - 5) hypergeom([n + 1, - n/2], [n/2 + 1], -1) + (2 n + 2) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) - ---------------------------------------------------------------------------------------------------------------------------- , n::odd} 2 n (5 n + 7) binomial(n - 1, n/2 - 1/2) "A260773" memory used=172254.6MB, alloc=3223.5MB, time=1371.93 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n {{ 16 2 ((-4 n - 5) hypergeom([n + 1, - n/2], [n/2 + 1], -1) + (2 n + 2) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) binomial(n, n/2) { ------------------------------------------------------------------------------------------------------------------------------------- , n::even { n (5 n + 7) (n + 1) - 8 2 2 (4 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) - 1/2 (3 n + 7) (7 n + 12) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) { { , { binomial(n + 1, n/2 + 1/2)/(n (n + 1) (5 n + 12)) , n::odd { { n 2 8 8 (4 (n + 2) hypergeom([n + 4, - n/2 - 3/2], [n/2 + 5/2], -1) - 1/2 (3 n + 7) (7 n + 12) hypergeom([n + 2, - n/2 - 1/2], [n/2 + 3/2], -1)) - ---------------------------------------------------------------------------------------------------------------------------------------------- 2 n (n + 1) (5 n + 12) binomial(n, n/2) , n::even (3 n - 3) 16 2 ((-4 n - 5) hypergeom([n + 1, - n/2], [n/2 + 1], -1) + (2 n + 2) hypergeom([n + 3, - n/2 - 1], [n/2 + 2], -1)) ---------------------------------------------------------------------------------------------------------------------------- , n::odd} 2 n (5 n + 7) binomial(n - 1, n/2 - 1/2) "A260774" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 3 ) 3 (LegendreP(n, 3 I) + 3 LegendreP(n + 1, 3 I) I), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 3 ) 3 (LegendreQ(n, 3 I) + 3 LegendreQ(n + 1, 3 I) I)} "A260878" {1, binomial(2 n, n)} "A261058" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A261193" {1, n!} "A261196" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n (-2 I) binomial(2 n, n) (2 I) binomial(2 n, n) {1/2 ------------------------, 1/2 -----------------------} n - 1/2 n - 1/2 "A261207" LREtools/SearchTable: "SearchTable successful" (n + 1) ((2 n + 1) LegendreP(n + 1, 3) + (-10 n - 3) LegendreP(n, 3)) (n + 1) ((2 n + 1) LegendreQ(n + 1, 3) + (-10 n - 3) LegendreQ(n, 3)) {---------------------------------------------------------------------, ---------------------------------------------------------------------} (n - 1) n (n - 1) n "A261266" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 /3 n1\ n - 1 /3 n1\ ----- |----| ----- |----| \ \ 2 / 1/2 1/2 1/2 \ \ 2 / 1/2 1/2 1/2 {1, ) 2 (-LegendreP(n1, 2 ) + 2 LegendreP(n1 + 1, 2 )), ) 2 (-LegendreQ(n1, 2 ) + 2 LegendreQ(n1 + 1, 2 ))} / / ----- ----- n1 = 0 n1 = 0 "A261317" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A261428" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A261429" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A261471" 2 {1, (n!) , n!} "A261588" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {5 } "A261589" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {6 } "A261668" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 8)| {1, (1/2) , (1/2) | ) ------------------------------------------------------------------------|} | / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A261681" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, 2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A261682" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 3/2 ------------------------ n::even { 8 binomial(n, n/2) (n + 1) n { (n + 1) binomial(n, n/2) { -------------------------- n::even {1, 2 , { , { n + 2 } { (2 n - 2) { { 2 2 (n + 1) { 6 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A262020" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I)} "A262033" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 (n/2)! { 4 binomial(n, n/2) (n/2)! { --------- n::even { ------------------------- n::even { n + 1 { n + 2 {{ , { } { (n - 1) { 2 (n/2 + 1/2)! binomial(n + 1, n/2 + 1/2) { 2 2 (n/2 - 1/2)! { ----------------------------------------- n::odd { ----------------------- n::odd { n + 1 { n + 2 "A262034" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 (n/2)! { 2 binomial(n, n/2) (n/2)! { --------------- n::even { ------------------------- n::even { (n + 1) (n + 3) { n + 2 {{ , { } { (n - 1) { 2 binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! { 2 (n/2 - 1/2)! { ----------------------------------------- n::odd { --------------------- n::odd { (n + 1) (n + 3) { n + 2 "A262376" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, (n + 1) n, ) (n1 + 1) n1!} / ----- n1 = 0 "A262407" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (-5 n - 2) hypergeom([-n, -n, -n], [1, 1], -1) {--------------------------------------------------------------------------------------------------------} n "A262441" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A262480" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { 2 (n/2)! n::even { (- n/2) { { 2 binomial(n, n/2) (n/2)! n::even {n!, { (n/2 + 1/2) , { } { 2 (n/2 + 1/2)! { (- n/2 + 1/2) { ------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n + 1 "A262601" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ n1! | {(n + 1) (n!) n, (n + 1) (n!) n | ) ---------------|} | / 2 | |----- ((n1 + 1)!) n1| \n1 = 0 / "A262607" LREtools/SearchTable: "SearchTable successful" (n + 2) LegendreP(n + 1, 3) + (-5 n - 6) LegendreP(n, 3) (n + 2) LegendreQ(n + 1, 3) + (-5 n - 6) LegendreQ(n, 3) {--------------------------------------------------------, --------------------------------------------------------} n n "A262664" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {(1/2) , (1/2) | ) 2 ((6 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -4) + (-6 n1 - 4) hypergeom([-1/2, -n1], [1], -4))|} | / | |----- | \n1 = 0 / "A262720" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 4 3 2 3 2 n (-1) ((n + 9 n + 37 n + 71 n + 54) hypergeom([1/2, -n - 1], [1], 4) + 3 (n + 12 n + 43 n + 50) (n + 1) hypergeom([1/2, -n], [1], 4)) {(-1) , ------------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A262732" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /12500\(n/2 - 1/2) {{ |-----| GAMMA(n/2 + 1/10) GAMMA(n/2 + 3/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + 7/10) , { \ 27 / { ------------------------------------------------------------------------------------------ n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) GAMMA(1/6 + n/2) { /12500\(n/2) { |-----| GAMMA(n/2 + 1/10) GAMMA(n/2 + 9/10) GAMMA(n/2 + 7/10) GAMMA(n/2 + 3/10) { \ 27 / { ------------------------------------------------------------------------------------ n::even} { GAMMA(n/2 + 1) GAMMA(1/6 + n/2) GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) { { 0 n::odd "A262957" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A262961" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A263064" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A263065" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A263102" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \ \\ | | |----- | || | | n2 | \ 1 | || | | (-1) (n2 + 1) | ) ---------| n2!|| n - 1 n - 1 |n1 - 1 | | / (n3 + 1)!| || n - 1 ----- ----- |----- | |----- | || ----- \ n1 \ n1 | \ | \n3 = 0 / || \ {1, ) (-1) n1!, ) (-1) n1! | ) |- --------------------------------------||, ) n1!} / / | / \ (n2 + 1)! /| / ----- ----- |----- | ----- n1 = 0 n1 = 0 \n2 = 0 / n1 = 0 "A263134" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) {1, ) ---------------------------------------------------} / (n1 + 1) (2 n1 + 3) (2 n1 + 1) ----- n1 = 0 "A263173" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A263316" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (2 n + 1)} "A263384" n n {(n + 3) (n + 2) (n + 1) 2 n!, (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) (3 n + 7)} "A263529" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I)} "A263656" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A263687" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 2 ), GAMMA(n + 2 )} "A263688" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 2 ), GAMMA(n + 2 )} "A263801" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 {1, ) (2 n1 + 1) (-1/2) n1! binomial(2 n1, n1)} / ----- n1 = 0 "A263823" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\n1\ / / 1/2 \n1\ |n - 1 | 5 | | |n - 1 |5 | | |----- |1/2 - ----| | |----- |---- + 1/2| | | \ \ 2 / | | \ \ 2 / | {n! | ) --------------|, n! | ) --------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A263841" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 4) hypergeom([1/2, -n - 1], [1], 4) + (-n - 2) hypergeom([1/2, -n], [1], 4))} "A263843" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 4 binomial(---, n/2) { 2 { --------------------- n::even { n + 1 {{ , { (2 n - 2) 3 n { 4 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { - ----------------------------------------------------- n::odd { n (n + 1) { 3 n 3 n { 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { - --------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { 3 n 3 n { binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ----------------------------------------------------------- n::odd { binomial(n + 1, n/2 + 1/2) (3 n + 2) "A263895" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ (-1) 2 (n1 + 1) | {2 n! (2 n + 1), 2 n! (2 n + 1) | ) -------------------------------|} | / (2 n1 + 1) (2 n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A263986" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2 \n | 5 | |5 | binomial(2 n, n) {|1/2 - ----| , |---- + 1/2| , ----------------} \ 2 / \ 2 / n + 1 "A264152" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { {{ (n/2 - 1/2) , { (2 n - 1) 2 (n/2 - 1/2)! binomial(2 n - 2, n - 1) { ------------------------------------------------------------ n::odd { n { (- n/2) 3 n { 2 binomial(2 n, n/2) binomial(---, n/2) (n/2)! n::even { 2 } { { 0 n::odd "A264557" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 3 { 3 { GAMMA(n/3 + 4/3) irem(n, 3) = 0 { 3 GAMMA(5/3 + n/3) { { ------------------- irem(n, 3) = 0 { 3 { 3 { n + 2 { 9 GAMMA(n/3 + 2) { ((n/3)!) (n/3 + 1) irem(n, 3) = 0 { { ----------------- irem(n, 3) = 1 { { 3 {{ 2 , { 3 2 , { GAMMA(n/3 + 4/3) irem(n, 3) = 1} { (n + 3) { 1/9 ((n/3 - 1/3)!) (n + 2) irem(n, 3) = 1 { { { { 3 { 3 { 3 3 { 9 GAMMA(n/3 + 2) { 3 GAMMA(5/3 + n/3) { 1/27 ((n/3 - 2/3)!) (n + 1) irem(n, 3) = 2 { ----------------- irem(n, 3) = 2 { ------------------- irem(n, 3) = 2 { 2 { n + 2 { (n + 3) "A264607" LREtools/SearchTable: "SearchTable successful" {(3 (6 n + 5) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) - (9 n + 7) (3 n + 1) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/((3 n + 1) (5 n + 4))} "A264608" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A264635" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 3 { 4 { 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 9/4) { (n/4 + 1) ((n/4)!) irem(n, 4) = 0 { ------------------------------------ irem(n, 4) = 0 { { n + 5 { 4 { { 16 ((n/4 + 3/4)!) { 3 { ------------------ irem(n, 4) = 1 { GAMMA(n/4 + 2) GAMMA(n/4 + 1) irem(n, 4) = 1 { 2 { {{ (n + 3) , { 3 , { { 64 GAMMA(n/4 + 7/4) GAMMA(n/4 + 11/4) { 4 { -------------------------------------- irem(n, 4) = 2 { 4 ((n/4 + 1/2)!) { 2 { ----------------- irem(n, 4) = 2 { (n + 3) (n + 7) { n + 2 { { { 3 { 4 { 16 GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/2) { ((n/4 + 1/4)!) irem(n, 4) = 3 { ------------------------------------- irem(n, 4) = 3 { (n + 2) (n + 6) { (-n) 3 4 4 { 1/2 4 (n + 2) binomial(n/2, n/4) ((n/4)!) irem(n, 4) = 0 { { (-2 n + 2) 4 4 4 { 1/8 2 (n + 1) binomial(n/2 - 1/2, n/4 - 1/4) ((n/4 - 1/4)!) irem(n, 4) = 1 { , { (-n) 4 4 4 { 1/2 4 n binomial(n/2 - 1, n/4 - 1/2) ((n/4 - 1/2)!) (n + 4) irem(n, 4) = 2 { { (-2 n - 2) 2 4 4 { 2 2 (n + 3) binomial(n/2 + 1/2, n/4 + 1/4) ((n/4 + 1/4)!) irem(n, 4) = 3 { 3 { 64 GAMMA(n/4 + 7/4) GAMMA(n/4 + 11/4) { -------------------------------------- irem(n, 4) = 0 { 2 { (n + 3) (n + 7) { { 3 { 16 GAMMA(n/4 + 3/2) GAMMA(n/4 + 5/2) { ------------------------------------- irem(n, 4) = 1} { (n + 2) (n + 6) { { 3 { 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 9/4) { ------------------------------------ irem(n, 4) = 2 { n + 5 { { 3 { GAMMA(n/4 + 2) GAMMA(n/4 + 1) irem(n, 4) = 3 "A264656" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { 5 { ((n/5)!) (n/5 + 1) irem(n, 5) = 0 { %4 irem(n, 5) = 0 { { %3 irem(n, 5) = 0 { %2 irem(n, 5) = 0 { { 5 2 { { { %5 irem(n, 5) = 1 { 1/25 ((n/5 - 1/5)!) (n + 4) irem(n, 5) = 1 { %4 irem(n, 5) = 1 { %3 irem(n, 5) = 1 { { { { {{ %1 irem(n, 5) = 2, { 5 3 , { %5 irem(n, 5) = 2, { %4 irem(n, 5) = 2, { { 1/125 ((n/5 - 2/5)!) (n + 3) irem(n, 5) = 2 { { { %2 irem(n, 5) = 3 { { %1 irem(n, 5) = 3 { %5 irem(n, 5) = 3 { { 5 4 { { { %3 irem(n, 5) = 4 { 1/625 ((n/5 - 3/5)!) (n + 2) irem(n, 5) = 3 { %2 irem(n, 5) = 4 { %1 irem(n, 5) = 4 { { 5 5 { 1/3125 ((n/5 - 4/5)!) (n + 1) irem(n, 5) = 4 { %1 irem(n, 5) = 0 { { %2 irem(n, 5) = 1 { { %3 irem(n, 5) = 2} { { %4 irem(n, 5) = 3 { { %5 irem(n, 5) = 4 5 125 GAMMA(n/5 + 9/5) %1 := --------------------- 3 (n + 4) 5 25 GAMMA(n/5 + 8/5) %2 := -------------------- 2 (n + 3) 5 5 GAMMA(n/5 + 7/5) %3 := ------------------- n + 2 5 %4 := GAMMA(n/5 + 6/5) 5 625 GAMMA(n/5 + 2) %5 := ------------------- 4 (n + 5) "A264717" LREtools/SearchTable: "SearchTable not successful" {} "A264955" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {1, (-2) } "A264960" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n { (4 n - 4) { 2 {4 , { 2 , { binomial(n, n/2) n::even} { ------------------------------ n::odd { { 2 2 { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A265033" memory used=173619.6MB, alloc=3227.3MB, time=1382.00 memory used=174389.2MB, alloc=3298.9MB, time=1392.36 memory used=175324.8MB, alloc=3291.3MB, time=1400.29 memory used=176220.1MB, alloc=3362.9MB, time=1411.15 LREtools/SearchTable: "SearchTable not successful" {} "A265118" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A265165" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 5 5 {GAMMA(n + 1/2 - ----), GAMMA(n + 1/2 + ----)} 2 2 "A265229" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 {1, ) (-1) (4 (n1 + 1) BesselI(n1 + 1/2, 1) + (-2 n1 - 3) BesselI(n1 - 1/2, 1)), / ----- n1 = 0 n - 1 ----- \ n1 2 ) (-1) (4 (n1 + 1) BesselK(n1 + 1/2, -1) + (-2 n1 - 3) BesselK(n1 - 1/2, -1))} / ----- n1 = 0 "A265233" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A265242" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A265376" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n (-I) n! (2 n + 1 + I) I n! (2 n + 1 - I) {----------------------, -------------------} n n "A265680" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" (2 n + 1) binomial(2 n, n) {--------------------------} n + 1 "A265864" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A265871" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A265919" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A265920" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {4 } "A265939" n {4 , binomial(2 n, n)} "A266083" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1| {n! | ) -----------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A266456" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A266656" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A266734" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A266922" memory used=177572.2MB, alloc=3355.3MB, time=1420.06 memory used=179031.3MB, alloc=3356.6MB, time=1428.86 memory used=180311.6MB, alloc=3435.2MB, time=1436.16 memory used=181669.0MB, alloc=3408.1MB, time=1445.13 memory used=183265.8MB, alloc=3457.8MB, time=1455.01 memory used=184664.4MB, alloc=3423.3MB, time=1464.19 memory used=186019.2MB, alloc=3423.6MB, time=1473.78 memory used=187065.0MB, alloc=3434.2MB, time=1480.97 memory used=188168.2MB, alloc=3515.7MB, time=1488.43 memory used=189008.4MB, alloc=3528.0MB, time=1496.24 memory used=189795.0MB, alloc=3465.1MB, time=1503.09 memory used=190885.8MB, alloc=3479.2MB, time=1511.52 memory used=193169.4MB, alloc=3417.4MB, time=1525.17 memory used=195672.3MB, alloc=3508.3MB, time=1540.60 memory used=198091.8MB, alloc=3457.4MB, time=1557.19 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=199732.5MB, alloc=3351.5MB, time=1568.76 memory used=201235.3MB, alloc=3359.4MB, time=1576.99 memory used=202622.3MB, alloc=3354.9MB, time=1584.51 memory used=203973.4MB, alloc=3356.8MB, time=1591.88 memory used=205233.0MB, alloc=3351.5MB, time=1598.38 memory used=206796.7MB, alloc=3351.5MB, time=1606.51 memory used=208418.4MB, alloc=3384.6MB, time=1615.16 memory used=209958.9MB, alloc=3373.3MB, time=1623.11 memory used=211430.3MB, alloc=3360.3MB, time=1631.21 memory used=212863.8MB, alloc=3362.5MB, time=1639.23 memory used=214377.8MB, alloc=3367.7MB, time=1647.62 memory used=215765.6MB, alloc=3366.7MB, time=1655.11 memory used=217159.9MB, alloc=3372.7MB, time=1662.74 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" { n { 4 4 { (n - 2) binomial(n, n/2) { ------------------ n::even { ------------------------ n::even { n binomial(n, n/2) {{ n + 2 , { } { { (2 n + 2) { 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 (n - 2) { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A267192" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A267532" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 {(1/2) (n!) binomial(2 n, n), n! binomial(2 n, n) hypergeom([-n], [n + 1], 1)} "A267991" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /n1 - 1 / /{ 0 n2::even\\\ ----- |----- | |{ ||| n n \ n1 | \ | n2 |{ n2 ||| {1, (-1) , 2 , ) (-1) | ) |-(-1) |{ 4 binomial(n2 - 1, ---- - 1/2) n2 (n2 + 2) |||, / | / | |{ 2 ||| ----- |----- | |{ ------------------------------------------ n2::odd ||| n1 = 0 \n2 = 0 \ \{ (n2 + 1) (n2 + 3) /// n - 1 /n1 - 1 / /{ n2 \\\ ----- |----- | |{ 4 (n2 + 2) ||| \ n1 | \ | n2 |{ 1/2 ------------------------------------ n2::even||| ) (-1) | ) |-(-1) |{ n2 |||} / | / | |{ (n2 + 1) (n2 + 3) binomial(n2, ----) ||| ----- |----- | |{ 2 ||| n1 = 0 |n2 = 0 | |{ ||| \ \ \{ 0 n2::odd /// "A268136" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | | \ 3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3)| {(2 n + 3) (2 n + 1), (2 n + 3) (2 n + 1) | ) -----------------------------------------|, | / (n1 + 2) (2 n1 + 5) (2 n1 + 1) | |----- | \n1 = 0 / /n - 1 \ |----- | | \ 3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3)| (2 n + 3) (2 n + 1) | ) -----------------------------------------|} | / (n1 + 2) (2 n1 + 5) (2 n1 + 1) | |----- | \n1 = 0 / "A268137" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" {LegendreP(n + 1, 3) - LegendreP(n, 3), LegendreQ(n + 1, 3) - LegendreQ(n, 3)} "A268138" LREtools/SolveLRE: "Reduced the order of" (n+5)*(2*n+5)*(n+4)^2*E^3-(2*n+7)*(n+4)*(35*n^2+177*n+224)*E^2+(2*n+5)*(n+2)*(35*n^2+243*n+422)*E-(2*n+ 7)*(n+1)*(n+2)^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E+n+1 LREtools/SearchTable: "SearchTable successful" 2 2 (n + 2) LegendreP(n + 1, 3) + (-6 n - 9) LegendreP(n, 3) LegendreP(n + 1, 3) + (n + 1) LegendreP(n, 3) {--------------------------------------------------------------------------------------------------------, n + 2 2 2 (n + 2) LegendreQ(n + 1, 3) + (-6 n - 9) LegendreQ(n, 3) LegendreQ(n + 1, 3) + (n + 1) LegendreQ(n, 3) --------------------------------------------------------------------------------------------------------, ( n + 2 (2 n + 4) LegendreP(n + 1, 3) LegendreQ(n + 1, 3) + (-6 n - 9) LegendreP(n, 3) LegendreQ(n + 1, 3) + (-6 n - 9) LegendreQ(n, 3) LegendreP(n + 1, 3) + (2 n + 2) LegendreP(n, 3) LegendreQ(n, 3))/(n + 2)} "A268163" LREtools/SearchTable: "SearchTable successful" / 1/2\n |12 9 3 | {|-- - ------| \11 11 / / 1/2 1/2 \ | 54 24 3 1/2 54 24 3 | |(33 n + 44) hypergeom([5/6, - 2/3 - n], [5/3], -- + -------) + 12 (4 + 3 3 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], -- + -------)| \ 11 11 11 11 / / 1/2\ | 43 24 3 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- -- + -------|/GAMMA(n + 2/3)} \ 11 11 / "A268208" LREtools/SearchTable: "SearchTable successful" 2 2 (2 n + 2 n + 5) LegendreP(n + 1, 3) + (-6 n - 18 n - 15) LegendreP(n, 3) {--------------------------------------------------------------------------, n (n + 2) (n - 1) 2 2 (2 n + 2 n + 5) LegendreQ(n + 1, 3) + (-6 n - 18 n - 15) LegendreQ(n, 3) --------------------------------------------------------------------------} n (n + 2) (n - 1) "A268297" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A268363" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 ) n!, (-2 ) n!} "A268369" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268400" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\ 1/2\ / / 1/2\ 1/2\ 1/2 n | 2 | 34 5 | 137 5 | 1/2 n | 2 | 34 5 | 137 5 | 1/2 n {(2 - 5 ) |n + |21 - -------| n + 347/5 - --------|, (2 + 5 ) |n + |21 + -------| n + 347/5 + --------|, (2 - 5 ) \ \ 5 / 5 / \ \ 5 / 5 / /n - 1 /n1 - 1 |----- |----- 1/2 2 1/2 | \ 1/2 n1 1/2 (-n1 - 1) 4 3 2 | \ (-34 5 n + 5 n - 137 5 + 105 n + 347) | ) (2 + 5 ) (2 - 5 ) (n1 + 44 n1 + 399 n1 + 1298 n1 + 1396) | ) | / | / |----- |----- \n1 = 0 \n2 = 0 / 1/2 1/2\ 1/2 (-n2 - 1) | 2 34 5 (n2 + 1) 137 5 | 2 (2 + 5 ) (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) |(n2 + 1) - ---------------- + 21 n2 + 452/5 - --------| (7 n2 + 30 n2 + 22) \ 5 5 / / 4 3 2 binomial(2 n2, n2) / ((n2 + 1) (n2 + 2) (n2 + 4) ((n2 + 1) + 44 (n2 + 1) + 399 (n2 + 1) + 1298 n2 + 2694) / \ | 4 3 2 | / 2 1/2 1/2 (n2 + 44 n2 + 399 n2 + 1298 n2 + 1396))| / ((n1 - 34/5 5 n1 + 21 n1 + 347/5 - 137/5 5 ) | / | / \ | 1/2 2 1/2 | (-34 5 (n1 + 1) + 5 (n1 + 1) - 137 5 + 105 n1 + 452))|} | | / "A268401" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\ / 1/2\ 1/2\ 1/2 n | 3 | 63 5 | 2 | 798 5 | 1761 5 | {(2 - 5 ) |n + |39 - -------| n + |395 - --------| n + 831 - ---------|, \ \ 5 / \ 5 / 5 / / / 1/2\ / 1/2\ 1/2\ 1/2 n | 3 | 63 5 | 2 | 798 5 | 1761 5 | 1/2 n (2 + 5 ) |n + |39 + -------| n + |395 + --------| n + 831 + ---------|, (2 - 5 ) \ \ 5 / \ 5 / 5 / /n - 1 |----- 1/2 2 3 1/2 2 1/2 | \ 1/2 n1 1/2 (-n1 - 1) (-63 5 n + 5 n - 798 5 n + 195 n - 1761 5 + 1975 n + 4155) | ) (2 + 5 ) (2 - 5 ) (n1 + 4) | / |----- \n1 = 0 /n1 - 1 |----- 5 4 3 2 | \ 1/2 (-n2 - 1) (n1 + 77 n1 + 1382 n1 + 9337 n1 + 26661 n1 + 27090) | ) (2 + 5 ) (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) (2 n2 + 9) | / |----- \n2 = 0 / 1/2 2 1/2 1/2\ | 3 63 5 (n2 + 1) 2 798 5 (n2 + 1) 1761 5 | 2 / |(n2 + 1) - ----------------- + 39 (n2 + 1) - ----------------- + 395 n2 + 1226 - ---------| (3 n2 + 15 n2 + 11) binomial(2 n2, n2) / ( \ 5 5 5 / / 5 4 3 2 (n2 + 1) (n2 + 2) (n2 + 4) (n2 + 5) ((n2 + 1) + 77 (n2 + 1) + 1382 (n2 + 1) + 9337 (n2 + 1) + 26661 n2 + 53751) \ | 5 4 3 2 | / 3 1/2 2 2 1/2 1/2 (n2 + 77 n2 + 1382 n2 + 9337 n2 + 26661 n2 + 27090))| / ((n1 - 63/5 5 n1 + 39 n1 - 798/5 5 n1 + 395 n1 + 831 - 1761/5 5 ) | / | / \ | 1/2 2 3 1/2 2 1/2 | (-63 5 (n1 + 1) + 5 (n1 + 1) - 798 5 (n1 + 1) + 195 (n1 + 1) - 1761 5 + 1975 n1 + 6130))|} | | / "A268402" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\ 1/2\ 1/2 n | 4 1/2 3 1/2 2 | 15002 5 | 24396 5 | {(2 - 5 ) |n + (62 - 20 5 ) n + (1277 - 522 5 ) n + |7012 - ----------| n + 56076/5 - ----------|, \ \ 5 / 5 / / / 1/2\ 1/2\ 1/2 n | 4 1/2 3 1/2 2 | 15002 5 | 24396 5 | 1/2 n (2 + 5 ) |n + (62 + 20 5 ) n + (1277 + 522 5 ) n + |7012 + ----------| n + 56076/5 + ----------|, (2 - 5 ) \ \ 5 / 5 / /n - 1 / |----- | 1/2 3 4 1/2 2 3 1/2 2 1/2 | \ | 1/2 n1 1/2 (-n1 - 1) (-100 5 n + 5 n - 2610 5 n + 310 n - 15002 5 n + 6385 n - 24396 5 + 35060 n + 56076) | ) |- (2 + 5 ) (2 - 5 ) | / | |----- | \n1 = 0 \ /n1 - 1 |----- 6 5 4 3 2 | \ / 1/2 (-n2 - 1) (n1 + 4) (n1 + 5) (n1 + 119 n1 + 3707 n1 + 43777 n1 + 239736 n1 + 615828 n1 + 597600) | ) |- (2 + 5 ) (2 n2 + 1) (2 n2 + 3) | / \ |----- \n2 = 0 (2 n2 + 5) (2 n2 + 7) (2 n2 + 9) (2 n2 + 11) 1/2 3 4 1/2 2 3 1/2 2 1/2 (100 5 (n2 + 1) - 5 (n2 + 1) + 2610 5 (n2 + 1) - 310 (n2 + 1) + 15002 5 (n2 + 1) - 6385 (n2 + 1) + 24396 5 - 35060 n2 - 91136) 2 / (11 n2 + 64 n2 + 48) binomial(2 n2, n2) / ((n2 + 1) (n2 + 2) (n2 + 4) (n2 + 5) (n2 + 6) / 6 5 4 3 2 ((n2 + 1) + 119 (n2 + 1) + 3707 (n2 + 1) + 43777 (n2 + 1) + 239736 (n2 + 1) + 615828 n2 + 1213428) \ | 6 5 4 3 2 \| / (n2 + 119 n2 + 3707 n2 + 43777 n2 + 239736 n2 + 615828 n2 + 597600))|| / ( /| / | / 1/2 3 4 1/2 2 3 1/2 2 1/2 (100 5 n1 - 5 n1 + 2610 5 n1 - 310 n1 + 15002 5 n1 - 6385 n1 + 24396 5 - 35060 n1 - 56076) 1/2 3 4 1/2 2 3 1/2 2 1/2 (-100 5 (n1 + 1) + 5 (n1 + 1) - 2610 5 (n1 + 1) + 310 (n1 + 1) - 15002 5 (n1 + 1) + 6385 (n1 + 1) - 24396 5 + 35060 n1 + 91136)) \\ || || ||} || || // "A268407" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (13 n2 + 8)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) --------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A268429" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / |n - 1 / 1/2\ / 1/2\ |----- 1/2 n | 13 5 | 1/2 n | 13 5 | 1/2 n 1/2 | \ 1/2 n1 1/2 (-n1 - 1) {(2 - 5 ) |n + 8 - -------|, (2 + 5 ) |n + 8 + -------|, (2 - 5 ) (5 n + 40 - 13 5 ) | ) (2 + 5 ) (2 - 5 ) \ 5 / \ 5 / | / |----- \n1 = 0 / / 1/2\ \ |n1 - 1 1/2 (-n2 - 1) | 13 5 | 2 | |----- (2 + 5 ) (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) |n2 + 9 - -------| (5 n2 + 19 n2 + 15) binomial(2 n2, n2)| / 2 | \ \ 5 / | / | (n1 + 17 n1 + 33) | ) ---------------------------------------------------------------------------------------------------------------| / | | / 2 2 | / \ |----- (n2 + 1) (n2 + 2) (n2 + 3) (n2 + 4) ((n2 + 1) + 17 n2 + 50) (n2 + 17 n2 + 33) | \n2 = 0 / \ | / 1/2\ \| | 13 5 | 1/2 || |n1 + 8 - -------| (5 n1 + 45 - 13 5 )||} \ 5 / /| | / "A268430" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 + 2 2 ) , (-2 2 + 2) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2 2 + 2) binomial(2 n2, n2) (n2 + 10 n2 + 7)|| (2 + 2 2 ) | ) (-2 2 + 2) (2 + 2 2 ) | ) -----------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A268431" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 + 2 2 ) , (-2 2 + 2) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2 2 + 2) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| (2 + 2 2 ) | ) (-2 2 + 2) (2 + 2 2 ) | ) ---------------------------------------------------------------||} | / | / (n2 + 4) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A268473" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268542" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268543" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([-n, 2 n + 1], [1], -1)} "A268545" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) hypergeom([-n, 2 n + 1], [n + 1], -1)} "A268546" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable not successful" {} "A268548" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 2 |----- (3 n + 1) binomial(3 n, n) 2 | \ 2 {-------------------------------------, (3 n + 1) binomial(3 n, n) | ) (4 n1 + 1) (2 n1 + 1) binomial(2 n1, n1) binomial(4 n1, 2 n1) 2 | / (4 n + 1) (2 n + 1) binomial(4 n, n) |----- \n1 = 0 5 4 3 2 / 3 2 (37345 n1 + 203059 n1 + 434180 n1 + 455586 n1 + 234288 n1 + 47232) binomial(4 n1 + 4, n1 + 1) / ((n1 + 1) (3 n1 + 4) / \ | | / 2 binomial(3 n1 + 3, n1 + 1) (4 n1 + 3))| / ((4 n + 1) (2 n + 1) binomial(4 n, n))} | / | / "A268550" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268551" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268552" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A268555" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) LegendreP(n, 3), binomial(2 n, n) LegendreQ(n, 3)} "A268600" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (-1) binomial(2 n1, n1)| {(4/3) , (4/3) | ) -------------------------|, binomial(2 n, n)} | / (n1 + 1) | |----- (n1 + 1) (4/3) | \n1 = 0 / "A268601" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (-1) binomial(2 n1, n1)| {(4/3) , (4/3) | ) -------------------------|, binomial(2 n, n)} | / (n1 + 1) | |----- (n1 + 1) (4/3) | \n1 = 0 / "A268869" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 memory used=218685.1MB, alloc=3351.5MB, time=1671.90 LREtools/SearchTable: "SearchTable not successful" (2 n + 3) (2 n + 1) binomial(2 n, n) {------------------------------------} (n + 1) (n + 2) "A269113" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 {(1/6) (n!) binomial(2 n, n) binomial(3 n, n)} "A269121" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 binomial(3 n1, n1) (5 n1 + n1 - 2)| {8 , 8 | ) ------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A269122" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 |----- n1 (-4 n1 - 4) 3 2 | |----- n n | \ 8 3 (139 n1 + 51 n1 - 112 n1 - 48) binomial(4 n1, n1)| n | \ n1 (-4 n1 - 4) {81 , 81 | ) --------------------------------------------------------------------|, 81 | ) 8 3 | / (n1 + 1) (3 n1 + 1) (3 n1 + 2) | | / |----- | |----- \n1 = 0 / \n1 = 0 /n1 - 1 |----- 3 2 | \ (-3 n2 - 3) (139 n1 + 51 n1 - 112 n1 - 48) | ) 2 (4 n2 + 1) (4 n2 + 3) | / |----- \n2 = 0 7 6 5 4 3 2 / 2 (11815 n2 + 51178 n2 + 62211 n2 + 6970 n2 + 7386 n2 + 63472 n2 + 47072 n2 + 6720) binomial(3 n2, n2) binomial(4 n2, n2) / ((n2 + 1) / \ | 3 2 3 2 | (n2 + 2) (139 (n2 + 1) + 51 (n2 + 1) - 112 n2 - 160) (139 n2 + 51 n2 - 112 n2 - 48) binomial(4 n2 + 4, n2 + 1))| binomial(4 n1, n1)/((n1 + 1) | | / \ | | (3 n1 + 1) (3 n1 + 2))|} | | / "A269450" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A269473" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 1/2 n 1/2 n 1/2 n {(-12 3 ) , (12 3 ) , (-12 3 ) / / / { 0 n2::even\\\ | | | { ||| | | | { (3 n2 - 3) 3 n2 n2 ||| | | | { 2 (3 n2 - 1) (3 n2 + 1) (15 n2 + 29) binomial(---- - 3/2, ---- - 1/2) ||| |n - 1 | |n1 - 1 { 2 2 ||| |----- | |----- { 1/2 ------------------------------------------------------------------------------- n2::odd ||| | \ | 1/2 n1 | \ { n2 (n2 + 1) (n2 + 2) (n2 + 3) ||| | ) |-1/36 3 (-1) | ) -----------------------------------------------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (12 3 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 3 n2 3 n2 n2 \\\ | | | { 2 2 (3 n2 + 1) (15 n2 + 29) binomial(3 n2, ----) binomial(----, ----) ||| | | | { 2 2 2 ||| | | | { ----------------------------------------------------------------------- n2::even||| | | | { n2 ||| | | | { (n2 + 1) (n2 + 2) (n2 + 3) binomial(n2, ----) ||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { ||| 1/2 n | \ | 1/2 n1 | \ { 0 n2::odd ||| (-12 3 ) | ) |-1/36 3 (-1) | ) -----------------------------------------------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (12 3 ) ||| \n1 = 0 \ \n2 = 0 /// "A269474" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /n1 - 1 |----- | |----- { 1/2 n / 49\ 1/2 n 2 | \ | 1/2 n1 2 | \ { {(-12 3 ) |n + --| (n + 3), (-12 3 ) (25 n + 124 n + 147) | ) |-1/36 3 (-1) (25 n1 + 174 n1 + 296) | ) { \ 25/ | / | | / { 0 , n2::even |----- | |----- { \n1 = 0 \ \n2 = 0 (3 n2 - 3) 3 2 3 n2 n2 2 (3 n2 - 1) (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (2100 n2 + 21523 n2 + 70324 n2 + 71181) binomial(---- - 3/2, ---- - 1/2) 2 2 1/32 ---------------------------------------------------------------------------------------------------------------------------------- , n2::odd n2 (n2 + 1) (n2 + 2) \ \\ | || / 1/2 (n2 + 1) 2 | / 2 || 1/2 n / ((12 3 ) (25 (n2 + 1) + 174 n2 + 470) (n2 + 3) (25 n2 + 49))| / (25 (n1 + 1) + 124 n1 + 271)||, (-12 3 ) / | / || | || / // / / / | | | | | | | | | | | | | | | |n - 1 | |n1 - 1 |----- | |----- 2 | \ | 1/2 n1 2 | \ (25 n + 124 n + 147) | ) |-1/36 3 (-1) (25 n1 + 174 n1 + 296) | ) | / | | / |----- | |----- \n1 = 0 \ \n2 = 0 { n2 3 2 3 n2 3 n2 n2 \ { 2 2 (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (2100 n2 + 21523 n2 + 70324 n2 + 71181) binomial(3 n2, ----) binomial(----, ----) | { 2 2 2 | { -------------------------------------------------------------------------------------------------------------------------- n2::even| { n2 | { (n2 + 1) (n2 + 2) binomial(n2, ----) | { 2 | { | { 0 n2::odd | --------------------------------------------------------------------------------------------------------------------------------------------| 1/2 (n2 + 1) 2 | (12 3 ) (25 (n2 + 1) + 174 n2 + 470) (n2 + 3) (25 n2 + 49) | / \\ || || || || || || || / 2 || 1/2 n / 49\ / (25 (n1 + 1) + 124 n1 + 271)||, (12 3 ) |n + --| (n + 3)} / || \ 25/ || // "A269475" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 1/2 n / 3 5297 2 57877 44759\ 1/2 n / 3 5297 2 57877 44759\ 1/2 n {(-12 3 ) |n + ---- n + ----- n + -----| (n + 5) (n + 3), (12 3 ) |n + ---- n + ----- n + -----| (n + 5) (n + 3), (-12 3 ) \ 441 1323 945 / \ 441 1323 945 / /n - 1 / /n1 - 1 |----- | |----- 3 2 | \ | 1/2 n1 3 2 | \ { (6615 n + 79455 n + 289385 n + 313313) (n + 5) (n + 3) | ) |-1/36 3 (-1) (6615 n1 + 99300 n1 + 468140 n1 + 688768) | ) { | / | | / { |----- | |----- \n1 = 0 \ \n2 = 0 0 , n2::even (3 n2 - 3) 1/512 2 (3 n2 - 1) (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (3 n2 + 11) (3 n2 + 13) 5 4 3 2 3 n2 n2 (29502900 n2 + 688794885 n2 + 6212881050 n2 + 26925135700 n2 + 55610797618 n2 + 43133536551) binomial(---- - 3/2, ---- - 1/2)/(n2 (n2 + 1) 2 2 / 1/2 (n2 + 1) 3 2 (n2 + 2)) , n2::odd / ((12 3 ) (n2 + 5) (n2 + 7) (6615 (n2 + 1) + 99300 (n2 + 1) + 468140 n2 + 1156908) (n2 + 3) / \ \\ | || 3 2 | / 3 2 || 1/2 n (6615 n2 + 79455 n2 + 289385 n2 + 313313))| / (6615 (n1 + 1) + 79455 (n1 + 1) + 289385 n1 + 602698)||, (-12 3 ) | / || | || / // /n - 1 / /n1 - 1 |----- | |----- 3 2 | \ | 1/2 n1 3 2 | \ { n2 (6615 n + 79455 n + 289385 n + 313313) (n + 5) (n + 3) | ) |-1/36 3 (-1) (6615 n1 + 99300 n1 + 468140 n1 + 688768) | ) { 2 2 | / | | / { |----- | |----- \n1 = 0 \ \n2 = 0 (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (3 n2 + 11) (3 n2 + 13) 5 4 3 2 3 n2 3 n2 n2 / / (29502900 n2 + 688794885 n2 + 6212881050 n2 + 26925135700 n2 + 55610797618 n2 + 43133536551) binomial(3 n2, ----) binomial(----, ----) / | 2 2 2 / \ n2 \ (n2 + 1) (n2 + 2) binomial(n2, ----)| , n2::even 2 / / 1/2 (n2 + 1) 3 2 0 , n2::odd / ((12 3 ) (n2 + 5) (n2 + 7) (6615 (n2 + 1) + 99300 (n2 + 1) + 468140 n2 + 1156908) (n2 + 3) / \ \\ | || 3 2 | / 3 2 || (6615 n2 + 79455 n2 + 289385 n2 + 313313))| / (6615 (n1 + 1) + 79455 (n1 + 1) + 289385 n1 + 602698)||} | / || | || / // "A269476" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 1/2 n / 4 883390 3 216400 2 144038210 672346141\ {(-12 3 ) |n + ------ n + ------ n + --------- n + ---------| (n + 7) (n + 5) (n + 3), \ 47439 1757 426951 2134755 / 1/2 n / 4 883390 3 216400 2 144038210 672346141\ 1/2 n (12 3 ) |n + ------ n + ------ n + --------- n + ---------| (n + 7) (n + 5) (n + 3), (-12 3 ) \ 47439 1757 426951 2134755 / /n - 1 / |----- | 4 3 2 | \ | 1/2 n1 (2134755 n + 39752550 n + 262926000 n + 720191050 n + 672346141) (n + 7) (n + 5) (n + 3) | ) |-1/36 3 (-1) | / | |----- | \n1 = 0 \ /n1 - 1 |----- 4 3 2 | \ { (2134755 n1 + 48291570 n1 + 394992180 n1 + 1373839720 n1 + 1697350496) | ) { 0 , n2::even | / { |----- \n2 = 0 (3 n2 - 3) 7 1/4096 2 (3 n2 - 1) (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (3 n2 + 11) (3 n2 + 13) (3 n2 + 17) (3 n2 + 19) (8865313032240 n2 6 5 4 3 2 + 365545739182680 n2 + 6284185525676565 n2 + 58200447007969875 n2 + 312375868393795782 n2 + 966669145727796386 n2 + 1585464375369413429 n2 3 n2 n2 / 1/2 (n2 + 1) + 1050830203606847475) binomial(---- - 3/2, ---- - 1/2)/(n2 (n2 + 1) (n2 + 2)) , n2::odd / ((12 3 ) (n2 + 5) (n2 + 7) (n2 + 9) 2 2 / 4 3 2 (2134755 (n2 + 1) + 48291570 (n2 + 1) + 394992180 (n2 + 1) + 1373839720 n2 + 3071190216) (n2 + 3) \ | 4 3 2 | / (2134755 n2 + 39752550 n2 + 262926000 n2 + 720191050 n2 + 672346141))| / ( | / | / \\ || 4 3 2 || 1/2 n 2134755 (n1 + 1) + 39752550 (n1 + 1) + 262926000 (n1 + 1) + 720191050 n1 + 1392537191)||, (-12 3 ) || || // /n - 1 / |----- | 4 3 2 | \ | 1/2 n1 (2134755 n + 39752550 n + 262926000 n + 720191050 n + 672346141) (n + 7) (n + 5) (n + 3) | ) |-1/36 3 (-1) | / | |----- | \n1 = 0 \ /n1 - 1 |----- 4 3 2 | \ { n2 (2134755 n1 + 48291570 n1 + 394992180 n1 + 1373839720 n1 + 1697350496) | ) { 2 2 (3 n2 + 1) (3 n2 + 5) (3 n2 + 7) (3 n2 + 11) | / { |----- \n2 = 0 7 6 5 4 (3 n2 + 13) (3 n2 + 17) (3 n2 + 19) (8865313032240 n2 + 365545739182680 n2 + 6284185525676565 n2 + 58200447007969875 n2 3 2 3 n2 3 n2 n2 / + 312375868393795782 n2 + 966669145727796386 n2 + 1585464375369413429 n2 + 1050830203606847475) binomial(3 n2, ----) binomial(----, ----) / 2 2 2 / / n2 \ |(n2 + 1) (n2 + 2) binomial(n2, ----)| , n2::even \ 2 / / 1/2 (n2 + 1) 0 , n2::odd / ((12 3 ) (n2 + 5) (n2 + 7) (n2 + 9) / 4 3 2 (2134755 (n2 + 1) + 48291570 (n2 + 1) + 394992180 (n2 + 1) + 1373839720 n2 + 3071190216) (n2 + 3) \ | 4 3 2 | / (2134755 n2 + 39752550 n2 + 262926000 n2 + 720191050 n2 + 672346141))| / ( | / | / \\ || 4 3 2 || 2134755 (n1 + 1) + 39752550 (n1 + 1) + 262926000 (n1 + 1) + 720191050 n1 + 1392537191)||} || || // "A269506" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n 3 2 n 3 2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , RootOf(_Z + _Z - _Z + 3, index = 1) , RootOf(_Z + _Z - _Z + 3, index = 2) , 3 2 n RootOf(_Z + _Z - _Z + 3, index = 3) } "A269509" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A269730" LREtools/SearchTable: "SearchTable successful" 5 LegendreP(n + 1, 5) - LegendreP(n, 5) 5 LegendreQ(n + 1, 5) - LegendreQ(n, 5) {---------------------------------------, ---------------------------------------} n + 2 n + 2 "A269731" LREtools/SearchTable: "SearchTable successful" 7 LegendreP(n + 1, 7) - LegendreP(n, 7) 7 LegendreQ(n + 1, 7) - LegendreQ(n, 7) {---------------------------------------, ---------------------------------------} n + 2 n + 2 "A269732" LREtools/SearchTable: "SearchTable successful" 9 LegendreP(n + 1, 9) - LegendreP(n, 9) 9 LegendreQ(n + 1, 9) - LegendreQ(n, 9) {---------------------------------------, ---------------------------------------} n + 2 n + 2 "A269820" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A270049" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I)} "A270229" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A270363" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A270386" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A270447" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) {--------------------------, n + 1 2 (3 (2 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) - 4 (4 n + 1) (n + 1) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) -------------------------------------------------------------------------------------------------------------------------------------} (n + 1) (2 n - 1) "A270489" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A270490" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" (2 n + 1) binomial(2 n, n) {--------------------------} n + 2 "A270530" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) {--------------------------, n + 1 2 2 ((32 n + 59 n + 25) hypergeom([1/2, -n - 1], [n + 2], -4) - 50 (n + 1) hypergeom([1/2, -n], [n + 1], -4)) binomial(2 n, n) (2 n + 1) --------------------------------------------------------------------------------------------------------------------------------------} (n + 1) (2 n + 3) (3 n + 1) "A270560" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" (2 n + 1) binomial(2 n, n) {--------------------------} n + 2 "A270561" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" (2 n + 1) binomial(2 n, n) n {----------------------------} (n + 1) (n + 2) "A270595" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A270661" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A270709" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A270724" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A270784" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1, (n + 4) (n + 3)} "A270785" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 6 _Z + 9 _Z - 4 _Z - 1 "A270786" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := 2 _Z - 13 _Z + 19 _Z - 8 _Z - 1 "A270787" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := 3 _Z - 22 _Z + 33 _Z - 14 _Z - 1 "A270822" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n 1/2 n 1/2 n {(-2) , 2 , (-2 2 ) , (2 2 ) , /n - 1 / /n1 - 1 / 3 n2\ /{ 0 n2::even\\\\ |----- | |----- |- 3/2 - ----| |{ |||| 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ (2 n2 - 2) |||| (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ 2 (n2 + 1) ||||, | / | | / |{ 1/2 ---------------------------------------- n2::odd |||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) |||| \ \ \ \{ 2 //// /n - 1 / /n1 - 1 / 3 n2\ /{ n2 \\\\ |----- | |----- |- 3/2 - ----| |{ 2 binomial(n2, ----) (n2 + 1) |||| 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ 2 |||| (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ ----------------------------- n2::even||||} | / | | / |{ n2 + 2 |||| |----- | |----- |{ |||| \n1 = 0 \ \n2 = 0 \{ 0 n2::odd //// "A271028" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 4 3 2 n \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 31 n1 + 113 n1 + 149 n1 + 40) {1, 2 , ) ---------------------------------------------------------------------------------} / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A271197" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) | n n | \ 6 ((n1 - 1) LegendreP(n1 + 1, 3) + (-n1 - 1) LegendreP(n1, 3))| {6 , 6 | ) -----------------------------------------------------------------------|, | / n1 + 2 | |----- | \n1 = 0 / /n - 1 \ |----- (-n1 - 1) | n | \ 6 ((n1 - 1) LegendreQ(n1 + 1, 3) + (-n1 - 1) LegendreQ(n1, 3))| 6 | ) -----------------------------------------------------------------------|} | / n1 + 2 | |----- | \n1 = 0 / "A271212" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n | \ (-1) 2 (n1 + 1) | 2 n! (2 n + 1) | ) -------------------------------| | / (2 n1 + 1) (2 n1 + 3) (n1 + 1)!| n |----- | 2 n! (2 n + 1) \n1 = 0 / {---------------, --------------------------------------------------------} n n "A271214" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n | \ (-1) 2 (n1 + 1) | 2 n! (2 n + 1) | ) -------------------------------| | / (2 n1 + 1) (2 n1 + 3) (n1 + 1)!| { n n |----- | { 2 (2 n - 1) (n/2)! 2 n! (2 n + 1) \n1 = 0 / { ------------------- n::even {---------------, --------------------------------------------------------, { n , n n { { (n - 1) { 4 2 (n/2 - 1/2)! n::odd { 4 (n/2)! binomial(n, n/2) n::even { { (2 n - 1) (n/2 + 1/2)! binomial(n + 1, n/2 + 1/2) } { ------------------------------------------------- n::odd { n "A271216" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 (n/2)! n::even { { (n/2)! binomial(n, n/2) n::even {{ (n + 1) , { } { 2 (n/2 + 1/2)! { n (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { --------------------- n::odd { n + 1 "A271217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { n { 2 (2 n - 1) (n/2)! { 4 (n/2)! binomial(n, n/2) n::even { ------------------- n::even { {{ n , { (2 n - 1) (n/2 + 1/2)! binomial(n + 1, n/2 + 1/2) } { { ------------------------------------------------- n::odd { (n - 1) { n { 4 2 (n/2 - 1/2)! n::odd "A271218" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A271318" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 2 %1 := _Z - 3 _Z - 3 "A271432" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A271475" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 2 {1, ) (n1 + 1) n1! (2 n1 + 5 n1 + 4)} / ----- n1 = 0 "A271476" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (2 n1 + 3)| {(n + 1) 2 n!, (n + 1) 2 n! | ) ---------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A271477" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ n1 2 {1, ) (n1 + 1) 2 n1! (2 n1 + 6 n1 + 5)} / ----- n1 = 0 "A271503" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 4 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even {1, (-1) , { , { (- n/2 + 1/2) { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::even { , { 1/2 1/2 (n/2 - 1/2) { (-1 - 3 I) n (-1/4 I 3 - 1/4) (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { 0 n::even { (n/2) { , { 2 (n/2)! n::even, { 1/2 1/2 (n/2 - 1/2) { { (3 I - 1) n (1/4 I 3 - 1/4) (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { 0 n::odd { 1/2 1/2 (n/2) { 1/2 1/2 (n/2) { (-1 - 3 I) (-1 - 3 I) (n/2)! n::even, { (3 I - 1) (3 I - 1) (n/2)! n::even} { { { 0 n::odd { 0 n::odd "A271622" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 1, index = 1) , RootOf(_Z - _Z - 1, index = 2) , RootOf(_Z - _Z - 1, index = 3) } "A271715" n n 2 {(1/2) n! binomial(2 n, n) binomial(3 n, n), (1/6) (n!) binomial(2 n, n) binomial(3 n, n)} "A271777" LREtools/SearchTable: "SearchTable successful" {(2 (n + 2) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) - (n + 1) (13 n + 6) hypergeom([-n, -n, -n], [1, -2 n], 1)) 2 binomial(2 n, n)/(n (n - 1))} "A271905" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" {1, (2 n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 2) hypergeom([-1/2, -n], [1], -8)} "A271941" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A271943" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A272261" LREtools/SearchTable: "SearchTable successful" n n {(-2) BesselI(n + 1/2, 1), (-2) BesselK(n + 1/2, -1)} "A272391" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A272392" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A272393" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A272395" memory used=220107.5MB, alloc=3351.5MB, time=1681.54 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A272492" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | |2 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) |- ----| n!, (n + 2) (n + 1) |----| n!} \ 2 / \ 2 / "A272493" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | |2 | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) |- ----| n!, (n + 3) (n + 2) (n + 1) |----| n!, \ 2 / \ 2 / 3 2 n 3 2 n (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 6, index = 1) ) n!, (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 6, index = 2) ) n!, 3 2 n (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 6, index = 3) ) n!} "A272514" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n n { { 4 (n + 2) {1, 2 , { 4 binomial(n - 1, n/2 - 1/2) n (n + 2) , { 1/2 -------------------------------- n::even} { -------------------------------------- n::odd { (n + 1) (n + 3) binomial(n, n/2) { (n + 1) (n + 3) { { 0 n::odd "A272641" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 2 n 2 2 {(n + 1) (1/3 %3 ) n! (63 n - 210 %3 n + 513 n + 350 %3 - 840 %3 + 1044), 2 n 2 2 (n + 1) (1/3 %2 ) n! (63 n - 210 %2 n + 513 n + 350 %2 - 840 %2 + 1044), 2 n 2 2 (n + 1) (1/3 %1 ) n! (63 n - 210 %1 n + 513 n + 350 %1 - 840 %1 + 1044)} 3 %1 := RootOf(_Z + 3, index = 3) 3 %2 := RootOf(_Z + 3, index = 2) 3 %3 := RootOf(_Z + 3, index = 1) "A272646" LREtools/SearchTable: "SearchTable successful" 1/2 1/2 1/2 n 2 1/2 n 2 {(-2 ) BesselI(n + 1/4, ----), (-2 ) BesselK(n + 1/4, - ----)} 2 2 "A272686" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A272687" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A272988" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (n1 + 1) | n n | \ (-1) 2 | {(n + 1) (1/2) n!, (n + 1) (1/2) n! | ) ------------------|, | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | n1 (n1 + 1) | \ n2 || | (-1) 2 | ) (-(-1) (n2 + 2) (n2 + 1) n2!)|| |n - 1 | / || |----- |----- || n | \ \n2 = 0 /| (n + 1) (1/2) n! | ) ---------------------------------------------------------|, | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / / /n1 - 1 / /n2 - 1 \\\\ | |----- | |----- n3 |||| | n1 (n1 + 1) | \ | n2 | \ (-1) |||| | (-1) 2 | ) |-(-1) (n2 + 2) (n2 + 1) n2! | ) ---------------------------|||| |n - 1 | / | | / (n3 + 3) (n3 + 2) (n3 + 1)!|||| |----- |----- | |----- |||| n | \ \n2 = 0 \ \n3 = 0 ///| (n + 1) (1/2) n! | ) ----------------------------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A273019" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" hypergeom([1/2, -2 n - 2], [1], 4) + (-12 n - 6) hypergeom([1/2, -2 n], [1], 4) {-------------------------------------------------------------------------------} 4 n + 3 "A273020" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4)), hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4)} "A273055" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (2 n + 2) hypergeom([1/2, -2 n - 2], [1], 4) + (-6 n - 3) hypergeom([1/2, -2 n], [1], 4) {----------------------------------------------------------------------------------------} 4 n + 3 "A273343" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273345" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273347" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z - _Z - 1, index = 1) , RootOf(_Z - 3 _Z - _Z - 1, index = 2) , RootOf(_Z - 3 _Z - _Z - 1, index = 3) } "A273351" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273526" (2 n + 1) binomial(2 n, n) {n, --------------------------} (n + 1) (n + 2) "A273596" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A273714" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273716" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273718" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273720" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 1} "A273722" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273889" n 2 n 2 (2 n + 1) 4 (n!) binomial(2 n, n) (4 n + 1) (1/4) (n!) binomial(3 n, n) binomial(4 n, n) {-----------------------------------, --------------------------------------------------------} 4 n + 3 4 n + 3 "A273898" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273900" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273902" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A273904" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A273905" LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273939" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::even {{ , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::odd { (n/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::even { , { (n/2 + 1/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselJ(n/2 + 1, -2) + (4 n + 8) BesselJ(n/2, -2)) n::even { , { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselJ(n/2 + 1/2, -2) + (4 n + 12) BesselJ(n/2 - 1/2, -2)) n::odd { (n/2) { (-1) (2 (n + 4) n BesselY(n/2 + 1, -2) + (4 n + 8) BesselY(n/2, -2)) n::even { } { (n/2 - 1/2) 2 { (-1) ((2 n + 8 n - 2) BesselY(n/2 + 1/2, -2) + (4 n + 12) BesselY(n/2 - 1/2, -2)) n::odd "A273958" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A273983" n 2 n 2 (n + 1) (2 n + 1) 4 (n!) binomial(2 n, n) (4 n + 1) (4 n + 3) (1/4) (n!) binomial(3 n, n) binomial(4 n, n) {-------------------------------------------, ------------------------------------------------------------------} 4 n + 5 4 n + 5 "A273988" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A274104" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) (3 n1 + 5) {1, ) ----------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A274112" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 1, index = 1) , RootOf(_Z - _Z - 1, index = 2) , RootOf(_Z - _Z - 1, index = 3) } "A274114" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| } \ 2 / \ 2 / "A274115" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 2 _Z - 1, index = 1) , RootOf(_Z - _Z - 2 _Z - 1, index = 2) , RootOf(_Z - _Z - 2 _Z - 1, index = 3) } "A274117" n n 36 GAMMA(n + 2/3) GAMMA(n + 1/2) GAMMA(n + 7/6) 36 GAMMA(n + 4/3) GAMMA(n + 1/2) GAMMA(n + 5/6) {------------------------------------------------, ------------------------------------------------} GAMMA(n + 3/2) GAMMA(n + 3/2) "A274160" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A274246" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274295" (n - 1) n binomial(2 n, n) (5 n - 12) 2 {-------------------------------------, 2 n - 9 n + 12} (2 n - 5) (2 n - 3) (2 n - 1) "A274492" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274493" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A274495" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274634" LREtools/SearchTable: "SearchTable successful" n n {(-1) n! (n + 1) (2 n BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) n! (n + 1) (2 n BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A274665" LREtools/SearchTable: "SearchTable not successful" {} "A274666" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274667" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274668" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274669" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274670" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274671" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274672" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274673" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274674" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274707" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even { n binomial(n, n/2) { binomial(n, n/2) { ------------------ n::even {{ , { n + 2 } { (2 n + 2) { { 2 n { n binomial(n - 1, n/2 - 1/2) n::odd { ------------------------------------------ n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A274734" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274780" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274781" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274782" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A274786" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) hypergeom([-n, -n, -n], [1, -2 n], 1)} "A275027" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A275049" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) binomial(3 n, n) hypergeom([-n, 3 n + 1], [1], -1)} "A275050" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) (3 (3 n + 5) (2 n + 3) (3 n + 1) (n + 2) hypergeom([-n - 1, 3 n + 6], [n + 4], -1) 2 - (105 n + 217 n + 104) (2 n + 1) (n + 3) hypergeom([-n, 3 n + 3], [n + 3], -1)) binomial(3 n, n)/((n + 1) (n + 2) (n + 3) (22 n + 29))} "A275207" memory used=221448.9MB, alloc=3351.5MB, time=1690.79 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n (-1) ((4 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 15) hypergeom([1/2, -n], [1], 4)) (n + 1) {-----------------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) { 0 n::even { { (n/2 - 1/2) , { (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ----------------------------------------------------------------------------------------- n::odd { n + 4 { (n/2) { 2 (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ------------------------------------------------------------------------------------- n::even} { n + 4 { { 0 n::odd "A275208" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n (-1) ((4 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-6 n - 15) hypergeom([1/2, -n], [1], 4)) (n + 1) {-----------------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) { 0 n::even { { (n/2 - 1/2) , { (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ----------------------------------------------------------------------------------------- n::odd { n + 4 { (n/2) { 2 (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ------------------------------------------------------------------------------------- n::even} { n + 4 { { 0 n::odd "A275209" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n 2 2 (-1) ((n + 2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 7) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) { 0 n::even { { (n/2 - 1/2) , { (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ----------------------------------------------------------------------------------------- n::odd { n + 4 { (n/2) { 2 (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ------------------------------------------------------------------------------------- n::even} { n + 4 { { 0 n::odd "A275210" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n 2 2 (-1) ((n + 2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 7) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) { 0 n::even { { (n/2 - 1/2) , { (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ----------------------------------------------------------------------------------------- n::odd { n + 4 { (n/2) { 2 (-1) (hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) { ------------------------------------------------------------------------------------- n::even} { n + 4 { { 0 n::odd "A275286" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 2 2 \| 2 n 2 2 2 n 2 2 | \ | 4 (-1) (2 n1 + 1) binomial(2 n1, n1) (n1!) || {(2 n + 1) (1/4) (n!) binomial(2 n, n) , (2 n + 1) (1/4) (n!) binomial(2 n, n) | ) |----------------------------------------------------|| | / | 2 2 2|| |----- \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / } "A275289" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((3 n + 10 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) {(-1) , ---------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A275293" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n + 1 {------------------------------------, (2 n + 3) (2 n + 1) binomial(2 n, n) /n - 1 \ |----- 2 2 2 3 2 | | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) (21 n1 + 134 n1 + 281 n1 + 192) (2 n1 + 5) binomial(2 n1 + 2, n1 + 1)| (n + 1) | ) -------------------------------------------------------------------------------------------------------------------| | / 2 2 2 | |----- (n1 + 3) (n1 + 2) (n1 + 1) (4 n1 + 10) | \n1 = 0 / ------------------------------------------------------------------------------------------------------------------------------------} (2 n + 3) (2 n + 1) binomial(2 n, n) "A275324" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ---------------- n::even { binomial(n, n/2) { 4 binomial(n, n/2) n::even {{ , { } { (2 n - 2) { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { 4 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A275329" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 3) { ------------------------ n::even { 4 binomial(n, n/2) (n + 4) { (n + 1) binomial(n, n/2) { -------------------------- n::even {{ , { n + 2 } { (2 n - 2) { { 4 2 (n + 4) { binomial(n + 1, n/2 + 1/2) (n + 3) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A275423" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A275424" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A275425" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A275426" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A275448" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A275521" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) { 2 (n/2)! (2 n - 2) n::even { 3 2 binomial(n, n/2) (n/2)! n::even {{ , { } { (n/2 - 1/2) { (- n/2 - 1/2) { 3 2 (n/2 - 1/2)! n::odd { 2 (2 n - 2) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd "A275539" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (- n/2) {n!, { , { 1/2 2 n binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2) { { 2 (n/2 - 1/2)! n n::odd { 0 n::odd "A275651" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n!) binomial(2 n, n), (n!) binomial(2 n, n) | ) ---------------------------------------|} | / 2 | |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A275822" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 3 3\| n n | \ | (-1) (2 n1 + 1) binomial(2 n1, n1) || {(-1) , (-1) | ) |- --------------------------------------||} | / | 3 || |----- \ (n1 + 1) /| \n1 = 0 / "A275825" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A275912" n n 16 GAMMA(n - 1/2) GAMMA(1/6 + n) 16 GAMMA(n - 1/2) GAMMA(n + 5/6) {---------------------------------, ---------------------------------} GAMMA(n + 1/3) GAMMA(n + 1) GAMMA(n + 1) GAMMA(n + 2/3) "A275929" {n + 2, n!} "A275941" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (35 n1 + 127 n1 + 108)| (2 n + 1) binomial(2 n, n) {8 , 8 | ) -----------------------------------------------------------------------------|, --------------------------} | / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) | n + 1 |----- | \n1 = 0 / "A276013" LREtools/SearchTable: "SearchTable successful" n 2 {6 binomial(2 n, n) ((3 n + 3) hypergeom([1/3, - 1/3 - n], [5/3], -1) + (-3 n - 4) hypergeom([1/3, 2/3 - n], [5/3], -1))} "A276015" LREtools/SearchTable: "SearchTable successful" n n 36 GAMMA(n + 1/2) GAMMA(n + 1/3) LegendreP(n, 3) 36 GAMMA(n + 1/2) GAMMA(n + 1/3) LegendreQ(n, 3) {-------------------------------------------------, -------------------------------------------------} 2 2 GAMMA(n + 1) GAMMA(n + 1) "A276017" LREtools/SearchTable: "SearchTable successful" n {6 ((3 n + 3) hypergeom([1/3, - 1/3 - n], [5/3], -1) + (-3 n - 4) hypergeom([1/3, 2/3 - n], [5/3], -1)) binomial(2 n, n) binomial(3 n, n)} "A276019" LREtools/SearchTable: "SearchTable not successful" {} "A276020" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276021" LREtools/SearchTable: "SearchTable not successful" {} "A276022" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276032" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ ----- |{ | n \ |{ (2 n1 - 2) | {1, (-1) , ) |{ 2 (n1 + 1) (n1 + 3) |, / |{ 1/4 ---------------------------------------------------------- n1::odd | ----- |{ n1 | n1 = 0 |{ n1 (n1 + 2) (n1 + 4) (n1 + 6) binomial(n1 - 1, ---- - 1/2) | \{ 2 / n - 1 /{ n1 \ ----- |{ 8 binomial(n1, ----) (n1 + 1) (n1 + 3) | \ |{ 2 | ) |{ -------------------------------------- n1::even|} / |{ (n1 + 2) (n1 + 6) (n1 + 4) | ----- |{ | n1 = 0 \{ 0 n1::odd / "A276033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | /{ n1 \ |{ ------------------------------------ n1::even| |{ 12 binomial(n1, ----) (n1 + 1) | n - 1 |{ n1 | n - 1 |{ 2 | ----- |{ (n1 + 1) (n1 + 3) binomial(n1, ----) | ----- |{ ------------------------------ n1::even| \ |{ 2 | \ |{ (n1 + 6) (n1 + 4) | {1, ) |{ |, ) |{ |} / |{ (2 n1 + 2) | / |{ n1 | ----- |{ 3 2 | ----- |{ 4 binomial(n1 - 1, ---- - 1/2) n1 | n1 = 0 |{ ---------------------------------------------- n1::odd | n1 = 0 |{ 2 | |{ n1 | |{ --------------------------------- n1::odd | |{ (n1 + 4) (n1 + 6) binomial(n1 + 1, ---- + 1/2) | \{ (n1 + 1) (n1 + 3) / \{ 2 / "A276068" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A276080" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselJ(n, -2) + 2 BesselJ(n - 1, -2)), (-1) (n BesselY(n, -2) + 2 BesselY(n - 1, -2))} "A276100" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { {{ 0 , n::even { (n/2 - 1/2) 17 11 19 29 13 23 1007769600000 GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + 1/30) GAMMA(n/2 + --) 30 30 30 30 30 30 GAMMA(n/2 + 7/30)/(GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) GAMMA(n/2 + 1/5) GAMMA(n/2 + 2/3) { (n/2) 13 29 11 GAMMA(n/2 + 1/3)) , n::odd, { 1007769600000 GAMMA(n/2 + 7/30) GAMMA(n/2 + --) GAMMA(n/2 + 1/30) GAMMA(n/2 + --) GAMMA(n/2 + --) { 30 30 30 17 19 23 GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --)/(GAMMA(n/2 + 3/5) GAMMA(n/2 + 1/3) GAMMA(n/2 + 1/5) GAMMA(n/2 + 1) GAMMA(n/2 + 4/5) 30 30 30 GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/2) GAMMA(n/2 + 2/5)) , n::even 0 , n::odd} "A276177" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276179" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A276180" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A276310" LREtools/SearchTable: "SearchTable successful" / 1/2 \n 1/2 | 5 10 | 500 130 10 |- ------- + 13/3| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], --- + ---------) \ 3 / 81 81 {------------------------------------------------------------------------------------} GAMMA(n + 2) "A276314" LREtools/SearchTable: "SearchTable successful" / 1/2 \n 1/2 | 20 10 29| 8000 1160 10 |- -------- + --| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], ---- + ----------) \ 13 13/ 3159 3159 {-------------------------------------------------------------------------------------} GAMMA(n + 2) "A276315" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |108 30 15 | 250 60 15 |--- - --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], --- + --------) \17 17 / 17 17 {---------------------------------------------------------------------------------} GAMMA(n + 2) "A276316" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 | 13 13 | 4394 1196 13 |23/3 - --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], ---- + ----------) \ 6 / 81 81 {-------------------------------------------------------------------------------------} GAMMA(n + 2) "A276368" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 3 n { 3 3 binomial(---, n/2) n::even { 2 { {{ (n/2 + 1/2) 3 n , { 2 3 binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ----------------------------------------------------- n::odd { 3 n + 1 { (n/2) 3 n 3 n { 2 (3/16) binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------------- n::even { binomial(n, n/2) { } { (n/2 - 1/2) 3 n 3 n { 3 (3 n - 2) (3/16) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { ----------------------------------------------------------------------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A276536" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276537" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276657" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276852" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276893" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A276902" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A276903" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" (4 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4) {----------------------------------------------------------------------------------------} n "A276924" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 n {(1/24 RootOf(%1, index = 1) + 1/24 RootOf(%1, index = 1) + 7/24 RootOf(%1, index = 1) + 5/8) n!, 3 2 n (1/24 RootOf(%1, index = 2) + 1/24 RootOf(%1, index = 2) + 7/24 RootOf(%1, index = 2) + 5/8) n!, 3 2 n (1/24 RootOf(%1, index = 3) + 1/24 RootOf(%1, index = 3) + 7/24 RootOf(%1, index = 3) + 5/8) n!, 3 2 n (1/24 RootOf(%1, index = 4) + 1/24 RootOf(%1, index = 4) + 7/24 RootOf(%1, index = 4) + 5/8) n!} 4 2 %1 := _Z + 6 _Z + 8 _Z - 39 "A276925" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 4 3 11 2 31 19\n {|1/120 RootOf(%1, index = 1) + 1/120 RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 2) + 1/120 RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 3) + 1/120 RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 4) + 1/120 RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 5) + 1/120 RootOf(%1, index = 5) + --- RootOf(%1, index = 5) + --- RootOf(%1, index = 5) + --| n!} \ 120 120 30/ 5 3 2 %1 := _Z + 10 _Z + 20 _Z + 45 _Z - 196 "A276926" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 5 4 3 2 191 {|1/720 RootOf(%1, index = 1) + 1/720 RootOf(%1, index = 1) + 1/45 RootOf(%1, index = 1) + 7/90 RootOf(%1, index = 1) + --- RootOf(%1, index = 1) \ 720 91 \n / 5 4 3 2 + ---| n!, |1/720 RootOf(%1, index = 2) + 1/720 RootOf(%1, index = 2) + 1/45 RootOf(%1, index = 2) + 7/90 RootOf(%1, index = 2) 144/ \ 191 91 \n / 5 4 3 + --- RootOf(%1, index = 2) + ---| n!, |1/720 RootOf(%1, index = 3) + 1/720 RootOf(%1, index = 3) + 1/45 RootOf(%1, index = 3) 720 144/ \ 2 191 91 \n / 5 4 + 7/90 RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + ---| n!, |1/720 RootOf(%1, index = 4) + 1/720 RootOf(%1, index = 4) 720 144/ \ 3 2 191 91 \n / 5 + 1/45 RootOf(%1, index = 4) + 7/90 RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + ---| n!, |1/720 RootOf(%1, index = 5) 720 144/ \ 4 3 2 191 91 \n / + 1/720 RootOf(%1, index = 5) + 1/45 RootOf(%1, index = 5) + 7/90 RootOf(%1, index = 5) + --- RootOf(%1, index = 5) + ---| n!, | 720 144/ \ 5 4 3 2 1/720 RootOf(%1, index = 6) + 1/720 RootOf(%1, index = 6) + 1/45 RootOf(%1, index = 6) + 7/90 RootOf(%1, index = 6) 191 91 \n + --- RootOf(%1, index = 6) + ---| n!} 720 144/ 6 4 3 2 %1 := _Z + 15 _Z + 40 _Z + 135 _Z + 264 _Z - 1175 "A276927" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 6 5 11 4 23 3 {|1/5040 RootOf(%1, index = 1) + 1/5040 RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) \ 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) + ---| n!, |1/5040 RootOf(%1, index = 2) + 1/5040 RootOf(%1, index = 2) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 3) + 1/5040 RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) + ---| n!, |1/5040 RootOf(%1, index = 4) + 1/5040 RootOf(%1, index = 4) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 5) + 1/5040 RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) + ---| n!, |1/5040 RootOf(%1, index = 6) + 1/5040 RootOf(%1, index = 6) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 7) + 1/5040 RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) 2520 1260 407 2 1331 177\n + ---- RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) + ---| n!} 5040 5040 280/ 7 5 4 3 2 %1 := _Z + 21 _Z + 70 _Z + 315 _Z + 924 _Z + 1855 _Z - 8226 "A276928" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 7 6 11 5 4 {|1/315 RootOf(%1, index = 1) - 1/105 RootOf(%1, index = 1) + --- RootOf(%1, index = 1) - 1/105 RootOf(%1, index = 1) \ 315 32 3 46 2 19 136\n / 7 + --- RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + -- RootOf(%1, index = 1) + ---| n!, |1/315 RootOf(%1, index = 2) 315 315 63 315/ \ 6 11 5 4 32 3 - 1/105 RootOf(%1, index = 2) + --- RootOf(%1, index = 2) - 1/105 RootOf(%1, index = 2) + --- RootOf(%1, index = 2) 315 315 46 2 19 136\n / 7 6 + --- RootOf(%1, index = 2) + -- RootOf(%1, index = 2) + ---| n!, |1/315 RootOf(%1, index = 3) - 1/105 RootOf(%1, index = 3) 315 63 315/ \ 11 5 4 32 3 46 2 19 + --- RootOf(%1, index = 3) - 1/105 RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + -- RootOf(%1, index = 3) 315 315 315 63 136\n / 7 6 11 5 4 + ---| n!, |1/315 RootOf(%1, index = 4) - 1/105 RootOf(%1, index = 4) + --- RootOf(%1, index = 4) - 1/105 RootOf(%1, index = 4) 315/ \ 315 32 3 46 2 19 136\n / 7 + --- RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + -- RootOf(%1, index = 4) + ---| n!, |1/315 RootOf(%1, index = 5) 315 315 63 315/ \ 6 11 5 4 32 3 - 1/105 RootOf(%1, index = 5) + --- RootOf(%1, index = 5) - 1/105 RootOf(%1, index = 5) + --- RootOf(%1, index = 5) 315 315 46 2 19 136\n / 7 6 + --- RootOf(%1, index = 5) + -- RootOf(%1, index = 5) + ---| n!, |1/315 RootOf(%1, index = 6) - 1/105 RootOf(%1, index = 6) 315 63 315/ \ 11 5 4 32 3 46 2 19 + --- RootOf(%1, index = 6) - 1/105 RootOf(%1, index = 6) + --- RootOf(%1, index = 6) + --- RootOf(%1, index = 6) + -- RootOf(%1, index = 6) 315 315 315 63 136\n / 7 6 11 5 4 + ---| n!, |1/315 RootOf(%1, index = 7) - 1/105 RootOf(%1, index = 7) + --- RootOf(%1, index = 7) - 1/105 RootOf(%1, index = 7) 315/ \ 315 32 3 46 2 19 136\n / 7 + --- RootOf(%1, index = 7) + --- RootOf(%1, index = 7) + -- RootOf(%1, index = 7) + ---| n!, |1/315 RootOf(%1, index = 8) 315 315 63 315/ \ 6 11 5 4 32 3 - 1/105 RootOf(%1, index = 8) + --- RootOf(%1, index = 8) - 1/105 RootOf(%1, index = 8) + --- RootOf(%1, index = 8) 315 315 46 2 19 136\n + --- RootOf(%1, index = 8) + -- RootOf(%1, index = 8) + ---| n!} 315 63 315/ 8 7 6 5 4 3 2 %1 := 2 _Z - 8 _Z + 28 _Z - 28 _Z + 70 _Z + 28 _Z + 98 _Z + 82 _Z - 587 "A276965" memory used=222866.1MB, alloc=3351.5MB, time=1700.26 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A277060" LREtools/SearchTable: "SearchTable successful" hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + (-12 n - 5) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1) {---------------------------------------------------------------------------------------------------------------------} n "A277132" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A277175" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(n + 1) n!, (-1/4) binomial(2 n, n) n! (2 n + 1) ( 5 4 3 2 4 3 2 (n - 21 n + 141 n - 331 n + 134 n + 172) LaguerreL(n + 1/2, -n + 3, 4) + (4 n - 72 n + 404 n - 784 n + 368) LaguerreL(n - 1/2, -n + 4, 4)) n n 4 3 2 , (-1/4) binomial(2 n, n) n! (2 n + 1) (4 (-1) (n - 18 n + 101 n - 196 n + 92) (n + 1/2)! KummerU(-n + 1/2, -n + 5, 4) (n + 1) 5 4 3 2 + (-1) (n - 21 n + 141 n - 331 n + 134 n + 172) KummerU(-n - 1/2, -n + 4, 4) (n - 1/2)!)/((n + 1/2)! (n - 1/2)!)} "A277176" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) binomial(2 n1, n1)| {n! | ) -----------------------------|, n!} | / (n1 + 1) (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A277178" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (2 n1 + 1) binomial(2 n1, n1)} / ----- n1 = 0 "A277188" LREtools/SearchTable: "SearchTable successful" 2 (6 n + 19 n + 16) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (-8 n - 8) hypergeom([-n, -n, -n], [1, 1], -1) {-------------------------------------------------------------------------------------------------------------------} 2 (n + 2) "A277220" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 1/2 \n | (95 - 190 5 ) (-5 + 9 5 )| 1/2 1/2 1/2 {|- --------------------------------| (-(95 - 190 5 ) (5 + 10) LegendreP(n, %2) + 95 LegendreP(n + 1, %2)), \ 190 / / 1/2 1/2 1/2 \n | (95 - 190 5 ) (-5 + 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 - 190 5 ) (5 + 10) LegendreQ(n, %2) + 95 LegendreQ(n + 1, %2)), \ 190 / / 1/2 1/2 1/2 \n | (95 + 190 5 ) (-5 - 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 + 190 5 ) (10 - 5 ) LegendreP(n, %1) + 95 LegendreP(n + 1, %1)), \ 190 / / 1/2 1/2 1/2 \n | (95 + 190 5 ) (-5 - 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 + 190 5 ) (10 - 5 ) LegendreQ(n, %1) + 95 LegendreQ(n + 1, %1))} \ 190 / 1/2 1/2 (95 + 190 5 ) %1 := ------------------ 19 1/2 1/2 (95 - 190 5 ) %2 := ------------------ 19 "A277221" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| , \ 2 / \ 2 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 5 | ||| / 1/2\n |----- | |----- |3/2 + ----| (2 n2 + 1) binomial(2 n2, n2) n2||| | 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |3/2 - ----| | ) |2 (3 + 5 ) (3 - 5 ) | ) ------------------------------------------------------|||} \ 2 / | / | | / (n2 + 1) (n2 + 2) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A277222" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\n / 1/2\n / 1/2\ / 1/2\n / 1/2\ / 1/2\n |----- | | 5 | | 5 | | 5 | | 8 5 | | 5 | | 8 5 | | 5 | 1/2 | \ | {|3/2 - ----| , |3/2 + ----| , |3/2 - ----| |n + 7 - ------|, |3/2 + ----| |n + 7 + ------|, |3/2 - ----| (5 n + 35 - 8 5 ) | ) |2 \ 2 / \ 2 / \ 2 / \ 5 / \ 2 / \ 5 / \ 2 / | / | |----- | \n1 = 0 \ 1/2 n1 1/2 (-n1 - 1) 2 (3 + 5 ) (3 - 5 ) (5 n1 + 75 n1 + 192) / / 1/2\(-n2 - 1) / 1/2\ \ |n1 - 1 | 5 | | 8 5 | 5 4 3 2 | |----- |3/2 + ----| |n2 + 8 - ------| (64 n2 + 759 n2 + 870 n2 - 2177 n2 + 1916 n2 - 1756) binomial(2 n2, n2)| / | \ \ 2 / \ 5 / | / | | ) -------------------------------------------------------------------------------------------------------------------| / | | / 2 2 | / \ |----- (n2 + 1) (n2 + 2) (2 n2 - 1) (5 (n2 + 1) + 75 n2 + 267) (5 n2 + 75 n2 + 192) | \n2 = 0 / \\ || / 1/2\ \|| | 8 5 | 1/2 ||| |n1 + 7 - ------| (5 n1 + 40 - 8 5 )|||} \ 5 / /|| || // "A277251" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 1/2 \n | (95 - 190 5 ) (-5 + 9 5 )| 1/2 1/2 1/2 {|- --------------------------------| (-(95 - 190 5 ) (5 + 10) LegendreP(n, %2) + 95 LegendreP(n + 1, %2)), \ 190 / / 1/2 1/2 1/2 \n | (95 - 190 5 ) (-5 + 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 - 190 5 ) (5 + 10) LegendreQ(n, %2) + 95 LegendreQ(n + 1, %2)), \ 190 / / 1/2 1/2 1/2 \n | (95 + 190 5 ) (-5 - 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 + 190 5 ) (10 - 5 ) LegendreP(n, %1) + 95 LegendreP(n + 1, %1)), \ 190 / / 1/2 1/2 1/2 \n | (95 + 190 5 ) (-5 - 9 5 )| 1/2 1/2 1/2 |- --------------------------------| (-(95 + 190 5 ) (10 - 5 ) LegendreQ(n, %1) + 95 LegendreQ(n + 1, %1))} \ 190 / 1/2 1/2 (95 + 190 5 ) %1 := ------------------ 19 1/2 1/2 (95 - 190 5 ) %2 := ------------------ 19 "A277287" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (n2 + 4 n2 + 2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A277296" 3 2 n (2 n + 3) (2 n + 1) binomial(2 n, n) (5 n + 75 n + 295 n + 348) {4 (17 n + 58), -----------------------------------------------------------------} (n + 3) (n + 2) (n + 1) "A277345" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\n1\ / / 1/2 \n1\ |n - 1 | 5 | | |n - 1 |5 | | |----- |1/2 - ----| | |----- |---- + 1/2| | | \ \ 2 / | | \ \ 2 / | {n! | ) --------------|, n! | ) --------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A277358" LREtools/SearchTable: "SearchTable successful" n {(-2) ((4 n + 4) LaguerreL(n + 1, - 5/4 - n, -1/4) - LaguerreL(n, - 1/4 - n, -1/4)) n!} "A277359" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" {(2 n + 1) binomial(2 n, n) (hypergeom([-n - 1], [n + 2], 1) - hypergeom([-n], [n + 1], 1))} "A277360" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 2 {(n + 1) 4 (n!) binomial(2 n, n) LaguerreL(2 n + 2, - 9/4 - 2 n, -1/4)} "A277374" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 2 ((n/2)!) n::even { (-n) 2 2 { { 2 2 binomial(n, n/2) ((n/2)!) n::even {{ (n + 1) 2 , { } { 2 2 ((n/2 + 1/2)!) { (-n + 1) 2 2 2 { -------------------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 2 { (n + 1) "A277378" LREtools/SolveLRE: "Reduced the order of" E^3+(-n-4)*E^2-(n+2)*(n+4)*E+(n+2)*(n+1)^2 "to two: Symmetric square" E^2-2*E-2*n-2 LREtools/SearchTable: "SearchTable successful" n 2 {(-1/2) HermiteH(n, I) } "A277382" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -3)} "A277393" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ | |{ || | (-n1 - 1) |{ / n1 \ || | 2 |{ |---- - 1/2| || |n - 1 |{ \ 2 / / n1 \ n1 || |----- |{ n1 (-1) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd || n n | \ \{ \ 2 / 2 /| {2 n!, 2 n! | ) ---------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / /{ / n1 \ \\ | |{ |----| || | (-n1 - 1) |{ \ 2 / / n1 \ || | 2 |{ (-4) |----|! n1::even|| |n - 1 |{ \ 2 / || |----- |{ || n | \ \{ 0 n1::odd /| 2 n! | ) -------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A277395" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / /3 n1 \ \ |n - 1 |---- + 1/2| | |----- \ 2 / 1/2 1/2 1/2 | | \ 2 ((-2 n1 - 2) LegendreP(n1, 2 ) + 2 n1 LegendreP(n1 + 1, 2 ))| {1, (n - 2) | ) ---------------------------------------------------------------------------------|, | / (n1 + 2) (n1 - 1) (n1 - 2) | |----- | \n1 = 0 / / /3 n1 \ \ |n - 1 |---- + 1/2| | |----- \ 2 / 1/2 1/2 1/2 | | \ 2 ((-2 n1 - 2) LegendreQ(n1, 2 ) + 2 n1 LegendreQ(n1 + 1, 2 ))| (n - 2) | ) ---------------------------------------------------------------------------------|, n - 2} | / (n1 + 2) (n1 - 1) (n1 - 2) | |----- | \n1 = 0 / "A277424" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-2) ((4 n + 4) LaguerreL(n + 1, - 5/4 - n, -1/4) - LaguerreL(n, - 1/4 - n, -1/4)) n!} "A277431" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / n1 \ \ |n - 1 |----| | /n - 1 \ |----- \ 2 / | |----- 1/2 n1| | \ 2 | | \ (-2 ) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A277432" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" / / n1 \ \ |n - 1 |----| | /n - 1 \ |----- \ 2 / | |----- 1/2 n1| | \ 2 | | \ (-2 ) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A277472" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ 0 n1::even\\ | (-n1 - 1) |{ || |n - 1 2 |{ n1 / n1 \ || |----- |{ n1 binomial(n1 - 1, ---- - 1/2) |---- - 1/2|! n1::odd || n n | \ \{ 2 \ 2 / /| {2 n!, 2 n! | ) ----------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / /{ n1 / n1 \ \\ | (-n1 - 1) |{ 2 |----|! n1::even|| |n - 1 2 |{ \ 2 / || |----- |{ || n | \ \{ 0 n1::odd /| 2 n! | ) ------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A277563" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 n! (n + 2 n - 7 n - 4 n + 5) {-------------------------------, n (n - 2) (n - 1) /n - 1 \ |----- n1 | 4 3 2 | \ (-1) (n1 + 1) (n1 - 1) n1 | n! (n + 2 n - 7 n - 4 n + 5) | ) ---------------------------------------------------------------------------------------------| | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 2 (n1 + 1) - 7 (n1 + 1) - 4 n1 + 1) (n1 + 2 n1 - 7 n1 - 4 n1 + 5)| \n1 = 0 / --------------------------------------------------------------------------------------------------------------------------------------} n (n - 2) (n - 1) "A277609" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 | \ (-1) n1 (n1 + 1) | n! (n + 3 n - n - 2) | ) -------------------------------------------------------------------| | / 3 2 3 2 | 3 2 |----- (n1 + 1)! ((n1 + 1) + 3 (n1 + 1) - n1 - 3) (n1 + 3 n1 - n1 - 2)| n! (n + 3 n - n - 2) \n1 = 0 / {----------------------, ---------------------------------------------------------------------------------------------------} (n - 1) n (n - 1) n "A277637" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) {1, ) -----------------------------------------------------------} / 2 ----- (n1 + 2) (n1 + 1) n1 = 0 "A277638" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A277652" LREtools/SearchTable: "SearchTable successful" 2 2 2 2 (n + n + 1) LegendreP(n + 1, 3) + (-3 n - 7 n - 3) LegendreP(n, 3) (n + n + 1) LegendreQ(n + 1, 3) + (-3 n - 7 n - 3) LegendreQ(n, 3) {--------------------------------------------------------------------, --------------------------------------------------------------------} n n "A277660" LREtools/SearchTable: "SearchTable successful" {(n + 1) (3 LegendreP(n + 1, 3) + (-4 n - 9) LegendreP(n, 3)), (n + 1) (3 LegendreQ(n + 1, 3) + (-4 n - 9) LegendreQ(n, 3))} "A277661" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(3 + 2 2 ) , (-2 2 + 3) , LegendreP(n, 3), LegendreQ(n, 3)} "A277662" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 n {(3 + 2 2 ) n, (-2 2 + 3) n, (n + 1) (3 LegendreP(n + 1, 3) + (-28 n - 9) LegendreP(n, 3)), (n + 1) (3 LegendreQ(n + 1, 3) + (-28 n - 9) LegendreQ(n, 3))} "A277663" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 2 2 {(n + 1) ((21 n + 3 n + 3) LegendreP(n + 1, 3) + (-7 n - 5 n - 9) LegendreP(n, 3)), 2 2 (n + 1) ((21 n + 3 n + 3) LegendreQ(n + 1, 3) + (-7 n - 5 n - 9) LegendreQ(n, 3)), // 1/2\ / 1/2\ \ // 1/2 \ / 1/2 \ \ 1/2 n ||1376 576 2 | 2 |2448 522 2 | | 1/2 n || 576 2 1376| 2 | 522 2 2448| | (3 + 2 2 ) ||---- + --------| n + |---- + --------| n + 1| n, (-2 2 + 3) ||- -------- + ----| n + |- -------- + ----| n + 1| n} \\1201 1201 / \1201 1201 / / \\ 1201 1201/ \ 1201 1201/ / "A277664" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 {(n + 1) ((555 n + 1422 n + 1092 n + 207 n + 27) LegendreP(n + 1, 3) + (-185 n - 738 n - 1156 n - 633 n - 81) LegendreP(n, 3)), 4 3 2 4 3 2 (n + 1) ((555 n + 1422 n + 1092 n + 207 n + 27) LegendreQ(n + 1, 3) + (-185 n - 738 n - 1156 n - 633 n - 81) LegendreQ(n, 3)), / / 1/2\ / 1/2\ 1/2\ 1/2 n | 3 | 27 2 | 2 |247 81 2 | 135 765 2 | (3 + 2 2 ) |n + |15/4 - -------| n + |--- - -------| n + --- - --------| n, \ \ 16 / \32 16 / 16 128 / / / 1/2 \ / 1/2 \ 1/2 \ 1/2 n | 3 |27 2 | 2 |81 2 247| 765 2 135| (-2 2 + 3) |n + |------- + 15/4| n + |------- + ---| n + -------- + ---| n} \ \ 16 / \ 16 32 / 128 16 / "A277665" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" {(n + 1) ( 5 4 3 2 5 4 3 2 (4440 n + 10155 n + 7110 n - 525 n - 300 n - 630) LegendreP(n + 1, 3) + (-740 n - 1129 n + 74 n + 2431 n + 1884 n + 1890) LegendreP(n, 3) ), (n + 1) ( 5 4 3 2 5 4 3 2 (4440 n + 10155 n + 7110 n - 525 n - 300 n - 630) LegendreQ(n + 1, 3) + (-740 n - 1129 n + 74 n + 2431 n + 1884 n + 1890) LegendreQ(n, 3) / / 1/2\ / 1/2\ / 1/2\ / 1/2\ 1/2\ 1/2 n | 5 |81 99 2 | 4 |3755 495 2 | 3 |897 9351 2 | 2 |55957 9837 2 | 56649 325107 2 | ), (3 + 2 2 ) |n + |-- - -------| n + |---- - --------| n + |--- - ---------| n + |----- - ---------| n + ----- - -----------| n, \ \10 40 / \128 32 / \16 256 / \1024 256 / 2560 20480 / / / 1/2 \ / 1/2 \ / 1/2 \ / 1/2 \ 1/2 \ 1/2 n | 5 |99 2 81| 4 |495 2 3755| 3 |9351 2 897| 2 |9837 2 55957| 325107 2 56649| (-2 2 + 3) |n + |------- + --| n + |-------- + ----| n + |--------- + ---| n + |--------- + -----| n + ----------- + -----| n} \ \ 40 10/ \ 32 128 / \ 256 16 / \ 256 1024 / 20480 2560 / "A277860" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 48 (4 n1 + 1) (4 n1 + 3) binomial(3 n1, n1) binomial(4 n1, n1)| {48 , 48 | ) -----------------------------------------------------------------------|} | / (2 n1 + 3) (n1 + 1) | |----- | \n1 = 0 / "A277871" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (3 (2 n + 1) n hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) - 2 (n + 1) (4 n + 1) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------, (n + 2) (n + 1) (2 n + 1) binomial(2 n, n) n ----------------------------} (n + 1) (n + 2) "A277876" n 2 (-1) n! n! (2 n + 8 n + 7) {---------------, -------------------} (n + 2) (n + 3) (n + 2) (n + 3) "A277920" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A277922" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A277924" n binomial(2 n, n) (3 n + 1) {4 , --------------------------} n + 1 "A277956" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 4 3 2 || | \ (1/2 + 1/2 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (91 n2 + 860 n2 + 2841 n2 + 3920 n2 + 1920)|| | ) -------------------------------------------------------------------------------------------------------------------||} | / (n2 + 3) (n2 + 1) (n2 + 2) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) || |----- || \n2 = 0 // "A277957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 4 3 2 || | \ (1/2 + 1/2 I 3 ) (3 n2 + 4) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (91 n2 + 1068 n2 + 4385 n2 + 7548 n2 + 4620)|| | ) -------------------------------------------------------------------------------------------------------------------------------||} | / (2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) (n2 + 3) (n2 + 2) (n2 + 1) || |----- || \n2 = 0 // "A277969" LREtools/SearchTable: "SearchTable successful" 2 3 2 (2 n - 1) (16 n - 56 n + 53) hypergeom([-1/2, -n - 1], [1], -4) + (-32 n + 112 n - 92 n - 53) hypergeom([-1/2, -n], [1], -4) {-------------------------------------------------------------------------------------------------------------------------------} n (n - 2) (n - 1) "A277973" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A278023" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + _Z + 1, index = 1) , RootOf(_Z + _Z + 1, index = 2) , RootOf(_Z + _Z + 1, index = 3) } "A278069" LREtools/SearchTable: "SearchTable successful" {BesselI(n + 1/2, 1/2) + BesselI(n - 1/2, 1/2), BesselK(n + 1/2, -1/2) + BesselK(n - 1/2, -1/2)} "A278070" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) (BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A278301" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) n {------------------, hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4)} (n + 1) (n + 2) "A278391" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" 2 (n + 1) (8 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 11 n - 1) hypergeom([-1/2, -n], [1], -4) {--------------------------------------------------------------------------------------------------------} n "A278394" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (4 (2 n + 1) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) - (9 n + 5) n hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(n (5 n + 3)), binomial(2 n, n)} "A278405" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A278415" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A278429" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n 2 n 2 | \ | (-1) 2 n1 (2 n1 + 1) (3 n1 + 1) binomial(2 n1, n1)|| {(-1) (9 n - n - 2), (-1) (9 n - n - 2) | ) |- ------------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (9 (n1 + 1) - n1 - 3) (9 n1 - n1 - 2) /| \n1 = 0 / "A278459" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A278461" memory used=224272.8MB, alloc=3351.5MB, time=1709.84 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A278472" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 - 5 ) , (2 + 5 ) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2) (n2 + 2 n2 - 2)|| (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -----------------------------------------------------------------------------||} | / | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A278618" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) n {--------------------------, (-1) n + 1 ((8 n + 4) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (3 n + 3) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n) (2 n + 1)/((5 n + 3) (n + 1))} "A278646" LREtools/SearchTable: "SearchTable successful" n {(-1) (2 n + 1) (hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4))} "A278745" LREtools/SearchTable: "SearchTable not successful" {} "A278934" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A278990" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1), (-1) BesselK(n + 1/2, -1)} "A278991" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A278992" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 3)} "A279014" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (n2 - 5 n2 - 2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A279055" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 3 2 | \ (n1 + 1) (n1!) (2 n1 + 5) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1)| (n + 1) (n!) | ) -----------------------------------------------------------------| | / 2 2 | 3 2 |----- (n1 + 2) ((n1 + 1)!) (4 n1 + 10) | (n + 1) (n!) \n1 = 0 / {------------------------------------, -----------------------------------------------------------------------------------------} (2 n + 3) (2 n + 1) binomial(2 n, n) (2 n + 3) (2 n + 1) binomial(2 n, n) "A279127" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" BesselJ(2 n + 1, -2) + BesselJ(2 n - 1, -2) BesselY(2 n + 1, -2) + BesselY(2 n - 1, -2) {-------------------------------------------, -------------------------------------------} n n "A279136" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n {1, (-1) , (-1) hypergeom([1/2, -n], [1], 4)} "A279159" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {1, RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 %1 := _Z - _Z + 1 "A279444" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A279553" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A279557" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) binomial(2 n1, n1) n1 {1, ) --------------------------------} / (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A279560" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | | \ binomial(2 n1, n1) (9 n1 - 7 n1 - 42 n1 + 22)| {1, (3 n - 5) | ) -----------------------------------------------|, 3 n - 5} | / (n1 + 1) (2 n1 - 1) (3 n1 - 2) (3 n1 - 5) | |----- | \n1 = 0 / "A279561" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) n1 {1, ) ---------------------} / n1 + 1 ----- n1 = 0 "A279563" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ binomial(2 n1, n1)| {1, n, 2 , n | ) ------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A279565" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ / n1 {(-1) , (-1) | ) |- (-1) | / \ |----- \n1 = 0 / n1 n1 n1 n1 \ |(20 n1 + 10) hypergeom([- ----, - ---- - 1/2], [-2 n1 - 2], -4) + (5 n1 - 2) hypergeom([- ----, - ---- + 1/2], [-2 n1], -4)| binomial(2 n1, n1)/ \ 2 2 2 2 / \ | \| ((n1 + 1) (3 n1 + 1))||} /| | / "A279619" LREtools/SearchTable: "SearchTable not successful" {} "A279683" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1! | | \ 2 n1! | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A279927" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- ----- \ n1 \ n1 {1, ) (-I) n1!, ) I n1!} / / ----- ----- n1 = 0 n1 = 0 "A280425" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 5 3 2 n! (n - 15 n + 10 n + 29 n - 14) 5 3 2 {-----------------------------------, n! (n - 15 n + 10 n + 29 n - 14) n (n - 1) (n - 2) (n - 3) /n - 1 \ |----- n1 | | \ (-1) (n1 + 1) n1 (n1 - 1) (n1 - 2) | | ) -----------------------------------------------------------------------------------------------------|/(n (n - 1) (n - 2) (n - 3))} | / 5 3 2 5 3 2 | |----- (n1 + 1)! ((n1 + 1) - 15 (n1 + 1) + 10 (n1 + 1) + 29 n1 + 15) (n1 - 15 n1 + 10 n1 + 29 n1 - 14)| \n1 = 0 / "A280556" {1, (n + 2) (n + 1) n! (n - 1)} "A280920" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 6 5 4 3 2 |----- n! (n - 3 n - 20 n + 65 n + 34 n - 161 n + 37) 6 5 4 3 2 | \ n1 {---------------------------------------------------, n! (n - 3 n - 20 n + 65 n + 34 n - 161 n + 37) | ) (-1) (n1 + 1) n1 (n1 - 1) n (n - 1) (n - 2) (n - 3) (n - 4) | / |----- \n1 = 0 / 6 5 4 3 2 (n1 - 2) (n1 - 3) / ((n1 + 1)! ((n1 + 1) - 3 (n1 + 1) - 20 (n1 + 1) + 65 (n1 + 1) + 34 (n1 + 1) - 161 n1 - 124) / \ | 6 5 4 3 2 | (n1 - 3 n1 - 20 n1 + 65 n1 + 34 n1 - 161 n1 + 37))|/(n (n - 1) (n - 2) (n - 3) (n - 4))} | | / "A280970" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 n1! | {n! | ) ---------|, (n + 1) n! (n - 6), n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A281262" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (2 n1 + 1) (-1) binomial(2 n1, n1) | {n! binomial(2 n, n), n! binomial(2 n, n) | ) ---------------------------------------------|} | / (n1 + 1) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A281433" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A281593" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) n1 {1, ) ---------------------} / n1 + 1 ----- n1 = 0 "A281861" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / / { n2 \\\ | | | { 4 ||| | | | { ------------------------------------ n2::even||| | | | { n2 ||| | | | { (n2 + 1) (n2 + 3) binomial(n2, ----) ||| | | | { 2 ||| | | | { ||| | | | { (2 n2 - 2) ||| | | | { 2 ||| | | | { - ---------------------------------------- n2::odd ||| |n - 1 | |n1 - 1 { n2 ||| |----- | |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) ||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ { 2 ||| {(5 ) , (-5 ) , (5 ) | ) |1/5 5 (-1) | ) ------------------------------------------------------------|||, | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// / / / { n2 \\\ | | | { 2 binomial(n2, ----) ||| | | | { 2 ||| | | | { - -------------------- n2::even||| | | | { n2 + 2 ||| | | | { ||| | | | { n2 ||| | | | { 2 binomial(n2 + 1, ---- + 1/2) ||| |n - 1 | |n1 - 1 { 2 ||| |----- | |----- { ------------------------------ n2::odd ||| 1/2 n | \ | 1/2 n1 | \ { n2 + 3 ||| (5 ) | ) |1/5 5 (-1) | ) ------------------------------------------------|||} | / | | / 1/2 (n2 + 1) ||| |----- | |----- (-5 ) ||| \n1 = 0 \ \n2 = 0 /// "A281912" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n! (n + n + 1), (n + 1) n! (n + n + 1) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + n1 + 2) (n1 + n1 + 1)| \n1 = 0 / "A281946" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {(n + 1) n! (3 n + 10 n + 27 n + 20 n + 11), (n + 1) n! (3 n + 10 n + 27 n + 20 n + 11) /n - 1 \ |----- | | \ 6 n1 + 17 | | ) ---------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! (3 (n1 + 1) + 10 (n1 + 1) + 27 (n1 + 1) + 20 n1 + 31) (3 n1 + 10 n1 + 27 n1 + 20 n1 + 11)| \n1 = 0 / "A281964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (n1 + 1) (-I) n1!| | \ (n1 + 1) I n1! | {(n + 1) n!, (n + 1) n! | ) -------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)! | | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A282132" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (n1 + 1) (-I) n1!| | \ (n1 + 1) I n1! | {(n + 1) n!, (n + 1) n! | ) -------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)! | | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A282733" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 3 2 | n n | \ 2 n1 (n1 - 1) (22 n1 - 18 n1 - 109 n1 - 66) binomial(4 n1, n1)| {n binomial(4 n, 2 n), 16 (n + 2), 16 (n + 2) | ) ---------------------------------------------------------------------------|} | / (n1 + 1) (n1 + 2) (n1 + 3) (3 n1 + 1) (3 n1 + 2) (3 n1 + 4) | |----- | \n1 = 0 / "A282735" LREtools/SearchTable: "SearchTable not successful" {} "A282736" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A282822" {n! (n - 4)} "A282876" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ 1/2 n | 2 1/2 2 | (-2 2 ) |(2 n + 10) LegendreP(n, ----) - 2 (2 n + 7) LegendreP(n + 1, ----)| \ 2 2 / {- ----------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) / 1/2 1/2 \ 1/2 n | 2 1/2 2 | (-2 2 ) |(2 n + 10) LegendreQ(n, ----) - 2 (2 n + 7) LegendreQ(n + 1, ----)| \ 2 2 / - ----------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A283184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A283298" LREtools/SearchTable: "SearchTable not successful" {} "A283322" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {(n + 1) n!, n! LaguerreL(n, -1)} "A283667" LREtools/SearchTable: "SearchTable successful" 2 ((8 n + 19 n + 9) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + 3 (n + 1) n hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) {------------------------------------------------------------------------------------------------------------------------------------------} (13 n + 9) (n + 1) "A283799" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 16 binomial(2 n, n/2) (2 n + 1) { ------------------------------- n::even { 3 n + 2 {{ , { 6 (n + 1) binomial(2 n + 2, n/2 + 1/2) { -------------------------------------- n::odd { 2 n + 1 { n { 3 16 binomial(2 n, n/2) { --------------------------------------------- n::even { 3 n { (3 n + 1) binomial(n, n/2) binomial(3 n, ---) { 2 { } { (4 n - 4) { 8 2 (2 n - 1) (2 n + 1) binomial(2 n - 2, n/2 - 1/2) { ----------------------------------------------------------------------------- n::odd { 3 n { n (3 n - 2) (3 n + 2) binomial(n - 1, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 "A284039" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | 2 2 2 2 | \ (n1 + 1) (n1!) (n1 + 4 n1 + 1)| {(n + 1) (n!) , (n + 1) (n!) | ) --------------------------------|} | / 2 | |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A284216" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A284230" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A284449" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {n + 2} "A284461" LREtools/SearchTable: "SearchTable not successful" {} "A284714" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n 2 n 2 {(-1) (27 n + 63 n + 34), (-1) (675 n + 1035 n + 382), /n - 1 \ |----- / n1 n1 \| n 2 | \ | (-1) 2 (2 n1 + 1) (2 n1 + 3) binomial(2 n1, n1) || n 2 (-1) (27 n + 63 n + 34) | ) |- -------------------------------------------------------------------||, (-1) (675 n + 1035 n + 382) | / | 2 2 || |----- \ (n1 + 1) (n1 + 2) (27 (n1 + 1) + 63 n1 + 97) (27 n1 + 63 n1 + 34)/| \n1 = 0 / /n - 1 \ |----- / n1 n1 2 \| | \ | (-1) 4 ((400 n1 - 66 n1 - 467) hypergeom([-1/2, -n1 - 1], [1], -1) - 4 (100 n1 - 53) (n1 + 1) hypergeom([-1/2, -n1], [1], -1))|| | ) |- -----------------------------------------------------------------------------------------------------------------------------------||, | / | 2 2 || |----- \ (n1 + 2) (675 (n1 + 1) + 1035 n1 + 1417) (675 n1 + 1035 n1 + 382) /| \n1 = 0 / /n - 1 \ |----- / n1 3 2 \| n 2 | \ | (-1) (459 n1 + 1557 n1 + 452 n1 - 640) binomial(2 n1, n1) || (-1) (675 n + 1035 n + 382) | ) |- ----------------------------------------------------------------------------||} | / | 2 2 || |----- \ (n1 + 1) (n1 + 2) (675 (n1 + 1) + 1035 n1 + 1417) (675 n1 + 1035 n1 + 382)/| \n1 = 0 / "A284716" LREtools/SearchTable: "SearchTable successful" 2 3 2 (2 n + 1) (n - 3 n - 9) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (6 n + 57 n + 99 n + 27) hypergeom([1/2, -n, -n], [1, 1], 4) {-------------------------------------------------------------------------------------------------------------------------------------} 2 2 2 n (n + 2) (n + 3) "A284717" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" 2 2 binomial(2 n, n) (n + 17 n + 27) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 63 n - 81) hypergeom([1/2, -n, -n], [1, 1], 4) {1, ----------------, ----------------------------------------------------------------------------------------------------------------------} n + 1 2 2 (n + 2) (n + 3) "A284756" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A284778" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((2 n + 7 n + 9) hypergeom([1/2, -n - 1], [1], 4) + 3 (2 n + 7) (n + 1) hypergeom([1/2, -n], [1], 4)) {(-1) , ------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A284843" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 1) n! (n + 3) , (n + 1) n! (n + 3) | ) ------------------------------------|} | / (n1 + 3) (n1 + 4) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A284844" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 1) n! (n + 4) (n + 7 n + 11), (n + 1) n! (n + 4) (n + 7 n + 11) | ) ------------------------------------------------------------|} | / 2 2 | |----- (n1 + 2) (n1 + 1)! ((n1 + 1) + 7 n1 + 18) (n1 + 7 n1 + 11)| \n1 = 0 / "A284845" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 {(n + 1) n! (n + 5) (n + 12 n + 44 n + 49), /n - 1 \ |----- n1 | 3 2 | \ (-1) | (n + 1) n! (n + 5) (n + 12 n + 44 n + 49) | ) --------------------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 2) (n1 + 1)! ((n1 + 1) + 12 (n1 + 1) + 44 n1 + 93) (n1 + 12 n1 + 44 n1 + 49)| \n1 = 0 / "A284968" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, 2 n + 3, ) ----------------------------------------} / (n1 + 3) (n1 + 2) (n1 + 1) ----- n1 = 0 "A285146" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {n + 2} "A285166" LREtools/SearchTable: "SearchTable successful" n {2 ((4 n + 7) (4 n + 5) hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) + 3 (n + 2) (3 n + 5) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (7 n + 13))} "A285167" memory used=225674.4MB, alloc=3351.5MB, time=1719.41 LREtools/SearchTable: "SearchTable successful" n 2 {2 ((4 n + 5) (4 n + 7) (n + 1) (181 n + 795 n + 864) hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) - 3 (3 n + 5) (5 n + 11) (3 n + 7) (n + 6) (n + 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (n + 3) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (7 n + 13))} "A285168" LREtools/SearchTable: "SearchTable successful" n 6 5 4 3 2 {2 ((4 n + 7) (4 n + 5) (53203 n + 624509 n + 2830875 n + 6124027 n + 6134546 n + 1933704 n - 396864) hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) + 5 4 3 2 3 (3 n + 5) (3 n + 7) (n + 2) (2185 n + 24086 n + 109263 n + 268690 n + 372896 n + 230880) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (n + 3) (n + 4) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (3 n + 10) (3 n + 11) (7 n + 13))} "A285169" memory used=226336.3MB, alloc=3351.5MB, time=1724.85 LREtools/SearchTable: "SearchTable successful" n 9 8 7 6 5 4 3 {2 ((4 n + 7) (4 n + 5) (13103771 n + 268309731 n + 2324593674 n + 11122498434 n + 32263248639 n + 59056689639 n + 70091339356 n 2 + 56246610996 n + 31238933760 n + 9476611200) hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) + 3 (3 n + 5) (3 n + 7) (n + 2) 8 7 6 5 4 3 2 (338915 n + 5550999 n + 34092744 n + 83709840 n - 38605455 n - 719690019 n - 1792868444 n - 2084094660 n - 1017273600) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (3 n + 10) (3 n + 11) (3 n + 13) (3 n + 14) (7 n + 13))} "A285170" memory used=226956.4MB, alloc=3351.5MB, time=1729.92 memory used=227423.7MB, alloc=3351.5MB, time=1734.20 memory used=227755.2MB, alloc=3351.5MB, time=1738.25 memory used=228161.3MB, alloc=3351.5MB, time=1742.71 LREtools/SearchTable: "SearchTable successful" n 12 11 10 9 8 7 {2 ((4 n + 7) (4 n + 5) (928945669 n + 28623995618 n + 387240650757 n + 3029050265634 n + 15196352686035 n + 51345060182430 n 6 5 4 3 2 + 119674415323223 n + 194394123684886 n + 219356582482476 n + 165181012285512 n + 68755001703840 n + 920844529920 n - 8938833408000) 11 10 9 8 hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) + 3 (3 n + 5) (3 n + 7) (n + 2) (26509831 n + 744117155 n + 9049036272 n + 63002761134 n 7 6 5 4 3 2 + 282231437211 n + 883583563971 n + 2111813568278 n + 4174827242956 n + 6843266835288 n + 8437180020384 n + 6692997467520 n + 2538317260800) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (3 n + 10) (3 n + 11) (3 n + 13) (3 n + 14) (3 n + 16) (3 n + 17) (7 n + 13))} "A285171" memory used=228785.7MB, alloc=3351.5MB, time=1748.00 memory used=229260.0MB, alloc=3351.5MB, time=1752.21 memory used=229598.9MB, alloc=3351.5MB, time=1756.20 memory used=230009.8MB, alloc=3351.5MB, time=1760.39 memory used=230504.3MB, alloc=3351.5MB, time=1764.84 memory used=230880.7MB, alloc=3351.5MB, time=1769.30 memory used=231113.6MB, alloc=3351.5MB, time=1773.36 memory used=231385.4MB, alloc=3351.5MB, time=1777.75 memory used=231635.8MB, alloc=3351.5MB, time=1781.98 memory used=231918.9MB, alloc=3351.5MB, time=1786.46 memory used=232219.3MB, alloc=3351.5MB, time=1790.96 memory used=232533.5MB, alloc=3351.5MB, time=1795.76 LREtools/SearchTable: "SearchTable successful" n 15 14 13 12 11 {2 ((4 n + 7) (4 n + 5) (595757534261 n + 25644301068670 n + 496306481548040 n + 5715594576075000 n + 43720983857538222 n 10 9 8 7 6 + 235121973558032220 n + 919555265456767540 n + 2676624051504617840 n + 5906461400218943653 n + 10035877000740877110 n 5 4 3 2 + 13252655129193620220 n + 13552090711902333960 n + 10568062075351846464 n + 6223324770286243200 n + 2715912225092505600 n 14 13 + 691956044236800000) hypergeom([- n/2, - n/2 - 1/2], [- 7/2 - 2 n], -1) + 3 (3 n + 5) (3 n + 7) (n + 2) (16735842947 n + 669208990477 n 12 11 10 9 8 7 + 11866450042727 n + 123044184798057 n + 828419645544561 n + 3800044713365631 n + 12078273387872941 n + 26092367870200451 n 6 5 4 3 2 + 33642599237357992 n + 3607990065907992 n - 95078998740385968 n - 239272652911851408 n - 320162169215577600 n - 244893204143481600 n - 86041075792896000) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1)) binomial(4 n, n) (2 n + 1) (4 n + 1) (4 n + 3)/((n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (3 n + 7) (3 n + 8) (3 n + 10) (3 n + 11) (3 n + 13) (3 n + 14) (3 n + 16) (3 n + 17) (3 n + 19) (3 n + 20) (7 n + 13))} "A285195" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 |----- | |----- n n | \ (2 n1 + 1) binomial(2 n1, n1) | n | \ {(-1/6) , (-1/6) | ) --------------------------------|, (-1/6) | ) | / (n1 + 1)| | / |----- (n1 + 1) (n1 + 2) (-1/6) | |----- \n1 = 0 / \n1 = 0 ((2 n1 + 1) (59 n1 + 28) hypergeom([n1 + 1, -n1 - 1], [-n1 - 1/2], 1/4) - (4 n1 + 1) (43 n1 + 36) hypergeom([n1, -n1], [-n1 + 1/2], 1/4)) \ | (n1 + 1) | binomial(2 n1, n1)/((n1 + 1) (n1 + 2) (2 n1 - 1) (-1/6) )|} | | / "A285199" LREtools/SearchTable: "SearchTable successful" {n! LegendreP(n, 2), n! LegendreQ(n, 2)} "A285201" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- | ----- \ | \ n2 + 2 | \ {1, ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A285231" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 2) (n1 + 1) n1! (n1 + 5)| {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------------------|} | / (n1 + 3) (n1 + 1)! | |----- | \n1 = 0 / "A285232" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 4) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 3) (n2 + 1) n2! (n2 + 6)|| | (n1 + 1) n1! | ) ------------------------------|| |n - 1 | / (n2 + 4) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------------------|} | / (n1 + 4) (n1 + 1)! | |----- | \n1 = 0 / "A285233" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 5) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 4) (n3 + 1) n3! (n3 + 7)||| | | (n2 + 1) n2! | ) ------------------------------||| | |n1 - 1 | / (n3 + 5) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------------------|| |n - 1 | / (n2 + 5) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------------------|} | / (n1 + 5) (n1 + 1)! | |----- | \n1 = 0 / "A285234" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 6) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 6) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, (n + 5) (n + 4) (n + 3) | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 5) (n4 + 1) n4! (n4 + 8)|||| | | | (n3 + 1) n3! | ) ------------------------------|||| | | |n2 - 1 | / (n4 + 6) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------------------||| | |n1 - 1 | / (n3 + 6) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------------------|| |n - 1 | / (n2 + 6) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------------------|} | / (n1 + 6) (n1 + 1)! | |----- | \n1 = 0 / "A285235" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 7) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 7) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 6) (n5 + 1) n5! (n5 + 9)||||| | | | | (n4 + 1) n4! | ) ------------------------------||||| | | | |n3 - 1 | / (n5 + 7) (n5 + 1)! ||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------------------|||| | | |n2 - 1 | / (n4 + 7) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------------------||| | |n1 - 1 | / (n3 + 7) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 7) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------------------|} | / (n1 + 7) (n1 + 1)! | |----- | \n1 = 0 / "A285236" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! | {(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------|, | / (n1 + 8) (n1 + 1)!| |----- | \n1 = 0 / / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| |n - 1 | / (n2 + 8) (n2 + 1)!|| |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ----------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / /n2 - 1 \\\ | | |----- ||| | | | \ (n3 + 1) n3! ||| | | (n2 + 1) n2! | ) ------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)!||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ----------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) --------------------------------------------------------------|, (n + 7) | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / / / /n3 - 1 \\\\ | | | |----- |||| | | | | \ (n4 + 1) n4! |||| | | | (n3 + 1) n3! | ) ------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)!|||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) ----------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) --------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------------------------------------|, | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! / / / / /n4 - 1 \\\\\ | | | | |----- ||||| | | | | | \ (n5 + 1) n5! ||||| | | | | (n4 + 1) n4! | ) ------------------||||| | | | |n3 - 1 | / (n5 + 8) (n5 + 1)!||||| | | | |----- |----- ||||| | | | | \ \n5 = 0 /|||| | | | (n3 + 1) n3! | ) ----------------------------------------|||| | | |n2 - 1 | / (n4 + 8) (n4 + 1)! |||| | | |----- |----- |||| | | | \ \n4 = 0 /||| | | (n2 + 1) n2! | ) --------------------------------------------------------------||| | |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| | |----- |----- ||| | | \ \n3 = 0 /|| | (n1 + 1) n1! | ) ------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 8) (n2 + 1)! || |----- |----- || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------|, (n + 7) (n + 6) (n + 5) | / (n1 + 8) (n1 + 1)! | |----- | \n1 = 0 / / | | | | | | | | | | | | | | | | | | | |n - 1 |----- | \ (n + 4) (n + 3) (n + 2) (n + 1) n! | ) | / |----- \n1 = 0 / / / / /n5 - 1 \\\\\\ | | | | |----- |||||| | | | | | \ (n6 + 7) (n6 + 1) n6! (n6 + 10)|||||| | | | | (n5 + 1) n5! | ) -------------------------------|||||| | | | |n4 - 1 | / (n6 + 8) (n6 + 1)! |||||| | | | |----- |----- |||||| | | | | \ \n6 = 0 /||||| | | | (n4 + 1) n4! | ) -----------------------------------------------------||||| | | |n3 - 1 | / (n5 + 8) (n5 + 1)! ||||| | | |----- |----- ||||| | | | \ \n5 = 0 /|||| | | (n3 + 1) n3! | ) ---------------------------------------------------------------------------|||| | |n2 - 1 | / (n4 + 8) (n4 + 1)! |||| | |----- |----- |||| | | \ \n4 = 0 /||| | (n2 + 1) n2! | ) -------------------------------------------------------------------------------------------------||| |n1 - 1 | / (n3 + 8) (n3 + 1)! ||| |----- |----- ||| | \ \n3 = 0 /|| (n1 + 1) n1! | ) -----------------------------------------------------------------------------------------------------------------------|| | / (n2 + 8) (n2 + 1)! || |----- || \n2 = 0 /| ---------------------------------------------------------------------------------------------------------------------------------------------|} (n1 + 8) (n1 + 1)! | | / "A285382" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! (n1 + 4) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A285489" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) n1! (n1 + 5 n1 + 8)| {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) -----------------------------|} | / (n1 + 2) (n1 + 1)! | |----- | \n1 = 0 / "A285673" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable not successful" {} "A285795" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 3) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A285850" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! (BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-2) n! (BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A285853" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ (n1 + 1) n1! (2 n1 + 7) | (n + 1) (-1) n! {(n + 1) n!, (n + 1) n! | ) ---------------------------|, ----------------} | / (n1 + 3) (n1 + 2) (n1 + 1)!| n + 2 |----- | \n1 = 0 / "A285917" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 binomial(n, n/2) (n + 1) n { { -------------------------- n::even {1, 2 , { (2 n - 2) , { n + 2 } { 2 (n + 1) { { 1/2 ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A286032" LREtools/SearchTable: "SearchTable successful" 1/2 (- n/2) 2 {2 HermiteH(n + 1, ----)} 2 "A286033" n {(-1) , binomial(2 n, n)} "A286038" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! (n + 1) ((2 n + 1) BesselJ(n + 1/2, -1) + BesselJ(n - 1/2, -1)), (-2) n! (n + 1) ((2 n + 1) BesselY(n + 1/2, -1) + BesselY(n - 1/2, -1))} "A286076" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A286209" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 2 {n + 3 n + 4} "A286282" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ n1 - n1 + 1| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A286286" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A286723" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 7/4) \| {4 GAMMA(n + 7/4), 4 GAMMA(n + 7/4) | ) |1/4 ----------------||} | / \ GAMMA(n1 + 11/4)/| |----- | \n1 = 0 / "A287041" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n), /n - 1 \ |----- (n1 + 1) | n | \ 2 (n1 + 1) (n1 + 2) (n1 + 3) n1! | (2 n + 5) (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n) | ) ----------------------------------------------------------|, (2 n + 5) (2 n + 3) | / (2 n1 + 3) (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / n (2 n + 1) (1/2) n! binomial(2 n, n) / /n1 - 1 \ \ | |----- (-n2) | | | (n1 + 1) | \ 2 (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) binomial(2 n2, n2) n2!| | | 2 (n1 + 1) (n1 + 2) (n1 + 3) | ) --------------------------------------------------------------| n1!| |n - 1 | / (n2 + 2) (n2 + 3) (n2 + 4) (n2 + 1)! | | |----- |----- | | | \ \n2 = 0 / | | ) ----------------------------------------------------------------------------------------------------------------|} | / (2 n1 + 3) (2 n1 + 5) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! | |----- | \n1 = 0 / "A287042" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n), /n - 1 \ |----- (n1 + 1) | n | \ 2 (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) n1! | (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n) | ) ---------------------------------------------------------------------| | / (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / n , (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n) / /n1 - 1 \ \ | |----- (-n2) | | | (n1 + 1) | \ 2 (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) binomial(2 n2, n2) n2!| | | 2 (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) | ) -------------------------------------------------------------------------| n1!| |n - 1 | / (n2 + 2) (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 1)! | | |----- |----- | | | \ \n2 = 0 / | | ) ------------------------------------------------------------------------------------------------------------------------------------|, | / (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! | |----- | \n1 = 0 / / | | | |n - 1 |----- n | \ (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n) | ) | / |----- \n1 = 0 /n1 - 1 \ \ |----- (-n2) | | (n1 + 1) | \ 6 (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) (2 n2 + 9) binomial(2 n2, n2) n2!| | 2 (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) | ) ------------------------------------------------------------------------------------| n1!| | / (n2 + 2) (n2 + 3) (n2 + 4) (n2 + 1)! | | |----- | | \n2 = 0 / | -----------------------------------------------------------------------------------------------------------------------------------------------|, (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)! | | / /n - 1 /n1 - 1 |----- |----- n | \ (n1 + 1) | \ (-n2) (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n) | ) 2 (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) | ) 2 | / | / |----- |----- \n1 = 0 \n2 = 0 /n2 - 1 / |----- | | \ | (-n3) 2 (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) | ) |2 3 (2 n3 + 1) (4 n3 + 34 n3 + 69) | / | |----- | \n3 = 0 \ /n3 - 1 \ |----- (n4 + 1) 2 | | \ 6 (n4 + 1) (n4 + 2) (n4 + 3) (n4 + 4) (n4 + 5) (4 n4 + 42 n4 + 125) n4! | | ) ------------------------------------------------------------------------------------------------------------------------------------| | / 2 2 | |----- (2 n4 + 3) (2 n4 + 5) (2 n4 + 7) (2 n4 + 9) (4 (n4 + 1) + 34 n4 + 103) (12 n4 + 102 n4 + 207) binomial(2 n4 + 2, n4 + 1) (n4 + 1)!| \n4 = 0 / \\ \ || | || | binomial(2 n3, n3) n3!/(binomial(2 n3 + 2, n3 + 1) (n3 + 1)!)|| binomial(2 n2, n2) n2!/((n2 + 2) (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 1)!)| n1!/( || | || | // / \ / | | | n | (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!)|, (2 n + 5) (2 n + 3) (2 n + 1) (2 n + 7) (1/2) n! binomial(2 n, n) | | | | | / \ n - 1 /n1 - 1 /n2 - 1 / ----- |----- |----- | \ (n1 + 1) | \ (-n2) | \ | (-n3) ) 2 (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) | ) 2 (2 n2 + 1) (2 n2 + 3) (2 n2 + 5) (2 n2 + 7) | ) |2 3 (2 n3 + 1) / | / | / | ----- |----- |----- | n1 = 0 \n2 = 0 \n3 = 0 \ /n3 - 1 |----- 2 | \ (n4 + 1) 2 (4 n3 + 34 n3 + 69) | ) 6 (n4 + 1) (n4 + 2) (n4 + 3) (n4 + 4) (n4 + 5) (4 n4 + 42 n4 + 125) | / |----- \n4 = 0 /n4 - 1 \ |----- (-n5) 2 2 | | \ 2 (2 n5 + 1) (2 n5 + 3) (2 n5 + 5) (2 n5 + 7) (2 n5 + 9) (4 n5 + 42 n5 + 107) (8 n5 + 58 n5 + 111) binomial(2 n5, n5) n5!| | ) --------------------------------------------------------------------------------------------------------------------------------| n4! | / 2 2 | |----- (n5 + 2) (n5 + 3) (n5 + 4) (n5 + 5) (n5 + 6) (4 (n5 + 1) + 42 n5 + 167) (4 n5 + 42 n5 + 125) (n5 + 1)! | \n5 = 0 / \ | / 2 2 | / ((2 n4 + 3) (2 n4 + 5) (2 n4 + 7) (2 n4 + 9) (4 (n4 + 1) + 34 n4 + 103) (12 n4 + 102 n4 + 207) binomial(2 n4 + 2, n4 + 1) (n4 + 1)!)| / | | / \\ \ || | || | binomial(2 n3, n3) n3!/(binomial(2 n3 + 2, n3 + 1) (n3 + 1)!)|| binomial(2 n2, n2) n2!/((n2 + 2) (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 1)!)| n1!/( || | || | // / \ | | (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!)|} | | / "A287880" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n { 0 n::even {1, 3 , { , { (n/2 - 1/2) { 3 (LegendreP(n/2 + 1/2, 5/3) - LegendreP(n/2 - 1/2, 5/3)) n::odd { 0 n::even { , { (n/2 - 1/2) { 3 (LegendreQ(n/2 + 1/2, 5/3) - LegendreQ(n/2 - 1/2, 5/3)) n::odd { (n/2) { 3 (LegendreP(n/2 + 1/2, 5/3) - LegendreP(n/2 - 1/2, 5/3)) n::even, { { 0 n::odd { (n/2) { 3 (LegendreQ(n/2 + 1/2, 5/3) - LegendreQ(n/2 - 1/2, 5/3)) n::even} { { 0 n::odd "A288035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 2 2 | \ n1 + 2 | {(n + 1) (n!) , (n + 1) (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A288268" LREtools/SearchTable: "SearchTable successful" ((2 n + 2) LaguerreL(n + 1, -1) + (-3 n - 4) LaguerreL(n, -1)) n! {-----------------------------------------------------------------} (n - 1) n "A288269" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A288270" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A288327" LREtools/SolveLRE: "Absolute Factorization reduced the order from 10 to 1 (Liouvillian solutions)" { 0 irem(n, 10) = 0 { 0 irem(n, 10) = 0 { { { 0 irem(n, 10) = 1 { 0 irem(n, 10) = 1 { { { 0 irem(n, 10) = 2 { 0 irem(n, 10) = 2 { { { 0 irem(n, 10) = 3 { 0 irem(n, 10) = 3 { { { 0 irem(n, 10) = 4 { 0 irem(n, 10) = 4 { { { 0 irem(n, 10) = 5 { 0 irem(n, 10) = 5 {{ , { , { 0 irem(n, 10) = 6 { 0 irem(n, 10) = 6 { { { 0 irem(n, 10) = 7 { 0 irem(n, 10) = 7 { { { 0 irem(n, 10) = 8 { / n \ { { |---- - 4/5| { / n \ { \ 10 / n { |---- - 9/10| { 10 GAMMA(---- + 1) irem(n, 10) = 8 { \ 10 / n { 10 { 10 GAMMA(---- + 1) irem(n, 10) = 9 { { 10 { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 0 { 0 irem(n, 10) = 0 { { { 0 irem(n, 10) = 1 { 0 irem(n, 10) = 1 { { { 0 irem(n, 10) = 2 { 0 irem(n, 10) = 2 { { { 0 irem(n, 10) = 3 { 0 irem(n, 10) = 3 { { { 0 irem(n, 10) = 4 { 0 irem(n, 10) = 4 { { { 0 irem(n, 10) = 5 { 0 irem(n, 10) = 5 { , { , { 0 irem(n, 10) = 6 { / n \ { { |---- - 3/5| { / n \ { \ 10 / n { |---- - 7/10| { 10 GAMMA(---- + 1) irem(n, 10) = 6 { \ 10 / n { 10 { 10 GAMMA(---- + 1) irem(n, 10) = 7 { { 10 { 0 irem(n, 10) = 7 { { { 0 irem(n, 10) = 8 { 0 irem(n, 10) = 8 { { { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 0 { { 0 irem(n, 10) = 1 { { 0 irem(n, 10) = 2 { { 0 irem(n, 10) = 3 { { 0 irem(n, 10) = 4 { { / n \ { |---- - 1/2| , { \ 10 / / n \ n { 1/5 n (5/2) |---- - 1/2|! binomial(n/5 - 1, ---- - 1/2) irem(n, 10) = 5 { \ 10 / 10 { { 0 irem(n, 10) = 6 { { 0 irem(n, 10) = 7 { { 0 irem(n, 10) = 8 { { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 0 { 0 irem(n, 10) = 0 { { { 0 irem(n, 10) = 1 { 0 irem(n, 10) = 1 { { { 0 irem(n, 10) = 2 { 0 irem(n, 10) = 2 { { { 0 irem(n, 10) = 3 { / n \ { { |---- - 3/10| { / n \ { \ 10 / n { |---- - 2/5| { 10 GAMMA(---- + 1) irem(n, 10) = 3 { \ 10 / n { 10 { 10 GAMMA(---- + 1) irem(n, 10) = 4, { , { 10 { 0 irem(n, 10) = 4 { { { 0 irem(n, 10) = 5 { 0 irem(n, 10) = 5 { { { 0 irem(n, 10) = 6 { 0 irem(n, 10) = 6 { { { 0 irem(n, 10) = 7 { 0 irem(n, 10) = 7 { { { 0 irem(n, 10) = 8 { 0 irem(n, 10) = 8 { { { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 0 { 0 irem(n, 10) = 0 { { { 0 irem(n, 10) = 1 { / n \ { { |---- - 1/10| { / n \ { \ 10 / n { |---- - 1/5| { 10 GAMMA(---- + 1) irem(n, 10) = 1 { \ 10 / n { 10 { 10 GAMMA(---- + 1) irem(n, 10) = 2 { { 10 { 0 irem(n, 10) = 2 { { { 0 irem(n, 10) = 3 { 0 irem(n, 10) = 3 { , { , { 0 irem(n, 10) = 4 { 0 irem(n, 10) = 4 { { { 0 irem(n, 10) = 5 { 0 irem(n, 10) = 5 { { { 0 irem(n, 10) = 6 { 0 irem(n, 10) = 6 { { { 0 irem(n, 10) = 7 { 0 irem(n, 10) = 7 { { { 0 irem(n, 10) = 8 { 0 irem(n, 10) = 8 { { { 0 irem(n, 10) = 9 { 0 irem(n, 10) = 9 { / n \ { |----| { \ 10 / / n \ { 10 |----|! irem(n, 10) = 0 { \ 10 / { { 0 irem(n, 10) = 1 { { 0 irem(n, 10) = 2 { { 0 irem(n, 10) = 3 { } { 0 irem(n, 10) = 4 { { 0 irem(n, 10) = 5 { { 0 irem(n, 10) = 6 { { 0 irem(n, 10) = 7 { { 0 irem(n, 10) = 8 { { 0 irem(n, 10) = 9 "A288454" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 64 64 { - -------------------- n::even { n 2 { 2 2 { 4 binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { (2 n - 2) 2 { (6 n + 6) { -16 2 binomial(n - 1, n/2 - 1/2) n::odd { 4 2 { ------------------------------------ n::odd { 2 2 { (n + 1) binomial(n + 1, n/2 + 1/2) "A288470" LREtools/SearchTable: "SearchTable successful" {((2 n + 5) (5 n + 4) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) 3 2 + (-86 n - 330 n - 394 n - 144) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1))/((n + 2) (2 n + 1))} "A288703" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | 2 2 | \ (n1 + 1) n1! (n1 + n1 - 3)| {(n!) , (n!) | ) ---------------------------|, | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / / /n1 - 1 \\ | |----- 6 5 4 3 2 || | 2 | \ n2 + 8 n2 + 22 n2 + 22 n2 + 4 n2 + 14 n2 + 29 || | (n1 + 1) n1! (n1 + n1 - 3) | ) ------------------------------------------------------|| |n - 1 | / 2 2 || |----- |----- (n2 + 2) (n2 + 1)! ((n2 + 1) + n2 - 2) (n2 + n2 - 3)|| 2 | \ \n2 = 0 /| (n!) | ) -------------------------------------------------------------------------------------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A288780" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {n, n | ) n1!|} | / | |----- | \n1 = 0 / "A288841" LREtools/SolveLRE: "Reduced the order of" (n^3+5*n^2+9*n+4)*E^3+(-n^4-9*n^3-26*n^2-27*n-1)*E^2-(n+2)*(n^4+10*n^3+38*n^2+71*n+52)*E+(n+2)*(n^3+8*n ^2+22*n+19)*(n+1)^2 "to two: Symmetric square" E^2-E-n-1 LREtools/SearchTable: "SearchTable successful" n 1/2 2 1/2 2 {(-1/2) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) )} "A288869" LREtools/SearchTable: "SearchTable successful" n (-1) n! (n + 1) (LaguerreL(n + 1, -1) - 2 LaguerreL(n, -1)) {------------------------------------------------------------} n "A288952" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 5 {|1/2 - ----| n! hypergeom([-n, 1/2 - ----], [1], 5/2 + ----)} \ 2 / 10 2 "A288953" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (-n) 2 2 {n! (n + 2), { , { 2 binomial(n, n/2) ((n/2)!) n::even} { (n - 1) 2 { { 2 ((n/2 - 1/2)!) n::odd { 0 n::odd "A288964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! n1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A288965" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 3 2 | | \ (-1) n1! (16 n1 - 66 n1 + 62 n1 - 3) | {n! (8 n + 3), n! (72 n + 67), n! (8 n + 3) | ) --------------------------------------------------------------|, | / (n1 - 3) (n1 - 2) (n1 - 1) n1 (n1 + 1)! (8 n1 + 11) (8 n1 + 3)| |----- | \n1 = 0 / /n - 1 \ |----- 6 4 3 2 | | \ n1! (10368 n1 - 208030 n1 + 315060 n1 + 189622 n1 - 306750 n1 + 255)| n! (72 n + 67) | ) ------------------------------------------------------------------------|} | / (n1 - 3) (n1 - 2) (n1 - 1) n1 (n1 + 1)! (72 n1 + 139) (72 n1 + 67) | |----- | \n1 = 0 / "A289030" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A289031" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 9) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1) {1, n + 4, ) -------------------------------------------------------------------------} / (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) ----- n1 = 0 "A289147" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -4)} "A289211" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -5)} "A289212" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -6)} "A289213" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -7)} "A289214" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -8)} "A289215" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -9)} "A289216" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -10)} "A289615" binomial(2 n, n) {n, ----------------} n + 1 "A289616" binomial(2 n, n) {n, ----------------} n + 1 "A289652" binomial(2 n, n) {1, ----------------} n + 1 "A289653" binomial(2 n, n) {1, ----------------} n + 1 "A289683" LREtools/SearchTable: "SearchTable successful" n! ((7 n + 4) LegendreP(n, 2) + (-2 n - 2) LegendreP(n + 1, 2)) n! ((7 n + 4) LegendreQ(n, 2) + (-2 n - 2) LegendreQ(n + 1, 2)) {- ---------------------------------------------------------------, - ---------------------------------------------------------------} (n - 1) n (n - 1) n "A289684" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(2 + 2 2 ) , (-2 2 + 2) , /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (-2 2 + 2) (2 n2 + 1) binomial(2 n2, n2)|| (2 + 2 2 ) | ) (-2 2 + 2) (2 + 2 2 ) | ) ----------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A289715" {(n + 1) n, binomial(2 n, n) (2 n + 1)} "A289718" 3 2 {(n + n + 4 n + 2) (n + 1), binomial(2 n, n) (2 n + 1)} "A289719" memory used=233954.2MB, alloc=3351.5MB, time=1805.49 {1, n binomial(2 n, n)} "A289720" 3 2 {n binomial(2 n, n), (n - 1) (n - n + 2 n + 2)} "A289800" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 8 _Z + 13 _Z + 12 _Z + 1 "A289810" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 8 _Z + 13 _Z + 12 _Z + 1 "A289830" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ n1! (n1 + 4 n1 + 5) | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A289834" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | | \ binomial(2 n1, n1) (41 n1 + 59 n1 - 46 n1 - 42)| {1, (3 n + 1) | ) -------------------------------------------------|, 3 n + 1} | / (n1 + 3) (n1 + 2) (n1 + 1) (3 n1 + 4) (3 n1 + 1) | |----- | \n1 = 0 / "A289896" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ /n - 1 \ |----- |----- || |----- | | \ | \ (n2 + 2) (n2 + 1)|| | \ | {(n + 1) | ) n1! | ) -----------------||, (n + 1) | ) n1!|, n + 1} | / | / (n2 + 1)! || | / | |----- |----- || |----- | \n1 = 0 \n2 = 0 // \n1 = 0 / "A289945" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 4 4 4 {1, ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A289946" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 6 6 6 {1, ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A289948" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 3 3 {1, ) (n1 + 1) (n1!) } / ----- n1 = 0 "A289949" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 4 4 {1, ) (n1 + 1) (n1!) } / ----- n1 = 0 "A289950" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ 1 | {1, ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!, ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ------------------------------------|, / / | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / / n2 - 1 \ | ----- | | \ | | ) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3!| n - 1 |n1 - 1 / | n - 1 ----- |----- ----- | ----- \ | \ n3 = 0 | \ ) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ----------------------------------------------|, ) / | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! | / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 / n2 - 1 /n3 - 1 \\ | ----- |----- || | \ | \ 1 || | ) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | ) ------------------------------------|| |n1 - 1 / | / (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)!|| |----- ----- |----- || | \ n3 = 0 \n4 = 0 /| (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) --------------------------------------------------------------------------------------------|} | / (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! | |----- | \n2 = 0 / "A289951" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!, / ----- n1 = 0 n - 1 /n1 - 1 \ ----- |----- | \ | \ 1 | ) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ------------------------------------------------------|, / | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!| ----- |----- | n1 = 0 \n2 = 0 / / n2 - 1 \ | ----- | | \ | | ) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3!| n - 1 |n1 - 1 / | ----- |----- ----- | \ | \ n3 = 0 | ) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ----------------------------------------------------------------|, / | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! | ----- |----- | n1 = 0 \n2 = 0 / n - 1 ----- \ ) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! / ----- n1 = 0 / n2 - 1 /n3 - 1 \\ | ----- |----- || | \ | \ 1 || | ) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | ) ------------------------------------------------------|| |n1 - 1 / | / (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)!|| n - 1 |----- ----- |----- || ----- | \ n3 = 0 \n4 = 0 /| \ | ) --------------------------------------------------------------------------------------------------------------------------------|, ) | / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! | / |----- | ----- \n2 = 0 / n1 = 0 / | | | | | | | | | | (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | | | \ / n4 - 1 \ | ----- | | \ | | ) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) (n5 + 1) n5!| n2 - 1 |n3 - 1 / | ----- |----- ----- | \ | \ n5 = 0 | ) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | ) ----------------------------------------------------------------| n1 - 1 / | / (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)! | ----- ----- |----- | \ n3 = 0 \n4 = 0 / ) ------------------------------------------------------------------------------------------------------------------------------------------ / (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! ----- n2 = 0 \ | | | | / | | | | | | | n - 1 |n1 - 1 n2 - 1 | ----- |----- ----- | \ | \ \ |, ) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | / | / / | ----- |----- ----- / n1 = 0 \n2 = 0 n3 = 0 / n4 - 1 /n5 - 1 \\ | ----- |----- || | \ | \ 1 || | ) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) (n5 + 1) n5! | ) ------------------------------------------------------|| |n3 - 1 / | / (n6 + 6) (n6 + 5) (n6 + 4) (n6 + 3) (n6 + 2) (n6 + 1)!|| |----- ----- |----- || | \ n5 = 0 \n6 = 0 /| | ) --------------------------------------------------------------------------------------------------------------------------------|/( | / (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)! | |----- | \n4 = 0 / \ | | | | | | (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!)|} | | / "A289952" memory used=235068.5MB, alloc=3351.5MB, time=1813.92 memory used=235967.6MB, alloc=3351.5MB, time=1821.11 memory used=236947.9MB, alloc=3351.5MB, time=1829.15 memory used=237995.7MB, alloc=3351.5MB, time=1837.53 memory used=238871.7MB, alloc=3351.5MB, time=1844.93 memory used=239747.4MB, alloc=3351.5MB, time=1852.35 memory used=240620.9MB, alloc=3351.5MB, time=1859.63 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 8" memory used=241763.7MB, alloc=3351.5MB, time=1867.79 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- ----- \ \ {1, ) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1!, ) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) / / ----- ----- n1 = 0 n1 = 0 /n1 - 1 \ n - 1 |----- | ----- | \ 1 | \ (n1 + 2) (n1 + 1) n1! | ) ------------------------------------------------------------------------|, ) (n1 + 8) (n1 + 7) (n1 + 6) | / (n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!| / |----- | ----- \n2 = 0 / n1 = 0 / n2 - 1 \ | ----- | | \ | | ) (n3 + 8) (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3!| |n1 - 1 / | |----- ----- | | \ n3 = 0 | (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ----------------------------------------------------------------------------------|, | / (n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)! | |----- | \n2 = 0 / n - 1 /n1 - 1 n2 - 1 ----- |----- ----- \ | \ \ ) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ) (n3 + 8) (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) / | / / ----- |----- ----- n1 = 0 \n2 = 0 n3 = 0 /n3 - 1 \ |----- | | \ 1 | (n3 + 3) (n3 + 2) (n3 + 1) n3! | ) ------------------------------------------------------------------------|/((n2 + 8) (n2 + 7) (n2 + 6) | / (n4 + 8) (n4 + 7) (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)!| |----- | \n4 = 0 / / | | | \ n - 1 |n1 - 1 | ----- |----- | \ | \ (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!)|, ) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) | / | / | ----- |----- / n1 = 0 \n2 = 0 n2 - 1 ----- \ ) (n3 + 8) (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! / ----- n3 = 0 / n4 - 1 \ | ----- | | \ | | ) (n5 + 8) (n5 + 7) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) (n5 + 1) n5!| |n3 - 1 / | |----- ----- | | \ n5 = 0 | | ) ----------------------------------------------------------------------------------|/((n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) | / (n4 + 8) (n4 + 7) (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)! | |----- | \n4 = 0 / \ | | | | n - 1 /n1 - 1 n2 - 1 | ----- |----- ----- | \ | \ \ (n2 + 3) (n2 + 2) (n2 + 1)!)|, ) (n1 + 8) (n1 + 7) (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ) (n3 + 8) | / | / / | ----- |----- ----- / n1 = 0 \n2 = 0 n3 = 0 /n3 - 1 n4 - 1 |----- ----- | \ \ (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | ) ) (n5 + 8) (n5 + 7) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) | / / |----- ----- \n4 = 0 n5 = 0 /n5 - 1 \ |----- | | \ 1 | (n5 + 1) n5! | ) ------------------------------------------------------------------------|/((n4 + 8) (n4 + 7) (n4 + 6) (n4 + 5) (n4 + 4) | / (n6 + 8) (n6 + 7) (n6 + 6) (n6 + 5) (n6 + 4) (n6 + 3) (n6 + 2) (n6 + 1)!| |----- | \n6 = 0 / \ \ n - 1 | | ----- | | \ (n4 + 3) (n4 + 2) (n4 + 1)!)|/((n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!)|, ) (n1 + 8) (n1 + 7) (n1 + 6) | | / | | ----- / / n1 = 0 / / | | | | | | |n1 - 1 n2 - 1 | |----- ----- | | \ \ | (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ) (n3 + 8) (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | | / / | |----- ----- | \n2 = 0 n3 = 0 \ n3 - 1 n4 - 1 ----- ----- \ \ ) ) (n5 + 8) (n5 + 7) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) (n5 + 1) n5! / / ----- ----- n4 = 0 n5 = 0 / n6 - 1 \ | ----- | | \ | | ) (n7 + 8) (n7 + 7) (n7 + 6) (n7 + 5) (n7 + 4) (n7 + 3) (n7 + 2) (n7 + 1) n7!| |n5 - 1 / | |----- ----- | | \ n7 = 0 | | ) ----------------------------------------------------------------------------------|/((n4 + 8) (n4 + 7) (n4 + 6) (n4 + 5) (n4 + 4) | / (n6 + 8) (n6 + 7) (n6 + 6) (n6 + 5) (n6 + 4) (n6 + 3) (n6 + 2) (n6 + 1)! | |----- | \n6 = 0 / \ \ | | | | | | | | n - 1 | | ----- | | \ (n4 + 3) (n4 + 2) (n4 + 1)!)|/((n2 + 8) (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!)|, ) (n1 + 8) (n1 + 7) (n1 + 6) | | / | | ----- / / n1 = 0 /n1 - 1 n2 - 1 / |----- ----- | | \ \ | (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) n1! | ) ) (n3 + 8) (n3 + 7) (n3 + 6) (n3 + 5) (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) n3! | | / / | |----- ----- | \n2 = 0 n3 = 0 \ n3 - 1 n4 - 1 /n5 - 1 n6 - 1 ----- ----- |----- ----- \ \ | \ \ ) ) (n5 + 8) (n5 + 7) (n5 + 6) (n5 + 5) (n5 + 4) (n5 + 3) (n5 + 2) (n5 + 1) n5! | ) ) (n7 + 8) (n7 + 7) (n7 + 6) (n7 + 5) / / | / / ----- ----- |----- ----- n4 = 0 n5 = 0 \n6 = 0 n7 = 0 /n7 - 1 \ |----- | | \ 1 | (n7 + 4) (n7 + 3) (n7 + 2) (n7 + 1) n7! | ) ------------------------------------------------------------------------|/((n6 + 8) (n6 + 7) | / (n8 + 8) (n8 + 7) (n8 + 6) (n8 + 5) (n8 + 4) (n8 + 3) (n8 + 2) (n8 + 1)!| |----- | \n8 = 0 / \ \ | | | | (n6 + 6) (n6 + 5) (n6 + 4) (n6 + 3) (n6 + 2) (n6 + 1)!)|/((n4 + 8) (n4 + 7) (n4 + 6) (n4 + 5) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1)!)|/((n2 + 8) | | | | / / \ | | (n2 + 7) (n2 + 6) (n2 + 5) (n2 + 4) (n2 + 3) (n2 + 2) (n2 + 1)!)|} | | / "A290147" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (2 LegendreP(n + 1, 2 I) I + 2 LegendreP(n, 2 I)) {-------------------------------------------------------------------------------, n 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (2 LegendreQ(n + 1, 2 I) I + 2 LegendreQ(n, 2 I)) -------------------------------------------------------------------------------} n "A290380" LREtools/SearchTable: "SearchTable successful" n (-1) ((5 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) (n + 1) {----------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A290442" LREtools/SearchTable: "SearchTable successful" n (-1) ((5 n + 9) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 3) hypergeom([1/2, -n], [1], 4)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) (n + 1) "A290443" LREtools/SearchTable: "SearchTable successful" n (2 n + 1) (-1) binomial(2 n, n) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------------------} 2 (n + 1) (n + 2) "A290575" LREtools/SearchTable: "SearchTable not successful" {} "A290576" LREtools/SearchTable: "SearchTable not successful" {} "A290690" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A290953" (2 n + 1) binomial(2 n, n) {--------------------------, n - 1} (n + 1) (n + 2) "A290954" (2 n + 1) binomial(2 n, n) 3 {--------------------------, n + 5 n - 15} (n + 1) (n + 2) "A291088" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A291089" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A291090" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A291091" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A291286" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 5 {|1/2 - ----| n! hypergeom([-n + 1, 1 + ----], [2], 5/2 + ----)} \ 2 / 5 2 "A291287" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 1/2 | 5 | 5 5 {|1/2 - ----| n! hypergeom([-n + 1, 1 + ----], [2], 5/2 + ----)} \ 2 / 5 2 "A291292" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 binomial(2 n1, n1) (n1 - 2)| {3 , 3 | ) --------------------------------------|} | / (n1 + 1) (2 n1 - 1) | |----- | \n1 = 0 / "A291456" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 6 | 6 6 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 6| |----- ((n1 + 1)!) | \n1 = 0 / "A291482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A291483" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A291484" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \ \\ | | |----- | || | | n2 | \ 1 | || | | (-1) (n2 + 1) | ) ------------------| n2!|| n - 1 n - 1 |n1 - 1 | | / (n3 + 2) (n3 + 1)!| || n - 1 ----- ----- |----- | |----- | || ----- \ n1 \ n1 | \ | \n3 = 0 / || \ {1, ) (-1) n1!, ) (-1) n1! | ) |- -----------------------------------------------||, ) n1!} / / | / \ (n2 + 1)! /| / ----- ----- |----- | ----- n1 = 0 n1 = 0 \n2 = 0 / n1 = 0 "A291505" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 7 | 7 7 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 7| |----- ((n1 + 1)!) | \n1 = 0 / "A291506" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 8 | 8 8 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 8| |----- ((n1 + 1)!) | \n1 = 0 / "A291507" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 9 | 9 9 | \ (n1!) | {(n!) , (n!) | ) ------------|} | / 9| |----- ((n1 + 1)!) | \n1 = 0 / "A291508" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 10 | 10 10 | \ (n1!) | {(n!) , (n!) | ) -------------|} | / 10| |----- ((n1 + 1)!) | \n1 = 0 / "A291534" LREtools/SearchTable: "SearchTable successful" n (-1) ((3 n + 4) hypergeom([-2 n - 2, -n - 1], [1], -1) + (32 n + 16) hypergeom([-2 n, -n], [1], -1)) {-----------------------------------------------------------------------------------------------------} (2 n + 3) (7 n + 4) "A291535" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / (n1 + 1) 1/2 1/2 1/2 || n n | \ | 5 2 (5 LegendreP(n1 + 1, 5 ) - LegendreP(n1, 5 ))|| {(-1/2) , (-1/2) | ) |- ----------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / (n1 + 1) 1/2 1/2 1/2 || n | \ | 5 2 (5 LegendreQ(n1 + 1, 5 ) - LegendreQ(n1, 5 ))|| (-1/2) | ) |- ----------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A291585" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 3 3 \| n 3 3 n 3 3 | \ | 8 binomial(2 n1, n1) (n1!) || {(1/8) (n!) binomial(2 n, n) , (1/8) (n!) binomial(2 n, n) | ) |----------------------------------------||} | / | 3 3|| |----- \binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A291586" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 4 4 \| n 4 4 n 4 4 | \ | 16 binomial(2 n1, n1) (n1!) || {(1/16) (n!) binomial(2 n, n) , (1/16) (n!) binomial(2 n, n) | ) |----------------------------------------||} | / | 4 4|| |----- \binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A291587" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 5 5 \| n 5 5 n 5 5 | \ | 32 binomial(2 n1, n1) (n1!) || {(1/32) (n!) binomial(2 n, n) , (1/32) (n!) binomial(2 n, n) | ) |----------------------------------------||} | / | 5 5|| |----- \binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) /| \n1 = 0 / "A291632" LREtools/SearchTable: "SearchTable successful" n {-I (-I) (HermiteH(n + 1, 1/2 I) - HermiteH(n, 1/2 I) I)} "A291662" {n, binomial(3 n, n)} "A291822" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 3) (2 n + 1) binomial(2 n, n) 3 2 {------------------------------------, (2 (2 n + 1) (108 n + 422 n + 442 n + 105) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) (n + 3) (n + 2) (n + 1) 2 - (4 n + 1) (4 n + 3) (36 n + 128 n + 105) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n)/((n + 1) (n + 2) (n + 3) (2 n - 1))} "A291856" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 )} "A291885" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 16 binomial(2 n, n/2) (n + 1) 2 { ----------------------------- n::even binomial(4 n, n) (2 n + 3) (2 n + 1) { 3 n + 2 {-------------------------------------, { , (n + 1) (3 n + 1) (3 n + 2) { 2 (n + 2) binomial(2 n + 2, n/2 + 1/2) { -------------------------------------- n::odd { 2 n + 1 { n { 16 (n + 2) binomial(2 n, n/2) { 1/2 ----------------------------------------------------- n::even { 3 n { (n + 1) (3 n + 1) binomial(n, n/2) binomial(3 n, ---) { 2 { } { (4 n - 4) { 4 2 (n + 1) (2 n - 1) binomial(2 n - 2, n/2 - 1/2) { ----------------------------------------------------------------------------- n::odd { 3 n { n (3 n - 2) (3 n + 2) binomial(n - 1, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 "A291898" LREtools/SearchTable: "SearchTable successful" n {(-3) hypergeom([- n/3, - n/3 + 2/3, 1/3 - n/3], [1, 1], 1)} "A292029" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 (n + 2) { n { 1/2 ------------------------ n::even n { 2 (4 n + 4) binomial(n, n/2) n::even { (n + 1) binomial(n, n/2) {4 (n + 2), { , { } { (n + 1) { (3 n - 3) { 2 (n + 2) binomial(n + 1, n/2 + 1/2) n::odd { 2 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A292062" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | 2 2 2 2 | \ (n1 + 1) (n1!) | {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A292119" LREtools/SearchTable: "SearchTable successful" (-n) {2 (n + 1) (n + 2) (n + 3) ((2 n + 1) hypergeom([1/2, -n - 1, -n - 2], [2, -n - 1/2], -1) + (-4 n - 8) hypergeom([1/2, -n, -n - 1], [2, -n + 1/2], -1)) binomial(2 n, n) n!} "A292347" LREtools/SearchTable: "SearchTable successful" n n (-2) BesselI(n + 1/2, 1) (-2) BesselK(n + 1/2, -1) {-------------------------, --------------------------} n + 2 n + 2 "A292437" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 4 binomial(2 n, n) {-------------------, (3 n (6 n + 5) (6 n + 1) hypergeom([-2 n - 2, -3 n - 3], [-3 n - 5/2], 1/2) n + 1 + 2 (n + 1) (3 n + 1) (3 n + 2) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)) binomial(6 n, 3 n)/((n + 1) (3 n + 1) (3 n + 2) (5 n + 4))} "A292440" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (5 n + 1) hypergeom([1/2, -n], [1], 4)) {-----------------------------------------------------------------------------------------} (n - 1) n "A292460" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A292461" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A292751" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A292880" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ 2 {(n + 1) n! (n + 7 n + 32 n + 67 n + 98 n + 57 n + 21), (n + 1) n! (n + 7 n + 32 n + 67 n + 98 n + 57 n + 21) | ) (6 n1 + 38 n1 + 53) | / |----- \n1 = 0 / 6 5 4 3 2 / ((n1 + 1)! ((n1 + 1) + 7 (n1 + 1) + 32 (n1 + 1) + 67 (n1 + 1) + 98 (n1 + 1) + 57 n1 + 78) / \ | 6 5 4 3 2 | (n1 + 7 n1 + 32 n1 + 67 n1 + 98 n1 + 57 n1 + 21))|} | | / "A292897" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 | 2 2 | \ (-1) (5 n1 + 14 n1 + 12) | {n! (5 n + 19 n + 15), n! (5 n + 19 n + 15) | ) ---------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! (5 (n1 + 1) + 19 n1 + 34) (5 n1 + 19 n1 + 15)| \n1 = 0 / "A292998" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 3 2 3 2 | \ 1 | {(n + 1) n! (n + 3 n + 5 n + 2), (n + 1) n! (n + 3 n + 5 n + 2) | ) -----------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 3 (n1 + 1) + 5 n1 + 7) (n1 + 3 n1 + 5 n1 + 2)| \n1 = 0 / "A293049" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293050" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A293073" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293075" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293081" memory used=243047.3MB, alloc=3351.5MB, time=1877.29 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 2 {2 n + 7 n + 7} "A293116" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, 1) - n LaguerreL(n, 1)) n! {----------------------------------------------------} n "A293117" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293118" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293120" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293121" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293122" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293123" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293125" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------------} n "A293458" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A293468" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) (2 hypergeom([-n - 1], [n + 2], 1) - hypergeom([-n], [n + 1], 1)) {--------------------------------------------------------------------------------------------} n "A293469" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable not successful" {} "A293490" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) + (-4 n - 1) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------} 2 n - 1 "A293588" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A293604" LREtools/SearchTable: "SearchTable successful" {HermiteH(n, 1/2)} "A293653" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 3 3 GAMMA(n/2 + 1) GAMMA(n/2 + 2/3) n::even { 2 3 GAMMA(n/2 + 3/2) GAMMA(n/2 + 7/6) { { -------------------------------------- n::even {{ (n + 1) , { n + 1 } { 2 3 GAMMA(n/2 + 3/2) GAMMA(n/2 + 7/6) { { -------------------------------------------- n::odd { (n - 1) { n + 1 { 3 3 GAMMA(n/2 + 1) GAMMA(n/2 + 2/3) n::odd "A293656" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) {{ , { 1/2 2 (n/2)! (n + 1) (n + 2) n::even} { (- n/2 + 1/2) { { 1/2 2 n (n + 1) (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A293716" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293717" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293720" LREtools/SearchTable: "SearchTable successful" n {(-2 I) HermiteH(n, 1/4 I)} "A293721" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293723" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A293914" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) ((2 n + 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) ((2 n + 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1)), n (2 n + 1) (1/2) n! binomial(2 n, n)} "A293915" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n 2 n 2 {(-1) (4 (n + 1) BesselI(n + 1/2, 1) + (-2 n - 3) BesselI(n - 1/2, 1)), (-1) (4 (n + 1) BesselK(n + 1/2, -1) + (-2 n - 3) BesselK(n - 1/2, -1))} "A293946" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable not successful" /3125\n |----| GAMMA(n + 3/5) GAMMA(n + 1/5) GAMMA(n + 2/5) GAMMA(n - 1/5) \108 / {-------------------------------------------------------------------} GAMMA(n + 1/3) GAMMA(n + 1) GAMMA(n + 2/3) GAMMA(n + 1/2) "A293964" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n {(-1/2 I) ((2 n + 7) HermiteH(n + 1, I) I - (n + 1) (n + 5) HermiteH(n, I)), { 0 n::even { , { { 1/2 (n/2 - 1/2) 1/2 1/2 1/2 { 0 , n::even { (-1/2 I 2 ) ((-n - 1) HermiteH(n/2 - 1/2, 1/2 I 2 ) + 2 HermiteH(n/2 + 1/2, 1/2 I 2 ) I) n::odd 1/2 (n/2 - 1/2) 1/2 (n/2 + 1) 1/2 (n/2) 1/2 , (-1/2 I 2 ) (2 (-1) HermiteH(n/2 + 1/2, -1/2 I 2 ) I - (-1) (n + 1) HermiteH(n/2 - 1/2, -1/2 I 2 )) , n::odd { 1/2 (n/2) 1/2 1/2 1/2 { (-1/2 I 2 ) ((-n - 1) HermiteH(n/2 - 1/2, 1/2 I 2 ) + 2 HermiteH(n/2 + 1/2, 1/2 I 2 ) I) n::even, { { 0 n::odd { 1/2 (n/2) 1/2 (n/2 + 1) 1/2 (n/2) 1/2 { (-1/2 I 2 ) (2 (-1) HermiteH(n/2 + 1/2, -1/2 I 2 ) I - (-1) (n + 1) HermiteH(n/2 - 1/2, -1/2 I 2 )) n::even} { { 0 n::odd "A293986" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A294004" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n { 0 n::even {(-1/2 I) ((n + 4) HermiteH(n + 1, I) + 2 I (n + 1) HermiteH(n, I)), { , { 1/2 (n/2 - 1/2) 1/2 { (-1/2 I 2 ) HermiteH(n/2 + 1, 1/2 I 2 ) n::odd { 0 n::even { , { 1/2 (n/2 - 1/2) (n/2 + 1) 1/2 { (-1/2 I 2 ) (-1) HermiteH(n/2 + 1, -1/2 I 2 ) n::odd { 1/2 (n/2) 1/2 { (-1/2 I 2 ) HermiteH(n/2 + 1, 1/2 I 2 ) n::even, { { 0 n::odd { 1/2 (n/2) (n/2 + 1) 1/2 { (-1/2 I 2 ) (-1) HermiteH(n/2 + 1, -1/2 I 2 ) n::even} { { 0 n::odd "A294035" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294036" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294039" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ n1 + 2 | {(2 n + 1) (n!) binomial(2 n, n), (2 n + 1) (n!) binomial(2 n, n) | ) --------------------------------------------------|} | / 2 | |----- (2 n1 + 3) ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A294040" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ | |----- 3 2 | | | 2 | \ 8 n2 + 26 n2 + 24 n2 + 3 | | | (n1!) binomial(2 n1, n1) (4 n1 + 1) | ) -------------------------------------------------------------| (n1 + 1)| |n - 1 | / 2 | | |----- |----- ((n2 + 1)!) binomial(2 n2 + 2, n2 + 1) (4 n2 + 5) (4 n2 + 1)| | | \ \n2 = 0 / | n! | ) --------------------------------------------------------------------------------------------------------------------| | / (n1 + 1)! | |----- | n! \n1 = 0 / {----, --------------------------------------------------------------------------------------------------------------------------------, n n 2 (n!) binomial(2 n, n) ----------------------} n "A294050" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A294051" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A294084" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) | n n | \ 2 binomial(4 n1, n1) (11 n1 - 4)| {16 , 16 | ) -------------------------------------------|} | / (n1 + 1) (2 n1 - 1) (4 n1 - 1) | |----- | \n1 = 0 / "A294117" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- 2 2 || | 2 2 | \ (n2 + 1) (n2!) || | (n1 + 1) (n1!) | ) ---------------------|| /n - 1 \ |n - 1 | / 2|| |----- 2 2 | |----- |----- (n2 + 3) ((n2 + 1)!) || 2 2 2 2 | \ (n1 + 1) (n1!) | 2 2 | \ \n2 = 0 /| {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|, (n + 1) (n!) | ) -----------------------------------------------|, | / 2| | / 2 | |----- (n1 + 2) ((n1 + 1)!) | |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / \n1 = 0 / / /n1 - 1 \\ | |----- 2 n2 2 || | 2 2 | \ (n2 + 1) 2 (n2!) || | (n1 + 1) (n1!) | ) ---------------------|| |n - 1 | / 2|| |----- |----- (n2 + 3) ((n2 + 1)!) || 2 2 | \ \n2 = 0 /| (n + 1) (n!) | ) -----------------------------------------------|} | / 2 | |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A294119" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-I 2 ) HermiteH(n, 2 I)} "A294120" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294129" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ 1/2 n1 1/2 {1, ) (-1/2 I 2 ) HermiteH(n1 + 1, 1/2 I 2 )} / ----- n1 = 0 "A294159" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294167" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((2 n + 3) (4 n + 12 n + 7) BesselI(n + 1/2, 1) - 4 (n + 2) BesselI(n - 1/2, 1)), n 2 2 (-1) ((2 n + 3) (4 n + 12 n + 7) BesselK(n + 1/2, -1) - 4 (n + 2) BesselK(n - 1/2, -1))} "A294175" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n { (n + 1) binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A294189" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294190" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294193" 2 {n! n, (n!) (n + 2) n} "A294213" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A294251" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A294255" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A294256" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A294290" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A294435" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 |----- (-n1 - 2) n1 | |----- n n n | \ 2 (-1) | n | \ (-n1 - 2) {16 (n + 2), n 4 binomial(2 n, n), n 4 binomial(2 n, n) | ) --------------------------------------|, n 4 binomial(2 n, n) | ) 2 | / n1 (n1 + 1) binomial(2 n1 + 2, n1 + 1)| | / |----- | |----- \n1 = 0 / \n1 = 0 n1 (-1) /n1 - 1 \ |----- | | \ (n2 + 1) hypergeom([-n2 - 1, -n2 - 1, -n2 - 1, -n2 - 1], [1, 1, 1], 1) + (-46 n2 - 32) hypergeom([-n2, -n2, -n2, -n2], [1, 1, 1], 1)| | ) ------------------------------------------------------------------------------------------------------------------------------------|/(n1 | / (n2 + 1) | |----- (-2) | \n2 = 0 / \ | | (n1 + 1) binomial(2 n1 + 2, n1 + 1))|} | | / "A294436" memory used=244393.0MB, alloc=3351.5MB, time=1886.97 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 |----- (-n1 - 3) n1 | |----- n n n | \ 2 (-1) | n | \ (-n1 - 3) {32 (n + 2), n 8 binomial(2 n, n), n 8 binomial(2 n, n) | ) --------------------------------------|, n 8 binomial(2 n, n) | ) 2 | / n1 (n1 + 1) binomial(2 n1 + 2, n1 + 1)| | / |----- | |----- \n1 = 0 / \n1 = 0 / | n1 | (-1) | | | \ n1 - 1 ----- n2 \ 2 ((n2 + 1) hypergeom([-n2 - 1, -n2 - 1, -n2 - 1, -n2 - 1], [1, 1, 1], 1) + (-46 n2 - 32) hypergeom([-n2, -n2, -n2, -n2], [1, 1, 1], 1)) ) ------------------------------------------------------------------------------------------------------------------------------------------ / (n2 + 1) ----- (-4) n2 = 0 \ \ | | | | |/(n1 (n1 + 1) binomial(2 n1 + 2, n1 + 1))|} | | | | / / "A294527" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) n1 | {(3 n + 1) (3 n - 2) (-1/2) , (3 n + 1) (3 n - 2) (-1/2) | ) -----------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (3 n1 + 4) (3 n1 + 1) (-1/2) (6 n1 - 4)| \n1 = 0 / "A294790" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)| {1, (3 n + 4) | ) ----------------------------------------|, 3 n + 4} | / (n1 + 1) (n1 + 2) (3 n1 + 7) (3 n1 + 4) | |----- | \n1 = 0 / "A294819" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| , \ 2 / \ 2 / / / / / 1/2\(-n2 - 1) \\\ |n - 1 | |n1 - 1 | 5 | ||| / 1/2\n |----- | |----- |3/2 + ----| binomial(2 n2, n2) (13 n2 - 1) n2||| | 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ \ 2 / ||| |3/2 - ----| | ) |2 (3 + 5 ) (3 - 5 ) | ) -------------------------------------------------------|||} \ 2 / | / | | / (n2 + 3) (n2 + 2) (n2 + 1) ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A294823" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 2 || n n | \ (-n1 - 1) | \ (2 n2 + 1) binomial(2 n2, n2) (17 n2 + 10 n2 - 12)|| {1, 2 , 2 | ) 2 (3 n1 + 1) | ) ---------------------------------------------------||, n + 4/3} | / | / (n2 + 4) (n2 + 3) (n2 + 2) (3 n2 + 4) (3 n2 + 1) || |----- |----- || \n1 = 0 \n2 = 0 // "A294824" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 / 1/2\n / 1/2\n | 5 | | 5 | (2 n + 1) binomial(2 n, n) {|3/2 - ----| , |3/2 + ----| , --------------------------} \ 2 / \ 2 / (n + 1) (n + 2) "A294825" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) 2 | n n | \ 2 binomial(2 n1, n1) (n1 - 1) (11 n1 + 14 n1 + 48)| {2 , 2 | ) ------------------------------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) | |----- | \n1 = 0 / "A294982" (3 n + 1) binomial(2 n, n) binomial(3 n, n) (3 n + 1) binomial(3 n, n) n {-------------------------------------------, ----------------------------} n + 1 (n + 1) (2 n + 1) "A294983" 2 (4 n + 1) binomial(2 n, n) binomial(4 n, 2 n) n (4 n + 1) binomial(3 n, n) binomial(4 n, n) n (n - 1) (4 n + 1) binomial(4 n, n) {----------------------------------------------, ---------------------------------------------, ------------------------------------} n + 1 (n + 1) (2 n + 1) (n + 1) (3 n + 1) (3 n + 2) "A295112" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ (-1) hypergeom([1/2, -n1 - 1], [1], 4)| {n, n | ) ----------------------------------------|} | / (n1 + 1) n1 | |----- | \n1 = 0 / "A295113" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" 2 2 {(4 n + 10 n + 3) n, (4 n + 10 n + 3) n /n - 1 \ |----- n1 2 | | \ (-1) ((10 n1 + 19 n1 + 3) hypergeom([1/2, -n1 - 1], [1], 4) - 3 (n1 + 1) (2 n1 - 1) hypergeom([1/2, -n1], [1], 4)) (8 n1 + 17)| | ) ---------------------------------------------------------------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 2) (4 (n1 + 1) + 10 n1 + 13) (4 n1 + 10 n1 + 3) n1 | \n1 = 0 / "A295132" LREtools/SolveLRE: "Reduced the order of" (2*n+7)*(2*n+5)*(n+5)*(4*n+15)*(n+6)^2*E^3-(2*n+5)*(n+5)*(n+3)*(56*n^3+758*n^2+3393*n+4995)*E^2-3*(2*n+ 9)*(n+2)*(56*n^4+766*n^3+3869*n^2+8595*n+7128)*E+27*(2*n+9)*(2*n+7)*(4*n+19)*(n+1)*(n+3)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" {- (n hypergeom([1/2, -n - 1], [1], 4) + (3 n + 12) hypergeom([1/2, -n], [1], 4)) 3 2 2 / 2 ((4 n + 43 n + 123 n + 108) hypergeom([1/2, -n - 1], [1], 4) + 3 (n + 1) (4 n + 19 n + 24) hypergeom([1/2, -n], [1], 4)) / ((n + 2) / 2 (n + 3) )} "A295166" LREtools/SearchTable: "SearchTable successful" n n (-1) n! ((2 n - 1) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)) (-1) n! ((2 n - 1) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2)) {------------------------------------------------------------------, --------------------------------------------------------------------} n n "A295371" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+4)^2*E^3-(2*n+3)*(7*n^2+52*n+97)*E^2-3*(2*n+7)*(7*n^2+18*n+12)*E+27*(2*n+7)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" 2 2 {hypergeom([1/2, -n - 1], [1], 4) + 3 hypergeom([1/2, -n], [1], 4) } "A295382" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, 2)} "A295404" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A295504" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A295518" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- n2 | ----- \ | \ 2 | \ {1, ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A295519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ n - 1 ----- |----- n2 | ----- \ | \ 3 | \ {1, ) n1! | ) ---------|, ) n1!} / | / (n2 + 1)!| / ----- |----- | ----- n1 = 0 \n2 = 0 / n1 = 0 "A295537" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A295541" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 36 2 | {|- 45/7 + -------| \ 7 / / 1/2 1/2 \ | 40 2 1/2 40 2 | |(21 n + 28) hypergeom([5/6, - 2/3 - n], [5/3], 64/7 + -------) + 15 (5 + 4 2 ) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], 64/7 + -------)| \ 7 7 / / 1/2\ | 40 2 | GAMMA(n - 1/3) GAMMA(n + 1/3) |- 57/7 + -------|/(GAMMA(n + 1) GAMMA(n + 2/3))} \ 7 / "A295542" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A295543" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A295544" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A295808" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A295809" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A295810" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A295864" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 binomial(2 n, n) { ------------------- n::even { binomial(n, n/2) {{ , { (2 n - 2) { 8 2 (2 n - 1) binomial(2 n - 2, n - 1) { ----------------------------------------------- n::odd { 2 { n binomial(n - 1, n/2 - 1/2) { 3 n { 8 binomial(---, n/2) binomial(2 n, n/2) n::even { 2 { { 2 3 n } { (n + 1) binomial(--- + 3/2, n/2 + 1/2) binomial(2 n + 2, n/2 + 1/2) { 2 { -------------------------------------------------------------------- n::odd { 2 n + 1 "A295870" LREtools/SearchTable: "SearchTable successful" {binomial(3 n, n) binomial(6 n, 3 n) hypergeom([-2 n, -3 n], [-3 n + 1/2], 1/2)} "A295934" (2 n + 1) binomial(2 n, n) (n + 3) {1, ----------------------------------} n + 1 "A296050" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n!} "A296532" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (-n) 3 n 3 n { { 4 binomial(3 n, ---) binomial(---, n/2) binomial(3 n, n) { 3 n { 2 2 {-----------------, { binomial(--- - 3/2, n/2 - 1/2) (3 n - 1) , { ------------------------------------------- n::even} (n + 1) (2 n + 1) { 2 { binomial(n, n/2) (n + 1) { ---------------------------------------- n::odd { { (n + 1) n { 0 n::odd "A296533" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) { 2 { ------------------ n::even binomial(3 n, n) { n + 1 {-----------------, { , (n + 1) (2 n + 1) { 3 n { 2 binomial(--- + 3/2, n/2 + 1/2) { 2 { -------------------------------- n::odd { 3 n + 1 { (-n) 3 n 3 n { 4 4 binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { (-2 n + 2) 3 n 3 n { 2 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { 2 2 { ----------------------------------------------------------------------------------- n::odd { n (n + 1) binomial(n - 1, n/2 - 1/2) "A296618" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, -n - 1/2, -1/4)} "A296619" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) (5 n + 9) n 3 2 {------------------------------------, (-1) ((94 n + 374 n + 450 n + 162) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) (n + 3) (n + 2) (n + 1) 2 - 9 (4 n + 9) (n + 1) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n) (2 n + 1)/((n + 1) (n + 2) (n + 3) (2 n + 3) (5 n + 3))} "A296660" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 2) n1| n n | \ 2 (-1) | {4 n!, 4 n! | ) -----------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A296661" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-3) | | \ 3 | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A296663" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { binomial(n, n/2) (2 n + 4) n::even { ------------------------ n::even n { { (n + 1) binomial(n, n/2) {2 , { 4 binomial(n - 1, n/2 - 1/2) n (n + 2) , { } { -------------------------------------- n::odd { (2 n + 2) { n + 1 { 2 (n + 2) { 1/2 ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A296665" n {4 , binomial(2 n, n) (n + 1)} "A296726" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { (-n1) n1 2 // n1 \ \2 \ | { | | { 2 binomial(n1, ----) ||----|!| n1::even| |n - 1 { (n1 - 1) // n1 \ \2 | |n - 1 { 2 \\ 2 / / | |----- { 2 ||---- - 1/2|!| n1::odd | |----- { | | \ { \\ 2 / / | | \ { 0 n1::odd | {n! | ) --------------------------------------------|, n! | ) -------------------------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A296727" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / // n1 \ \2 n1 2 | | { |---- - 1/2| | | { (-1/4) ||----|!| binomial(n1, ----) n1::even| |n - 1 { \ 2 / // n1 \ \2 | |n - 1 { \\ 2 / / 2 | |----- { (-4) ||---- - 1/2|!| n1::odd | |----- { | | \ { \\ 2 / / | | \ { 0 n1::odd | {n! | ) ---------------------------------------------------|, n! | ) -------------------------------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A296769" n (2 n + 1) binomial(2 n, n) (2 n + 3) {4 , ------------------------------------} n + 1 "A296770" n (2 n + 1) binomial(2 n, n) (n + 2) {4 , ----------------------------------} n + 1 "A296771" n {4 , binomial(2 n, n) (2 n + 3)} "A296785" 2 2 3 3 {(n + 1) n!, (n + 1) (n!) , (n + 1) (n!) } "A296943" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A296944" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A296964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 + 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A297151" n n {8 n, n 2 binomial(2 n, n)} "A297470" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 2 2 { n 2 { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even { 2 2 ((n/2)!) n::even {{ , { } { (-n + 1) 2 2 2 { (n + 1) 2 { 2 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 2 ((n/2 + 1/2)!) n::odd "A297474" LREtools/SearchTable: "SearchTable successful" n {(n + 1) (-2) n! (2 LaguerreL(n + 1, -n - 3/2, -1/2) - LaguerreL(n, -n - 1/2, -1/2))} "A297487" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Cannot reduce the operator to order two" { 0 n::even { n 3 {{ , { 8 ((n/2)!) n::even} { 3 3 3 { { n ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) n::odd { 0 n::odd "A297527" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 { 3 3 3 { 4 8 ((n/2)!) n::even { 3 (n + 1) ((n/2)!) binomial(n, n/2) { { -------------------------------------- n::even {{ (3 n + 3) 3 , { n + 2 } { 3 2 ((n/2 + 1/2)!) { { ---------------------------- n::odd { 3 3 3 { n + 2 { 4 n ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) n::odd "A297670" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ n2 + 1 | {1, ) n1! (n1 + 1) (n1 + 2), ) n1! | ) ---------| (n1 + 1) (n1 + 2)} / / | / (n2 + 1)!| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A297705" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ 9 LegendreP(n1 + 1, 9) - LegendreP(n1, 9)| n | \ 9 LegendreQ(n1 + 1, 9) - LegendreQ(n1, 9)| {(-2/3) , (-2/3) | ) -----------------------------------------|, (-2/3) | ) -----------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (n1 + 2) (-2/3) | |----- (n1 + 2) (-2/3) | \n1 = 0 / \n1 = 0 / "A297708" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SearchTable: "SearchTable successful" memory used=245729.5MB, alloc=3351.5MB, time=1896.41 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), { (n/4) { (-1) HermiteH(n/2, -I) n::even { { (n/2 + 1/2) (n/2 + 3/2) (n/2 + 1/2) , { (-I) ((-1) HermiteH(n/2 + 3/2, -I) I + 2 (-1) HermiteH(n/2 + 1/2, -I)) { - ------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { (-I) HermiteH(n/2, I) n::even { { (n/2 + 1/2) , { (-I) (HermiteH(n/2 + 3/2, I) I + 2 HermiteH(n/2 + 1/2, I)) { - --------------------------------------------------------------------- n::odd { n + 1 { (n/2) { (-I) (HermiteH(n/2 + 3/2, I) I + 2 HermiteH(n/2 + 1/2, I)) { - --------------------------------------------------------------- n::even { n + 1 , { { (n/2 - 1/2) { (-I) HermiteH(n/2, I) n::odd { (n/2) (n/2 + 3/2) (n/2 + 1/2) { (-I) ((-1) HermiteH(n/2 + 3/2, -I) I + 2 (-1) HermiteH(n/2 + 1/2, -I)) { - ------------------------------------------------------------------------------------------------- n::even { n + 1 } { { (n/2 - 1/2) (n/2) { (-I) HermiteH(n/2, -I) (-1) n::odd "A298012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 1) n2!)|| /n - 1 \ |n - 1 | / || |----- n1 | |----- |----- || | \ (-1) | | \ \n2 = 0 /| n! | ) ---------| n! | ) --------------------------------------| | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | n! \n1 = 0 / \n1 = 0 / {-----, ---------------------, --------------------------------------------------, n - 1 n - 1 n - 1 / /n1 - 1 / /n2 - 1 \ \\\ | |----- | |----- n3 | ||| | n1 | \ | n2 | \ (-1) | ||| | (-1) | ) |-(-1) (n2 + 1) | ) ---------| n2!||| |n - 1 | / | | / (n3 + 1)!| ||| |----- |----- | |----- | ||| | \ \n2 = 0 \ \n3 = 0 / //| n! | ) ---------------------------------------------------------| | / (n1 + 1)! | |----- | \n1 = 0 / ---------------------------------------------------------------------} n - 1 "A298122" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 { 16 16 (7 n - 1) { 2 binomial(n, n/2) (7 n - 6) { ----------------------------- n::even { ----------------------------- n::even { 2 2 2 { (n + 2) (n - 1) { n (n + 1) binomial(n, n/2) {{ , { } { 2 { (4 n + 4) { 8 binomial(n - 1, n/2 - 1/2) (7 n - 1) { 4 2 (7 n - 6) { --------------------------------------- n::odd { ---------------------------------------------------- n::odd { 2 { 2 2 { (n + 1) { (n - 1) (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A298300" LREtools/SearchTable: "SearchTable successful" n (-1) ((3 n + 4) hypergeom([1/2, -n - 1], [1], 4) - 3 n hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------------------} n + 2 "A298308" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A298358" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A298567" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A298611" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A298646" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 16 { 2 { --------------------------------- n::even { 4 binomial(n, n/2) (n + 1) n { 2 { ----------------------------- n::even (2 n + 1) binomial(2 n, n) n { (n + 1) (n + 3) binomial(n, n/2) { 2 {----------------------------, { , { (n + 2) } (n + 1) (n + 2) { (4 n + 4) { { 2 n { 2 2 { -------------------------------------------- n::odd { 16 binomial(n - 1, n/2 - 1/2) n { 2 2 { --------------------------------- n::odd { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { (n + 1) (n + 3) "A298647" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { n { 16 2 ((3 n + 6) LegendreP(n/2 + 1, 3) + (-17 n - 18) LegendreP(n/2, 3)) { ------------------------------------------------------------------------ n::even { n - 2 {{ , { (n + 1) 2 { 8 2 ((n + 3) (7 n - 1) LegendreP(n/2 + 3/2, 3) + (-41 n - 76 n + 13) LegendreP(n/2 + 1/2, 3)) { - ----------------------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { n { 16 2 ((17 n + 18) LegendreQ(n/2, 3) + (-3 n - 6) LegendreQ(n/2 + 1, 3)) { - ------------------------------------------------------------------------ n::even { n - 2 { , { (n + 1) 2 { 8 2 ((41 n + 76 n - 13) LegendreQ(n/2 + 1/2, 3) - (n + 3) (7 n - 1) LegendreQ(n/2 + 3/2, 3)) { ---------------------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { n 2 { 4 2 ((41 n + 76 n - 13) LegendreQ(n/2 + 1/2, 3) - (n + 3) (7 n - 1) LegendreQ(n/2 + 3/2, 3)) { ---------------------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { , { (n - 1) { 8 2 ((17 n + 18) LegendreQ(n/2, 3) + (-3 n - 6) LegendreQ(n/2 + 1, 3)) { - ----------------------------------------------------------------------------- n::odd { n - 2 { n 2 { 4 2 ((n + 3) (7 n - 1) LegendreP(n/2 + 3/2, 3) + (-41 n - 76 n + 13) LegendreP(n/2 + 1/2, 3)) { - ----------------------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { } { (n - 1) { 8 2 ((3 n + 6) LegendreP(n/2 + 1, 3) + (-17 n - 18) LegendreP(n/2, 3)) { ----------------------------------------------------------------------------- n::odd { n - 2 "A298700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n | \ / n1 / n1 n1 {(-1) , (-1) | ) |- (-1) |2 (27 n1 + 44) (2 n1 + 1) hypergeom([- ----, - ---- - 1/2], [-2 n1 - 2], -4) | / \ \ 2 2 |----- \n1 = 0 \ | n1 n1 \ \| + 3 (3 n1 + 4) (3 n1 + 2) hypergeom([- ----, - ---- + 1/2], [-2 n1], -4)| binomial(2 n1, n1)/((n1 + 2) (n1 + 1))||} 2 2 / /| | / "A299270" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A299271" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {n - 1} "A299296" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) | n n | \ 6 ((5 n1 + 4) LegendreP(n1 + 1, 3) + (-27 n1 - 12) LegendreP(n1, 3))| {6 , 6 | ) -----------------------------------------------------------------------------|, | / n1 (n1 - 1) | |----- | \n1 = 0 / /n - 1 \ |----- (-n1 - 1) | n | \ 6 ((5 n1 + 4) LegendreQ(n1 + 1, 3) + (-27 n1 - 12) LegendreQ(n1, 3))| 6 | ) -----------------------------------------------------------------------------|} | / n1 (n1 - 1) | |----- | \n1 = 0 / "A299443" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A299501" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A299502" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A299506" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 2 1/2 2 (n + 1) (2 n + 1) LegendreP(2 n + 2, 5 ) + (-76 n - 72 n - 11) LegendreP(2 n, 5 ) {----------------------------------------------------------------------------------------, (4 n + 3) (2 n - 1) 1/2 2 1/2 2 (n + 1) (2 n + 1) LegendreQ(2 n + 2, 5 ) + (-76 n - 72 n - 11) LegendreQ(2 n, 5 ) ----------------------------------------------------------------------------------------} (4 n + 3) (2 n - 1) "A299507" LREtools/SearchTable: "SearchTable successful" (n + 1) LegendreP(n + 1, 7) + (-13 n - 7) LegendreP(n, 7) (n + 1) LegendreQ(n + 1, 7) + (-13 n - 7) LegendreQ(n, 7) {---------------------------------------------------------, ---------------------------------------------------------} n n "A299845" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (10 n - 1) LegendreP(n + 1, 9) + (-180 n - 71 n + 9) LegendreP(n, 9) {-----------------------------------------------------------------------------, n (n - 1) 2 (n + 1) (10 n - 1) LegendreQ(n + 1, 9) + (-180 n - 71 n + 9) LegendreQ(n, 9) -----------------------------------------------------------------------------} n (n - 1) "A299853" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 4 4 binomial(---, n/2) { 3 n 3 n { 2 {{ binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { ----------------------- n::even} { 2 2 { (3 n - 4) (3 n - 2) { ----------------------------------------------------------- n::odd { { n (3 n - 4) binomial(n - 1, n/2 - 1/2) { 0 n::odd "A299854" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 2 4 binomial(---, n/2) { 3 n 3 n { 2 {{ (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { ----------------------- n::even} { 2 2 { n + 2 { --------------------------------------------------------------------- n::odd { { (n + 2) n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A299855" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n 3 n { { 2 4 binomial(---, n/2) { 3 n 3 n { 2 {{ binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { ----------------------- n::even} { 2 2 { 3 n - 2 { ----------------------------------------------------------- n::odd { { n binomial(n - 1, n/2 - 1/2) { 0 n::odd "A299864" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 1/2 2 1/2 (-1) (2 (n + 1) (2 n + 1) LegendreP(2 n + 2, 3 I) + (52 n + 56 n + 13) LegendreP(2 n, 3 I)) {---------------------------------------------------------------------------------------------------, (4 n + 3) (2 n - 1) n 1/2 2 1/2 (-1) (2 (n + 1) (2 n + 1) LegendreQ(2 n + 2, 3 I) + (52 n + 56 n + 13) LegendreQ(2 n, 3 I)) ---------------------------------------------------------------------------------------------------} (4 n + 3) (2 n - 1) "A299958" LREtools/SolveLRE: "Absolute Factorization reduced the order from 5 to 1 (Liouvillian solutions)" { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { 0 irem(n, 5) = 2 {{ , { 0 irem(n, 5) = 3 { { (n/5 - 4/5) { (-800000) GAMMA(n/5 + 7/10) GAMMA(n/5 + 1/5) GAMMA(n/5 - 1/20) GAMMA(n/5 + 9/20) { ------------------------------------------------------------------------------------------- irem(n, 5) = 4 { GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 2/5) { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { 0 irem(n, 5) = 2 { , { (n/5 - 3/5) { (-800000) GAMMA(n/5 - 1/20) GAMMA(n/5 + 1/5) GAMMA(n/5 + 7/10) GAMMA(n/5 + 9/20) { ------------------------------------------------------------------------------------------- irem(n, 5) = 3 { GAMMA(n/5 + 2/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { { 0 irem(n, 5) = 1 { { (n/5 - 2/5) { (-800000) GAMMA(n/5 + 1/5) GAMMA(n/5 - 1/20) GAMMA(n/5 + 7/10) GAMMA(n/5 + 9/20) , { ------------------------------------------------------------------------------------------- irem(n, 5) = 2 { GAMMA(n/5 + 3/5) GAMMA(n/5 + 2/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 { 0 irem(n, 5) = 0 { { (n/5 - 1/5) { (-800000) GAMMA(n/5 - 1/20) GAMMA(n/5 + 9/20) GAMMA(n/5 + 7/10) GAMMA(n/5 + 1/5) { ------------------------------------------------------------------------------------------- irem(n, 5) = 1 { GAMMA(n/5 + 2/5) GAMMA(n/5 + 4/5) GAMMA(n/5 + 3/5) GAMMA(n/5 + 1) , { { 0 irem(n, 5) = 2 { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 { (n/5) { (-800000) GAMMA(n/5 + 1/5) GAMMA(n/5 + 9/20) GAMMA(n/5 + 7/10) GAMMA(n/5 - 1/20) { ------------------------------------------------------------------------------------- irem(n, 5) = 0 { GAMMA(n/5 + 3/5) GAMMA(n/5 + 1) GAMMA(n/5 + 4/5) GAMMA(n/5 + 2/5) { { 0 irem(n, 5) = 1} { { 0 irem(n, 5) = 2 { { 0 irem(n, 5) = 3 { { 0 irem(n, 5) = 4 "A300048" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A300057" LREtools/SearchTable: "SearchTable not successful" {} "A300116" LREtools/SearchTable: "SearchTable not successful" {} "A300126" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A300159" LREtools/SearchTable: "SearchTable successful" 2 3 ((n + 1) (n - 2 n - 1) LaguerreL(n + 1, -1) + (-n + 6 n + 2) LaguerreL(n, -1)) n! {-----------------------------------------------------------------------------------} n (n - 1) "A300482" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A300483" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A300484" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A300485" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A300490" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ n2 || | (-1) | ) (-(-1) (n2 + 1) n2!)|| /n - 1 \ |n - 1 | / || |----- n1 | |----- |----- || | \ (-1) | | \ \n2 = 0 /| {n! | ) ---------|, n! | ) --------------------------------------|, | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / / /n1 - 1 / /n2 - 1 \\\\ | |----- | |----- n3 |||| | n1 | \ | n2 | \ (-1) |||| | (-1) | ) |-(-1) (n2 + 1) n2! | ) ------------------|||| |n - 1 | / | | / (n3 + 2) (n3 + 1)!|||| |----- |----- | |----- |||| | \ \n2 = 0 \ \n3 = 0 ///| n! | ) ------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A300559" {1, (n + 1) n! n} "A300931" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" { (n/2) { 2 (n/2)! n::even 1/2 n 1/2 { {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), n!, { (n/2 + 1/2) , { 2 (n/2 + 1/2)! { ------------------------- n::odd { n + 1 { (- n/2) { 2 binomial(n, n/2) (n/2)! n::even { } { (- n/2 + 1/2) { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A301476" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 / n1 \ \ /n - 1 / n1 \ \ |----- |- ---- - 2| 1/2 | |----- |- ---- - 2| 1/2 | n n | \ \ 2 / 2 | n | \ \ 2 / 2 | {4 , 4 | ) 2 LegendreP(n1 + 1, ----)|, 4 | ) 2 LegendreQ(n1 + 1, ----)|} | / 2 | | / 2 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A301897" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A301990" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 3) | n n | \ 2 binomial(2 n1, n1) n1!| {8 n!, 8 n! | ) -----------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A302113" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 3) n1 | n | \ 2 (-1) (n1 + 1) binomial(2 n1, n1) n1!| 8 n! (2 n - 1) | ) ---------------------------------------------------| | / (2 n1 - 1) (n1 + 1)! | n |----- | 8 n! (2 n - 1) \n1 = 0 / {---------------, ----------------------------------------------------------------------------} n n "A302115" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-5 n1 - 4) n1 | n | \ 2 (-1) (n1 + 1) binomial(2 n1, n1) n1!| 16 n! | ) ---------------------------------------------------| | / (n1 - 1/2) (n1 + 1)! | n |----- | 16 n! \n1 = 0 / {------, -------------------------------------------------------------------} n n "A302117" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 2) | n | \ 2 (n1 + 1) binomial(2 n1, n1) n1!| 4 n! | ) --------------------------------------------| | / (n1 - 1/2) (n1 + 1)! | n |----- | 4 n! \n1 = 0 / {-----, -----------------------------------------------------------} n n "A302178" LREtools/SearchTable: "SearchTable successful" ((7 n + 11) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) + (-9 n - 9) hypergeom([1/2, -n, -n], [1, 1], 4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------} 2 (n + 2) (n + 1) "A302180" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302181" LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 4) {------------------------------------------------------------} n + 1 "A302183" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302184" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302186" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302188" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 5 | | 5 | {|- 3/2 - ----| n!, |- 3/2 + ----| n!} \ 2 / \ 2 / "A302195" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 2 | | 2 | {|-1 - ----| n!, |-1 + ----| n!} \ 2 / \ 2 / "A302196" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | | 2 | |-1 - ----| n! |-1 + ----| n! \ 2 / \ 2 / {---------------, ---------------} n n "A302198" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | |- 3/2 - ----| n! |- 3/2 + ----| n! \ 2 / \ 2 / {------------------, ------------------} n n "A302705" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A302734" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- n2 | \ \ | \ (-1) | {1, n, ) (n1 + 1) n1! (n1 + 3), ) (n1 + 1) n1! (n1 + 3) | ) ------------------------------------|} / / | / (n2 + 2) (n2 + 1)! (n2 + 4) (n2 + 3)| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A302749" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1) ((-2 n - 1) BesselI(n/2, 1) + 2 BesselI(n/2 - 1, 1)) n::even {{ , { (n/2 + 1/2) { (-1) ((-n - 2) BesselI(n/2 + 1/2, 1) + BesselI(n/2 - 1/2, 1)) n::odd { (n/2) { (-1) ((-2 n - 1) BesselK(n/2, -1) + 2 BesselK(n/2 - 1, -1)) n::even { , { (n/2 + 1/2) { (-1) ((-n - 2) BesselK(n/2 + 1/2, -1) + BesselK(n/2 - 1/2, -1)) n::odd { (n/2) { (-1) ((-n - 2) BesselI(n/2 + 1/2, 1) + BesselI(n/2 - 1/2, 1)) n::even { , { (n/2 - 1/2) { (-1) ((-2 n - 1) BesselI(n/2, 1) + 2 BesselI(n/2 - 1, 1)) n::odd { (n/2) { (-1) ((-n - 2) BesselK(n/2 + 1/2, -1) + BesselK(n/2 - 1/2, -1)) n::even { } { (n/2 - 1/2) { (-1) ((-2 n - 1) BesselK(n/2, -1) + 2 BesselK(n/2 - 1, -1)) n::odd "A302750" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { (-1) (n BesselI(n/2, 1) - BesselI(n/2 - 1, 1)) n::even {{ , { (n/2 + 1/2) { (-1) ((n + 2) BesselI(n/2 + 1/2, 1) - BesselI(n/2 - 1/2, 1)) n::odd { (n/2) { (-1) (n BesselK(n/2, -1) - BesselK(n/2 - 1, -1)) n::even { , { (n/2 + 1/2) { (-1) ((n + 2) BesselK(n/2 + 1/2, -1) - BesselK(n/2 - 1/2, -1)) n::odd { (n/2) { (-1) ((n + 2) BesselI(n/2 + 1/2, 1) - BesselI(n/2 - 1/2, 1)) n::even { , { (n/2 - 1/2) { (-1) (n BesselI(n/2, 1) - BesselI(n/2 - 1, 1)) n::odd { (n/2) { (-1) ((n + 2) BesselK(n/2 + 1/2, -1) - BesselK(n/2 - 1/2, -1)) n::even { } { (n/2 - 1/2) { (-1) (n BesselK(n/2, -1) - BesselK(n/2 - 1, -1)) n::odd "A302769" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) -----------------------------------------------|} | / (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A302827" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ | |----- 2 | | | 2 | \ (n2!) (2 n2 + 3)| 2| | (n1!) | ) -----------------| (n1 + 1) | |n - 1 | / 2 | | |----- |----- n2 ((n2 + 1)!) | | 2 | \ \n2 = 0 / | (n!) | ) -------------------------------------------| | / 2 | 2 2 |----- ((n1 + 1)!) | (n!) (n!) \n1 = 0 / {-----, -----, ----------------------------------------------------------} 3 2 3 n n n "A302865" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) n1! | {(2 n + 1) n! binomial(2 n, n), (2 n + 1) n! binomial(2 n, n) | ) -----------------------------------------------|} | / (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A302870" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 3 n {(-RootOf(_Z - 5 _Z + 3, index = 1) - 2 RootOf(_Z - 5 _Z + 3, index = 1) + 1) n!, 3 2 3 n (-RootOf(_Z - 5 _Z + 3, index = 2) - 2 RootOf(_Z - 5 _Z + 3, index = 2) + 1) n!, 3 2 3 n (-RootOf(_Z - 5 _Z + 3, index = 3) - 2 RootOf(_Z - 5 _Z + 3, index = 3) + 1) n!} "A302944" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ (-1) 2 n1! | {(2 n + 1) 2 n! binomial(2 n, n), (2 n + 1) 2 n! binomial(2 n, n) | ) -----------------------------------------------|} | / (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A302945" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) 2 | n 2 n 2 | \ (-1) 2 (n1!) | {(2 n + 1) 2 (n!) binomial(2 n, n), (2 n + 1) 2 (n!) binomial(2 n, n) | ) ---------------------------------------|} | / 2| |----- binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) | \n1 = 0 / "A302978" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A303108" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 n1! | {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) -----------------------------------------------|} | / (2 n1 + 3) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A303109" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) 2 | n 2 n 2 | \ 2 (n1!) | {(1/2) (n!) binomial(2 n, n), (1/2) (n!) binomial(2 n, n) | ) ---------------------------------------|} | / 2| |----- binomial(2 n1 + 2, n1 + 1) ((n1 + 1)!) | \n1 = 0 / "A303224" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 1/2 n 1/2 1/2 1/2 1/2 {(-3 ) 3 (-n BesselJ(n, -2 3 ) - 3 BesselJ(n - 1, -2 3 )), (-3 ) 3 (-n BesselY(n, -2 3 ) - 3 BesselY(n - 1, -2 3 ))} "A303259" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 3 n 3 n { 4 n binomial(3 n, ---) binomial(---, n/2) { 2 2 { 3/2 --------------------------------------------- n::even n { (3 n - 1) binomial(n, n/2) {n, (-1) n, { , { (-2 n - 2) 3 n 3 n { 2 (n + 1) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { ------------------------------------------------------------------------------- n::odd { (3 n + 2) binomial(n + 1, n/2 + 1/2) { 3 n { 2 binomial(---, n/2) n::even { 2 { } { 3 n { 3 binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 "A303271" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 2 2 (-1) ((41 n + 318 n + 775 n + 594) hypergeom([1/2, -n - 1], [1], 4) - 3 (13 n + 81 n + 122) (n + 1) hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------------------------------------------------------------------} (n + 5) (n + 4) (n + 3) (n + 2) "A303486" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { {{ 0 irem(n, 3) = 1, { { (n/3 - 2/3) { 6912 GAMMA(n/3 + 3/4) GAMMA(n/3 + 1/4) GAMMA(n/3 + 1/2) irem(n, 3) = 2 { 0 irem(n, 3) = 0 { { (n/3 - 1/3) , { 6912 GAMMA(n/3 + 3/4) GAMMA(n/3 + 1/4) GAMMA(n/3 + 1/2) irem(n, 3) = 1 { { 0 irem(n, 3) = 2 { n 2 n 2 3 4 n 2 n { 3 binomial(---, n/3) ((n/3)!) binomial(---, ---) irem(n, 3) = 0 { 3 3 3 { } { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A303537" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (2 n + 1) hypergeom([3/4, -n], [1], 2)) {----------------------------------------------------------------------------------------} n "A303538" LREtools/SearchTable: "SearchTable successful" n 8 ((4 n + 4) hypergeom([7/8, -n - 1], [1], 2) + (4 n + 3) hypergeom([7/8, -n], [1], 2)) {----------------------------------------------------------------------------------------} n "A303602" n {4 (2 n + 1), (2 n + 1) binomial(2 n, n)} "A303730" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1/4) } "A303790" memory used=247073.6MB, alloc=3351.5MB, time=1905.98 LREtools/SearchTable: "SearchTable not successful" {} "A303952" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (n1 + 1) | {1, (1/2) , (1/2) | ) 2 ((12 n1 + 7) hypergeom([-1/2, -n1 - 1], [1], -4) + (-12 n1 - 13) hypergeom([-1/2, -n1], [1], -4))|} | / | |----- | \n1 = 0 / "A304011" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((n - 3) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 3) hypergeom([1/2, -n], [1], 4)) {1, ------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A304202" n (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {4 (2 n + 3), ----------------------------------------------} (n + 4) (n + 2) (n + 1) "A304561" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) 3 2 { (27/2) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (32 n + 129 n + 33 n + 292) { ------------------------------------------------------------------------- irem(n, 3) = 0 { n + 3 { {{ (n/3 - 1/3) , { 486 (27/2) GAMMA(5/3 + n/3) GAMMA(n/3 + 1) irem(n, 3) = 1 { { (n/3 + 1/3) { 144 n (27/2) GAMMA(n/3 + 7/3) GAMMA(5/3 + n/3) { --------------------------------------------------------- irem(n, 3) = 2 { n + 4 { (n/3) { 486 (27/2) GAMMA(n/3 + 1) GAMMA(5/3 + n/3) irem(n, 3) = 0 { { (n/3 + 2/3) { 144 n (27/2) GAMMA(5/3 + n/3) GAMMA(n/3 + 7/3) { { --------------------------------------------------------- irem(n, 3) = 1, { { n + 4 { { { (n/3 + 1/3) 3 2 { (27/2) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) (32 n + 129 n + 33 n + 292) { ------------------------------------------------------------------------------- irem(n, 3) = 2 { n + 3 (- n/3) 2 2 n 144 2 n ((n/3)!) (n + 1) (n + 2) binomial(n, n/3) binomial(---, n/3) , irem(n, 3) = 0 3 (1/3 - n/3) 2 3 2 2 n 2 n ((n/3 - 1/3)!) (n + 1) (32 n + 129 n + 33 n + 292) binomial(n - 1, n/3 - 1/3) binomial(--- - 2/3, n/3 - 1/3) , irem(n, 3) = 1 3 (- n/3 + 2/3) 2 2 n 486 2 n ((n/3 - 2/3)!) (n - 1) (n + 2) binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) , irem(n, 3) = 2} 3 "A304564" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { (n/3) { (27/8) GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) %1 { --------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 13/6) (2 n + 13) (n + 3) { { (n/3 - 1/3) {{ 486 (27/8) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) , { ---------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 11/6) { { (n/3 + 1/3) { 72 (4 n + 15) n (27/8) GAMMA(n/3 + 7/3) GAMMA(n/3 + 2) GAMMA(5/3 + n/3) { ---------------------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 5/2) (n + 3) (n + 4) { (n/3) { 486 (27/8) GAMMA(n/3 + 1) GAMMA(5/3 + n/3) GAMMA(n/3 + 4/3) { ---------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 11/6) { { (n/3 + 2/3) { 72 (4 n + 15) n (27/8) GAMMA(5/3 + n/3) GAMMA(n/3 + 7/3) GAMMA(n/3 + 2) , { ---------------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 5/2) (n + 3) (n + 4) { { (n/3 + 1/3) { (27/8) GAMMA(5/3 + n/3) GAMMA(n/3 + 2) GAMMA(n/3 + 4/3) %1 { --------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 13/6) (2 n + 13) (n + 3) { (- n/3) 2 { 216 2 n ((n/3)!) (n + 1) (n + 2) (4 n + 15) binomial(n, n/3) { -------------------------------------------------------------------- irem(n, 3) = 0 { (2 n + 3) (2 n + 9) { { (1/3 - n/3) 2 { 3 2 n ((n/3 - 1/3)!) (n + 1) (n + 2) %1 binomial(n - 1, n/3 - 1/3) } { ------------------------------------------------------------------------------ irem(n, 3) = 1 { (2 n + 1) (2 n + 7) (2 n + 13) { { (- n/3 + 2/3) 2 { 1458 2 n ((n/3 - 2/3)!) (n - 1) (n + 1) (n + 2) binomial(n - 2, n/3 - 2/3) { ---------------------------------------------------------------------------------------- irem(n, 3) = 2 { (2 n - 1) (2 n + 5) 4 3 2 %1 := 64 n + 1106 n + 6603 n + 13649 n + 7738 "A304915" LREtools/SearchTable: "SearchTable successful" n / 15 15 \ 16 |(8 n + 8) hypergeom([--, -n - 1], [1], 2) + (8 n + 7) hypergeom([--, -n], [1], 2)| \ 16 16 / {---------------------------------------------------------------------------------------} n "A304933" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (2 n + 1) hypergeom([3/4, -n], [1], 2))} "A304934" LREtools/SearchTable: "SearchTable successful" n {8 ((4 n + 4) hypergeom([7/8, -n - 1], [1], 2) + (4 n + 3) hypergeom([7/8, -n], [1], 2))} "A304940" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 8 { n { ------------------ n::even { 2 binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { (n - 1) { (3 n + 3) { 4 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A304941" LREtools/SearchTable: "SearchTable successful" n (-4) ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (-2 n + 1) hypergeom([3/4, -n], [1], 2)) {--------------------------------------------------------------------------------------------} n "A304944" LREtools/SearchTable: "SearchTable successful" n {(-4) ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (-2 n + 1) hypergeom([3/4, -n], [1], 2))} "A305031" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 (4 n - 1) { ------------------ n::even { (4 n + 1) binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { binomial(n - 1, n/2 - 1/2) (8 n - 2) n::odd { (2 n + 2) { 2 2 (4 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A305032" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (4 n - 1) { ---------------- n::even { n binomial(n, n/2) (4 n + 1) n::even { binomial(n, n/2) {{ , { } { 1/2 (4 n - 1) (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (8 n + 2) { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A305118" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A305459" LREtools/SearchTable: "SearchTable successful" n n {(-1) (3 n BesselI(n, 2/3) - BesselI(n - 1, 2/3)), (-1) (3 n BesselK(n, -2/3) - BesselK(n - 1, -2/3))} "A305460" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 1/2 2 2 2 2 | 1/2 n 1/2 | 2 2 1/2 2 2 | {(-2 ) 2 |2 BesselK(n - 1, - ------) - 3 n BesselK(n, - ------)|, (-2 ) 2 |-3 n BesselI(n, ------) + 2 BesselI(n - 1, ------)|} \ 3 3 / \ 3 3 / "A305471" LREtools/SearchTable: "SearchTable successful" n n {(-1) (3 n BesselJ(n, -2/3) + BesselJ(n - 1, -2/3)), (-1) (3 n BesselY(n, -2/3) + BesselY(n - 1, -2/3))} "A305472" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n 1/2 | 2 2 1/2 2 2 | 1/2 n 1/2 | 2 2 1/2 2 2 | {(-2 ) 2 |-3 n BesselJ(n, - ------) - 2 BesselJ(n - 1, - ------)|, (-2 ) 2 |-3 n BesselY(n, - ------) - 2 BesselY(n - 1, - ------)| \ 3 3 / \ 3 3 / } "A305554" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 3 2 n (n + 1) (8 n - 5 n - 2) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7 n + 7 n - 2) hypergeom([-1/2, -n], [1], -4) {5 , ---------------------------------------------------------------------------------------------------------------------} n (n - 1) "A305555" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 3 2 n (n + 1) (16 n + 10 n - 19) hypergeom([-1/2, -n - 1], [1], -4) + (-16 n - 34 n - n - 19) hypergeom([-1/2, -n], [1], -4) {5 , -------------------------------------------------------------------------------------------------------------------------} n (n - 1) "A305561" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /n - 1 / /n1 - 1 / 3 n2\ /{ 0 n2::even\\\\ |----- | |----- |- 3/2 - ----| |{ |||| 1/2 n 1/2 n 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ (2 n2 - 2) |||| {(-2 2 ) , (2 2 ) , (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ 2 ||||, | / | | / |{ ---------------------------------------- n2::odd |||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) |||| \ \ \ \{ 2 //// /n - 1 / /n1 - 1 / 3 n2\ /{ n2 \\\\ |----- | |----- |- 3/2 - ----| |{ 2 binomial(n2, ----) |||| 1/2 n | \ | 1/2 n1 | \ \ 2 / |{ 2 |||| (-2 2 ) | ) |-1/4 2 (-1) | ) 2 |{ -------------------- n2::even||||} | / | | / |{ n2 + 2 |||| |----- | |----- |{ |||| \n1 = 0 \ \n2 = 0 \{ 0 n2::odd //// "A305573" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 (3 n1 + 1) (3 n1 + 2) (3 n1 + 4) binomial(3 n1, n1)| {8 (n + 4), 8 (n + 4) | ) ----------------------------------------------------------------|} | / (n1 + 1) (n1 + 4) (n1 + 5) (2 n1 + 1) (2 n1 + 3) | |----- | \n1 = 0 / "A305577" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2 + 1) (n/2)! n::even {{ , { } { (-n + 1) { 0 n::odd { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A305578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 1/2 (n/2)! binomial(n, n/2) n n::even {{ , { } { (n - 1) { 0 n::odd { 2 n (n/2 - 1/2)! n::odd "A305608" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n + 2) hypergeom([3/4, -n - 1], [1], 2) + (2 n + 1) hypergeom([3/4, -n], [1], 2)) {----------------------------------------------------------------------------------------} n "A305609" LREtools/SearchTable: "SearchTable successful" n 8 ((4 n + 4) hypergeom([7/8, -n - 1], [1], 2) + (4 n + 3) hypergeom([7/8, -n], [1], 2)) {----------------------------------------------------------------------------------------} n "A305612" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 (4 n - 1) { ------------------ n::even { (4 n + 1) binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { binomial(n - 1, n/2 - 1/2) (8 n - 2) n::odd { (2 n + 2) { 2 2 (4 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A305693" 2 n binomial(2 n, n) {-------------------, binomial(4 n, 2 n)} 2 n - 1 "A305730" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {(n + 1) n! n, (n + 1) n! n | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A305739" 2 {1, (n + 2) (n + 1) n!} "A306150" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ 1 | | \ (-1) | {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A306183" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 5 5 {GAMMA(n + 3/2 - ----), GAMMA(n + 3/2 + ----)} 2 2 "A306185" n n {(n + 1) 2 n!, (2 n + 3) (2 n + 1) (1/2) n! binomial(2 n, n)} "A306258" n 2 {(-1) n!, n! (2 n - 1)} "A306292" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) { -------------------------- n::even {--------------------------, { , { n + 2 } (n + 1) (n + 2) { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A306322" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- ----- \ 9 n1 + 2 \ {1, ) --------------------------------, ) / (2 n1 + 1) n1 binomial(2 n1, n1) / ----- ----- n1 = 0 n1 = 0 /n1 - 1 \ |----- 2 6 5 4 3 2 | | \ binomial(2 n2, n2) (7875 n2 + 29475 n2 + 33705 n2 + 8109 n2 - 6760 n2 - 3852 n2 - 512) binomial(2 n2 + 2, n2 + 1)| (9 n1 + 2) | ) -----------------------------------------------------------------------------------------------------------------------| | / 2 | |----- (n2 + 1) (n2 + 2) (2 n2 - 1) (9 n2 + 11) (9 n2 + 2) | \n2 = 0 / -------------------------------------------------------------------------------------------------------------------------------------------, (2 n1 + 1) n1 binomial(2 n1, n1) binomial(2 n, n)} "A306376" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n \ (2 n1 + 1) binomial(2 n1, n1) {1, 2 , ) -----------------------------} / n1 + 1 ----- n1 = 0 "A306386" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A306409" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ | ----- | | \ (2 n2 + 1) binomial(2 n2, n2) (3 n2 + 5)| | ) ----------------------------------------| |n - 1 / (n2 + 1) (n2 + 2) | |----- ----- | n n | \ n2 = 0 | {1, (-1/2) , (-1/2) | ) -----------------------------------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A306419" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A306432" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- n \ (-n1) {1, (-1) , ) 3 hypergeom([1/2, -n1 - 1], [1], 4)} / ----- n1 = 0 "A306455" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (2 n1 + 1) | {(n + 1) n!, n! (2 n + 3), n! (2 n + 3) | ) -------------------------------|} | / (n1 + 1)! (2 n1 + 5) (2 n1 + 3)| |----- | \n1 = 0 / "A306463" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A306495" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, n! n, n! n | ) ---------------------|} | / (n1 + 1)! (n1 + 1) n1| |----- | \n1 = 0 / "A306511" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 2 {1, ) (-1) ((n1 + 1) BesselI(n1, 2) + (-n1 - 2) BesselI(n1 - 1, 2)), / ----- n1 = 0 n - 1 ----- \ n1 2 ) (-1) ((n1 + 1) BesselK(n1, -2) + (-n1 - 2) BesselK(n1 - 1, -2)), n!} / ----- n1 = 0 "A306519" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- / n1 \| n n | \ (2 I) (-LegendreQ(n1 + 1, I) I + LegendreQ(n1, I))| n | \ | (2 I) (-LegendreP(n1, I) + LegendreP(n1 + 1, I) I)|| {(-1) , (-1) | ) ----------------------------------------------------|, (-1) | ) |- ----------------------------------------------------||} | / n1 + 2 | | / \ n1 + 2 /| |----- | |----- | \n1 = 0 / \n1 = 0 / "A306525" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselJ(n, -2) + BesselJ(n - 1, -2)), (-1) (n BesselY(n, -2) + BesselY(n - 1, -2)), n!} "A306535" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 2 n1 + 1 | {(n!) binomial(2 n, n), (n!) binomial(2 n, n) | ) ---------------------------------------|} | / 2 | |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A306609" n {16 (2 n + 1), binomial(4 n, 2 n) (2 n + 1)} "A306623" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A306642" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) binomial(3 n, n) hypergeom([-n, -n, -n], [1, 1], -1)} "A306668" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (4 (2 n + 1) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (31 n + 19) n hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/(n (2 n - 1) (5 n + 3))} "A306675" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 2 n1 + 1 | {(n!) binomial(2 n, n), (n!) binomial(2 n, n) | ) ---------------------------------------|} | / 2 | |----- ((n1 + 1)!) binomial(2 n1 + 2, n1 + 1)| \n1 = 0 / "A306948" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- / n2 2\|| | n1 | \ | (-1) (n2 + 2) ||| | (-1) n1! | ) |- ----------------||| /n - 1 \ |n - 1 | / \ (n2 + 1)! /|| |----- n1 | |----- |----- || | \ (-1) n1! | | \ \n2 = 0 /| {(n + 1) | ) -----------------|, (n + 1) | ) ----------------------------------------|, n + 1} | / (n1 + 2) (n1 + 1)| | / (n1 + 2) (n1 + 1) | |----- | |----- | \n1 = 0 / \n1 = 0 / "A307005" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2 \n / 1/2 \n | 3 | |3 | {|- ---- + 1/2| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A307006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A307158" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 1/2 n 1/2 {(1/2 - 1/2 I 3 ) LegendreP(n, 3 I), (1/2 - 1/2 I 3 ) LegendreQ(n, 3 I), (1/2 + 1/2 I 3 ) LegendreP(n, -I 3 ), { 0 n::even 1/2 n 1/2 { { (n/2) (1/2 + 1/2 I 3 ) LegendreQ(n, -I 3 ), { (n/2 - 1/2) , { (-1) binomial(n, n/2) n::even} { (-16) { { ---------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A307318" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 |----- n1 | |----- n n | \ (2 n1 + 1) (1/4) binomial(2 n1, n1)| n | \ n1 {1, (-1/3) , (-1/3) | ) -------------------------------------|, (-1/3) | ) (2 n1 + 1) (1/4) binomial(2 n1, n1) | / (n1 + 1) | | / |----- (n1 + 1) (-1/3) | |----- \n1 = 0 / \n1 = 0 /n1 - 1 \ \ |----- n2 (2 n2 + 2) 2 | | | \ (-1) 2 (3 n2 + 1) (3 n2 + 2) (35 n2 + 103 n2 + 75) binomial(2 n2, n2) binomial(3 n2, n2)| / (n1 + 1) | | ) -----------------------------------------------------------------------------------------------------| / ((n1 + 1) (-1/3) )|} | / 2 | / | |----- (n2 + 1) (2 n2 + 3) (2 n2 + 4) binomial(2 n2 + 2, n2 + 1) | | \n2 = 0 / / "A307349" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n n | \ (2 n1 + 1) binomial(2 n1, n1)| {1, (-1) , (-1/2) , (-1/2) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A307354" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / n1 - 1 \ | ----- | | \ (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)| | ) ----------------------------------------| |n - 1 / (n2 + 1) (n2 + 2) | |----- ----- | n n | \ n2 = 0 | {1, (-1/2) , (-1/2) | ) -----------------------------------------------|} | / (n1 + 1) | |----- (-1/2) | \n1 = 0 / "A307374" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307376" LREtools/SearchTable: "SearchTable successful" ((6 n + 3) hypergeom([-n, 2 n + 2], [], -1/2) - 2 hypergeom([2 n, -n + 1], [], -1/2)) binomial(2 n, n) {------------------------------------------------------------------------------------------------------} 9 n + 1 "A307495" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([-n - 1], [n + 2], 1) + (-4 n - 1) hypergeom([-n], [n + 1], 1)) binomial(2 n, n) {-----------------------------------------------------------------------------------------------------} n "A307557" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307572" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A307577" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A307642" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ (n1 + 1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A307678" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307695" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -3))} "A307696" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307733" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307768" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ------------------ n::even n { n binomial(n, n/2) { binomial(n, n/2) n::even {2 , { , { } { (2 n + 2) { 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A307788" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A307789" LREtools/SearchTable: "SearchTable successful" n (-1) ((n - 2) hypergeom([1/2, -n - 1], [1], 4) + (-n - 2) hypergeom([1/2, -n], [1], 4)) (n + 1) {------------------------------------------------------------------------------------------------} (n - 1) n "A307810" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-4*(2*n+5)*(84*n^2+336*n+311)*E^2+256*(2*n+3)*(84*n^2+336*n+311)*E-262144*(2*n+5)*(n +1)^2 "to two: Symmetric square" (4*n+8)*E^2+(-10*n-9)*E+4*n+4 LREtools/SearchTable: "SearchTable successful" n 2 {16 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -3) + (-2 n - 3) hypergeom([-1/2, -n], [1], -3)) } "A307811" memory used=248361.4MB, alloc=3383.5MB, time=1915.57 LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(151*n^2+604*n+555)*E^2+45*(2*n+3)*(151*n^2+604*n+555)*E-91125*(2*n+5)*(n+1 )^2 "to two: Symmetric square" (3*n+6)*E^2+(-14*n-13)*E+15*n+15 LREtools/SearchTable: "SearchTable successful" n 2 2 {25 (16 (n + 1) (n - 3) hypergeom([-3/2, -n - 1], [1], -4/5) + (-16 n + 8 n + 39) hypergeom([-3/2, -n], [1], -4/5)) } "A307935" LREtools/SearchTable: "SearchTable successful" n n {(-2) n! (n + 1) ((2 n + 1) BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-2) n! (n + 1) ((2 n + 1) BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A307969" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ 1/2 n | 1/2 3 3 | 1/2 n | 1/2 3 3 | (-2 3 ) |3 LegendreP(n + 1, ----) - 3 LegendreP(n, ----)| (-2 3 ) |3 LegendreQ(n + 1, ----) - 3 LegendreQ(n, ----)| \ 3 3 / \ 3 3 / {---------------------------------------------------------------, ---------------------------------------------------------------} n + 2 n + 2 "A307970" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A307971" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A307972" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A308036" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2) | 1/2 13 13 | (n/2) | 1/2 13 13 | 13 |13 LegendreP(n + 1, -----) - 13 LegendreP(n, -----)| 13 |-13 LegendreQ(n + 1, -----) + 13 LegendreQ(n, -----)| \ 13 13 / \ 13 13 / {----------------------------------------------------------------, - -----------------------------------------------------------------} n + 2 n + 2 "A308294" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A308295" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A308363" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A308435" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A308523" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ binomial(4 n1, n1) (24 n1 + 11) | {|---| , |---| | ) -----------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A308524" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- ||| n n n n | \ | (-n1) n1 | \ binomial(2 n2, n2) n2 ||| {4 , (-4/3) , (-1/2) , (-1/2) | ) |-2 3 8 | ) -----------------------|||} | / | | / (n2 + 1)||| |----- | |----- (n2 + 1) (-4/3) ||| \n1 = 0 \ \n2 = 0 /// "A308526" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 / /n1 - 1 /n2 - 1 / 1/2\n / 1/2\n / 1/2\n |----- | |----- |----- n |272 112 7 | |272 112 7 | |272 112 7 | | \ | 27 n1 1/2 (-n1 - 1) 1/2 n1 | \ n2 | \ / {(-2) , |--- - --------| , |--- + --------| , |--- - --------| | ) |- -- (-1) (7 7 - 17) (17 + 7 7 ) | ) (-2) | ) |- \27 27 / \27 27 / \27 27 / | / | 16 | / | / \ |----- | |----- |----- \n1 = 0 \ \n2 = 0 \n3 = 0 n3 / 1/2 (-1) (4 n3 + 1) (4 n3 + 3) | \ 5 4 3 2 n3 n3 2 (4 n3 + 7) (4 n3 + 5) (1722 n3 + 3493 n3 - 18834 n3 - 62833 n3 - 55548 n3 - 10080) hypergeom([- ----, - ---- - 1/2], [-2 n3 - 7/2], -1) 2 2 4 3 2 n3 n3 \ + 9 (3 n3 + 5) (3 n3 + 7) (n3 + 2) (1036 n3 + 5943 n3 + 11213 n3 + 7436 n3 + 1120) hypergeom([- ----, - ---- + 1/2], [-2 n3 - 3/2], -1)| 2 2 / \ | \| / binomial(4 n3, n3)/(n3 (n3 + 1) (n3 + 2) (3 n3 + 1) (3 n3 + 2) (3 n3 + 4) (3 n3 + 5) (3 n3 + 7) (3 n3 + 8) (7 n3 + 13))|| / /| / | / \\\ / 1/2\(n2 + 1)||| |272 112 7 | ||| |--- + --------| |||} \27 27 / ||| ||| /// "A308599" {1, (n + 2) (n + 1) n! (4 n + 3)} "A308606" 2 {1, (n + 1) n! (n + n + 4)} "A308616" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A308729" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | | \ (n2 + 1) n2! || | (n1 + 1) n1! | ) ------------------|| /n - 1 \ |n - 1 | / (n2 + 2) (n2 + 1)!|| |----- | |----- |----- || | \ (n1 + 1) n1! | | \ \n2 = 0 /| {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ----------------------------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A308815" LREtools/SearchTable: "SearchTable successful" n {16 ((2 n + 5) (8 n + 7) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) 3 2 + (-128 n - 498 n - 610 n - 234) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1))/((n + 2) (2 n + 1))} "A308876" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 n1| {2 n!, 2 n! | ) -------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A308939" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) (-4 n1 - 3) | n n | \ 3 2 binomial(2 n1, n1) n1!| {(8/3) n!, (8/3) n! | ) ---------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A309303" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) , (-1) ((2 n + 2) hypergeom([1/2, -n - 1], [1], 4) + (6 n + 3) hypergeom([1/2, -n], [1], 4))} "A309490" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {(n + 1) n!, (n + 1) n! | ) ---------|} | / (n1 + 1)!| |----- | \n1 = 0 / "A309579" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 1) n1! (n1 + 2) (n1 + 3)} / ----- n1 = 0 "A309618" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {(1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) + 2 (n + 1) HermiteH(n, 1/2 I 2 ) I) 2 I} "A309619" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(1/2 I) (HermiteH(n + 1, 1/2 I) + 2 I (n + 1) HermiteH(n, 1/2 I)) I} "A309976" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - 2 _Z - 1, index = 1) , RootOf(_Z - _Z - 2 _Z - 1, index = 2) , RootOf(_Z - _Z - 2 _Z - 1, index = 3) } "A316330" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (-n) 2 {8 (n!) binomial(3 n, n) binomial(4 n, n) hypergeom([-2 n], [1/2], -1/2)} "A316331" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (-n) 2 {8 (n!) (4 n + 1) ((4 n + 3) hypergeom([-2 n - 2], [1/2], -1/2) + (-4 n - 4) hypergeom([-2 n], [1/2], -1/2)) binomial(3 n, n) binomial(4 n, n)/( 4 n + 3)} "A316332" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (-n) 2 {8 (n!) (4 n + 1) ((4 n + 3) hypergeom([-2 n - 2], [1/2], -1/2) + (4 n + 2) hypergeom([-2 n], [1/2], -1/2)) binomial(3 n, n) binomial(4 n, n)/( 4 n + 3)} "A316333" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (-n) 2 {8 (n!) (4 n + 1) ((4 n + 3) hypergeom([-2 n - 2], [1/2], -1/2) + (-4 n - 2) hypergeom([-2 n], [1/2], -1/2)) binomial(3 n, n) binomial(4 n, n)} "A316363" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A316371" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A316403" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2, 4 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n + 2) { ------------------------ n::even n 1/2 n 1/2 n n { (n + 1) binomial(n, n/2) {(-2) , (2 ) , (-2 ) , 2 (2 n - 1), { , { (2 n + 2) { 2 (n + 1) { 1/2 ---------------------------------- n::odd { (n + 2) binomial(n + 1, n/2 + 1/2) { 0 irem(n, 4) = 0 { 2 { { 2 binomial(n, n/2) (n + 1) { 0 irem(n, 4) = 1 { --------------------------- n::even { { n + 2 , { 0 irem(n, 4) = 2, { { { 4 binomial(n - 1, n/2 - 1/2) (n + 2) n { (n/2 - 3/2) { -------------------------------------- n::odd { 2 GAMMA(n/4 + 1) { n + 1 { --------------------------- irem(n, 4) = 3 { GAMMA(n/4 + 3/2) { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1) { 2 binomial(n/2 - 1, n/4 - 1/2) n , { --------------------------- irem(n, 4) = 1, { -------------------------------- irem(n, 4) = 2 { GAMMA(n/4 + 3/2) { n + 2 { { { 0 irem(n, 4) = 2 { 0 irem(n, 4) = 3 { { 0 irem(n, 4) = 3 { n { 2 2 { -------------------------- irem(n, 4) = 0 { (n + 2) binomial(n/2, n/4) { } { 0 irem(n, 4) = 1 { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A316598" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n (2 n + 1) 3 binomial(2 n, n) {-----------------------------} (n + 1) (n + 2) "A316666" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! (n + 2), n! (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2)| |----- | \n1 = 0 / "A316673" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A316698" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable not successful" /n - 1 \ |----- n1 (-3 n1 - 3) 2 | n n | \ 2 3 (15 n1 + 52 n1 + 24) binomial(3 n1, n1)| {27 , 27 | ) ---------------------------------------------------------|} | / (n1 + 1) (n1 + 2) (2 n1 + 1) | |----- | \n1 = 0 / "A316777" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ 5 5 5 {1, ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A316987" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A317094" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 {(n + 1) , n! LaguerreL(n, -1)} "A317096" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1) LaguerreL(n1 + 1, 1) n1!| {2 n!, 2 n! | ) --------------------------------------------|} | / n1 (n1 + 1)! | |----- | \n1 = 0 / "A317111" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 3 n {(1/6 RootOf(_Z + 3 _Z + 2, index = 1) + 1/6 RootOf(_Z + 3 _Z + 2, index = 1) + 2/3) n!, 3 2 3 n (1/6 RootOf(_Z + 3 _Z + 2, index = 2) + 1/6 RootOf(_Z + 3 _Z + 2, index = 2) + 2/3) n!, 3 2 3 n (1/6 RootOf(_Z + 3 _Z + 2, index = 3) + 1/6 RootOf(_Z + 3 _Z + 2, index = 3) + 2/3) n!} "A317133" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A317276" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A317364" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) LaguerreL(n + 1, 2) + (-n + 1) LaguerreL(n, 2)) n! {-----------------------------------------------------------------} n "A317365" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) LaguerreL(n + 1, 1) - 2 n LaguerreL(n, 1)) n! {------------------------------------------------------------} n "A317409" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -I) + (-n - 1 - I) LaguerreL(n, -I)) n! ((n + 1) LaguerreL(n + 1, I) + (-n - 1 + I) LaguerreL(n, I)) n! {-----------------------------------------------------------------, ---------------------------------------------------------------} n n "A317483" {1, (n + 1) n! (n - 2)} "A317487" {(n + 1) (n - 1) (n - 2) n!} "A317527" {(n + 1) n! (n - 1)} "A317553" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A317618" LREtools/SearchTable: "SearchTable successful" ((4 n - 1) (n + 1) hypergeom([-1/2, -n - 1], [1], -2) - (4 n + 5) n hypergeom([-1/2, -n], [1], -2)) n! {------------------------------------------------------------------------------------------------------} n "A317639" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" 1/2 n 1/2 n 3 2 n 3 2 n {(- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , RootOf(_Z - 2 _Z + _Z - 1, index = 1) , RootOf(_Z - 2 _Z + _Z - 1, index = 2) , 3 2 n RootOf(_Z - 2 _Z + _Z - 1, index = 3) } "A317980" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n 19 { 5 GAMMA(n/2 + 1) GAMMA(n/2 + 7/5) n::even { 2 5 GAMMA(n/2 + 3/2) GAMMA(n/2 + --) { { 10 {{ (n + 1) 19 , { ------------------------------------- n::even} { 2 5 GAMMA(n/2 + 3/2) GAMMA(n/2 + --) { 5 n + 9 { 10 { { ------------------------------------------- n::odd { (n - 1) { 5 n + 9 { 5 GAMMA(n/2 + 1) GAMMA(n/2 + 7/5) n::odd "A317985" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A317998" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A318101" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A318104" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 11 10 9 8 7 6 {(-1) (n + 9) (56721229371 n + 4367534661567 n + 152481657562740 n + 3185456413349430 n + 44233119627174423 n + 428555611745766351 n 5 4 3 2 + 2955125062113102450 n + 14497117394423576220 n + 49562488158649997256 n + 112402417863997744032 n + 152102679600966869760 n + 92977176012653363200), n 6 5 4 3 2 n 8 (n + 9) (57638385 n + 2500218495 n + 43421396625 n + 389309902425 n + 1908529848462 n + 4863180842232 n + 5039530452544), (-1) (n + 9) ( 11 10 9 8 7 6 56721229371 n + 4367534661567 n + 152481657562740 n + 3185456413349430 n + 44233119627174423 n + 428555611745766351 n 5 4 3 2 + 2955125062113102450 n + 14497117394423576220 n + 49562488158649997256 n + 112402417863997744032 n + 152102679600966869760 n /n - 1 |----- | \ / n1 n1 + 92977176012653363200) | ) |- (-1) 2 (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) (2 n1 + 9) (2 n1 + 11) (2 n1 + 13) (2 n1 + 15) | / \ |----- \n1 = 0 15 14 13 12 11 (2 n1 + 17) (60585042168441 n1 + 8751866043069423 n1 + 563298987978045153 n1 + 21724200070836582267 n1 + 565951832929362251295 n1 10 9 8 7 + 10603776767220297322893 n1 + 148112263242942329441979 n1 + 1574221478030973306621801 n1 + 12857940867026447535051852 n1 6 5 4 3 + 80802924045999424330688880 n1 + 387830069948112103120411648 n1 + 1397007803421416891206346032 n1 + 3657104765800423418430486912 n1 2 / + 6569656008878455161060226304 n1 + 7242064709995848071952926720 n1 + 3692703873777445105960550400) binomial(2 n1, n1) / ((n1 + 1) (n1 + 2) / 11 10 9 8 (n1 + 3) (n1 + 4) (n1 + 9) (n1 + 10) (56721229371 (n1 + 1) + 4367534661567 (n1 + 1) + 152481657562740 (n1 + 1) + 3185456413349430 (n1 + 1) 7 6 5 4 + 44233119627174423 (n1 + 1) + 428555611745766351 (n1 + 1) + 2955125062113102450 (n1 + 1) + 14497117394423576220 (n1 + 1) 3 2 11 + 49562488158649997256 (n1 + 1) + 112402417863997744032 (n1 + 1) + 152102679600966869760 n1 + 245079855613620232960) (56721229371 n1 10 9 8 7 6 5 + 4367534661567 n1 + 152481657562740 n1 + 3185456413349430 n1 + 44233119627174423 n1 + 428555611745766351 n1 + 2955125062113102450 n1 \ | 4 3 2 \| + 14497117394423576220 n1 + 49562488158649997256 n1 + 112402417863997744032 n1 + 152102679600966869760 n1 + 92977176012653363200))||} /| | / "A318108" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A318109" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A318113" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A318114" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A318160" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 3 n 3 n { 4 n binomial(3 n, ---) binomial(---, n/2) { 2 2 { ---------------------------------------------- n::even n { (3 n - 1) binomial(n, n/2) {(n - 1) n, (-1) (n - 1) n, { , { (-2 n + 2) /3 n \ 3 n 3 n { 2 |--- - 3/2| binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) { \ 2 / 2 2 { ----------------------------------------------------------------------------------- n::odd { binomial(n - 1, n/2 - 1/2) { 3 n { 3 binomial(---, n/2) (n - 1) n { 2 { ------------------------------ n::even { 3 n - 2 { } { 3 n 2 { 2 binomial(--- + 3/2, n/2 + 1/2) (n + 1) n { 2 { ------------------------------------------- n::odd { (3 n + 1) (3 n - 1) "A318161" (2 n - 1) n binomial(3 n, n) {(2 n - 1) n, ----------------------------} 3 n - 1 "A318215" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A318223" LREtools/SearchTable: "SearchTable successful" n (-2) ((2 n + 2) LaguerreL(n + 1, 1/2) + (-2 n - 1) LaguerreL(n, 1/2)) n! {-------------------------------------------------------------------------} n "A318237" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A318245" memory used=249857.4MB, alloc=3479.5MB, time=1925.78 LREtools/SearchTable: "SearchTable not successful" {} "A318293" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A318355" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A318356" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A318417" LREtools/SearchTable: "SearchTable successful" n {4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 3) hypergeom([-1/2, -n], [1], -1)) binomial(2 n, n)} "A318495" LREtools/SearchTable: "SearchTable not successful" {} "A318496" LREtools/SearchTable: "SearchTable not successful" {} "A318591" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 {binomial(3 n, n), { , { (n/3 - 2/3) { (27/4) GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { --------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) { 0 irem(n, 3) = 0 { { binomial(n, n/3) irem(n, 3) = 0 { (n/3 - 1/3) { { (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1/3) , { 0 irem(n, 3) = 1} { --------------------------------------------------- irem(n, 3) = 1 { { GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) { 0 irem(n, 3) = 2 { { 0 irem(n, 3) = 2 "A318614" LREtools/SearchTable: "SearchTable successful" n 4 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 3) hypergeom([-1/2, -n], [1], -1)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------} n + 1 "A318618" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | | 2 | {|1 - ----| n!, |1 + ----| n!} \ 2 / \ 2 / "A318918" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" 1/2 n 1/2 n {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { , { (- n/3 + 2/3) 2 2 n { 3 n ((n/3 - 2/3)!) (n - 1) binomial(n - 2, n/3 - 2/3) binomial(--- - 4/3, n/3 - 2/3) irem(n, 3) = 2 { 3 { 0 irem(n, 3) = 0 { /2 n\ { { |---| { /2 n \ { \ 3 / { |--- - 2/3| , { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) irem(n, 3) = 0} { \ 3 / { { 3 GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) irem(n, 3) = 1 { 0 irem(n, 3) = 1 { { { 0 irem(n, 3) = 2 { 0 irem(n, 3) = 2 "A318976" LREtools/ReduceToOrderTwo: "Checking Symmetric Cube... (can be time consuming...)" LREtools/ReduceToOrderTwo: "Galois group is Sp4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A319013" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A319028" n (2 n + 1) binomial(2 n, n) n {4 , ----------------------------} (n + 1) (n + 2) "A319201" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A319202" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A319204" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A319364" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / { 0 irem(n1, 3) = 0\ | { | | { 0 irem(n1, 3) = 1| | { | | { / n1 \ | | { |- ---- + 2/3| | |n - 1 { \ 3 / // n1 \ \2 n1 2 n1 n1 | |----- { 3 n1 ||---- - 2/3|!| (n1 - 1) binomial(n1 - 2, ---- - 2/3) binomial(---- - 4/3, ---- - 2/3) irem(n1, 3) = 2| | \ { \\ 3 / / 3 3 3 | {n! | ) -----------------------------------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { 0 irem(n1, 3) = 0\ | { | | { /2 n1 \ | | { |---- - 2/3| | | { \ 3 / n1 n1 | | { 3 GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 3) = 1| |n - 1 { 3 3 | |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { /2 n1\ \ | { |----| | | { \ 3 / n1 n1 | | { 3 GAMMA(---- + 1) GAMMA(---- + 2/3) irem(n1, 3) = 0| | { 3 3 | | { | |n - 1 { 0 irem(n1, 3) = 1| |----- { | | \ { 0 irem(n1, 3) = 2| n! | ) ------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A319365" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" / | |n - 1 |----- | \ {n! | ) | / |----- |n1 = 0 | \ { 0 irem(n1, 4) = 0 { { 0 irem(n1, 4) = 1 { { 0 irem(n1, 4) = 2 { { / n1 \ { |- ---- + 3/2| { \ 2 / n1 n1 2 // n1 \ \3 n1 { 1/2 2 n1 binomial(---- - 3/2, ---- - 3/4) ||---- - 3/4|!| (n1 - 2) (n1 - 1) binomial(n1 - 3, ---- - 3/2) irem(n1, 4) = 3 { 2 4 \\ 4 / / 2 / { 0 irem(n1, 4) = 0\ | { | | { 0 irem(n1, 4) = 1| | { | | { /3 n1 \ | | { |---- - 3| | \ | { \ 2 / n1 n1 n1 | | | { 2 GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 3/4) irem(n1, 4) = 2| | |n - 1 { 4 4 4 | | |----- { | | | \ { 0 irem(n1, 4) = 3| /(n1 + 1)!|, n! | ) ----------------------------------------------------------------------------------------|, | | / (n1 + 1)! | | |----- | | \n1 = 0 / | / / { 0 irem(n1, 4) = 0\ | { | | { // n1 \ \3 3 n1 n1 n1 | | { n1 ||---- - 1/4|!| binomial(---- - 3/4, ---- - 1/4) binomial(n1 - 1, ---- - 1/4) irem(n1, 4) = 1| | { \\ 4 / / 4 4 4 | | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ----------------------------------------------------------------------------------------------------------|, | / (n1 + 1)! | |----- | \n1 = 0 / / { /3 n1\ \ | { |----| | | { \ 2 / n1 n1 n1 | | { 2 GAMMA(---- + 1) GAMMA(---- + 1/2) GAMMA(---- + 3/4) irem(n1, 4) = 0| | { 4 4 4 | | { | | { 0 irem(n1, 4) = 1| | { | |n - 1 { 0 irem(n1, 4) = 2| |----- { | | \ { 0 irem(n1, 4) = 3| n! | ) ------------------------------------------------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A319536" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n n | \ (-1) 2 | {2 n! (2 n + 3), (n + 1) 2 n!, 2 n! (2 n + 3) | ) -------------------------------|} | / (2 n1 + 3) (2 n1 + 5) (n1 + 1)!| |----- | \n1 = 0 / "A319636" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {1, RootOf(_Z - _Z + 1, index = 1) , RootOf(_Z - _Z + 1, index = 2) , RootOf(_Z - _Z + 1, index = 3) } "A319743" LREtools/SearchTable: "SearchTable successful" (n + 1) (3 n + 5) hypergeom([-n - 1, -n - 1, -n - 1, -n - 1], [1, 1, 1], 1) + 2 (4 n + 5) (4 n + 3) hypergeom([-n, -n, -n, -n], [1, 1, 1], 1) {---------------------------------------------------------------------------------------------------------------------------------------------} 3 (n + 1) (n + 2) "A319924" {1, n binomial(2 n, n)} "A319948" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { n { 3 GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) n::even { 2 3 GAMMA(n/2 + 3/2) GAMMA(n/2 + 11/6) { { --------------------------------------- n::even {{ (n + 1) , { 3 n + 5 } { 2 3 GAMMA(n/2 + 3/2) GAMMA(n/2 + 11/6) { { --------------------------------------------- n::odd { (n - 1) { 3 n + 5 { 3 GAMMA(n/2 + 1) GAMMA(n/2 + 4/3) n::odd "A319949" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { n { 4 4 GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/4) irem(n, 3) = 0 { { n 23 { 144 4 GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) GAMMA(n/3 + --) { 12 {{ -------------------------------------------------------- irem(n, 3) = 1, { (n + 2) (4 n + 11) { { (2 n + 2) 19 { 12 2 GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) GAMMA(n/3 + --) { 12 { --------------------------------------------------------------- irem(n, 3) = 2 { 4 n + 7 { n 19 { 12 4 GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) GAMMA(n/3 + --) { 12 { ------------------------------------------------------- irem(n, 3) = 0 { 4 n + 7 { { (2 n - 2) , { 4 2 GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/4) irem(n, 3) = 1 { { (2 n + 2) 23 { 9 2 GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) GAMMA(n/3 + --) { 12 { -------------------------------------------------------------- irem(n, 3) = 2 { (n + 2) (4 n + 11) { n 23 { 9 4 GAMMA(5/3 + n/3) GAMMA(n/3 + 7/6) GAMMA(n/3 + --) { 12 { ------------------------------------------------------ irem(n, 3) = 0 { (n + 2) (4 n + 11) { { (2 n - 2) 19 } { 12 2 GAMMA(n/3 + 4/3) GAMMA(n/3 + 5/6) GAMMA(n/3 + --) { 12 { --------------------------------------------------------------- irem(n, 3) = 1 { 4 n + 7 { { n { 1/4 4 GAMMA(n/3 + 1) GAMMA(n/3 + 1/2) GAMMA(n/3 + 5/4) irem(n, 3) = 2 "A320314" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" 3 2 n 3 2 n 3 2 n {RootOf(_Z - _Z - _Z - 1, index = 1) , RootOf(_Z - _Z - _Z - 1, index = 2) , RootOf(_Z - _Z - _Z - 1, index = 3) } "A320326" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A320329" n {2 n!, n! binomial(2 n, n), n + 1} "A320615" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 2) (n + 1) n!} "A320758" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 3 | | 3 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) |1/2 - ----| n!, (n + 2) (n + 1) |1/2 + ----| n!} \ 2 / \ 2 / "A320759" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 3 | | 3 | {(n + 3) (n + 2) (n + 1) |1/2 - ----| n!, (n + 3) (n + 2) (n + 1) |1/2 + ----| n!, \ 2 / \ 2 / 3 2 3 n (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 1) - 1/6 RootOf(_Z + 3 _Z + 10, index = 1) + 2/3) n!, 3 2 3 n (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 2) - 1/6 RootOf(_Z + 3 _Z + 10, index = 2) + 2/3) n!, 3 2 3 n (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 3) - 1/6 RootOf(_Z + 3 _Z + 10, index = 3) + 2/3) n!} "A320760" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 4 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 3 n {(n + 4) (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 1) - 1/6 RootOf(_Z + 3 _Z + 10, index = 1) + 2/3) n!, 3 2 3 n (n + 4) (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 2) - 1/6 RootOf(_Z + 3 _Z + 10, index = 2) + 2/3) n!, 3 2 3 n (n + 4) (n + 3) (n + 2) (n + 1) (1/6 RootOf(_Z + 3 _Z + 10, index = 3) - 1/6 RootOf(_Z + 3 _Z + 10, index = 3) + 2/3) n!, 3 2 n (n + 4) (n + 3) (n + 2) (n + 1) (1/24 RootOf(%1, index = 1) + 1/24 RootOf(%1, index = 1) + 7/24 RootOf(%1, index = 1) + 5/8) n!, 3 2 n (n + 4) (n + 3) (n + 2) (n + 1) (1/24 RootOf(%1, index = 2) + 1/24 RootOf(%1, index = 2) + 7/24 RootOf(%1, index = 2) + 5/8) n!, 3 2 n (n + 4) (n + 3) (n + 2) (n + 1) (1/24 RootOf(%1, index = 3) + 1/24 RootOf(%1, index = 3) + 7/24 RootOf(%1, index = 3) + 5/8) n!, 3 2 n (n + 4) (n + 3) (n + 2) (n + 1) (1/24 RootOf(%1, index = 4) + 1/24 RootOf(%1, index = 4) + 7/24 RootOf(%1, index = 4) + 5/8) n!} 4 2 %1 := _Z + 6 _Z + 8 _Z - 39 "A320825" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 binomial(2 n1, n1)| {3 (n - 3), 3 (n - 3) | ) -----------------------------|} | / (n1 + 1) (2 n1 - 1) | |----- | \n1 = 0 / "A320826" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 binomial(2 n1, n1)| {3 (n - 6), 3 (n - 6) | ) -----------------------------|} | / (n1 - 6) (n1 - 5) (n1 - 1/2) | |----- | \n1 = 0 / "A320827" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 binomial(2 n1, n1) (n1 - 2)| {3 , 3 | ) --------------------------------------|} | / (n1 + 1) (2 n1 - 1) | |----- | \n1 = 0 / "A321032" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 (3 n1 + 5) (3 n1 + 2) (3 n1 + 4) (3 n1 + 1) binomial(3 n1, n1) (5 n1 + 13)| {1, 8 , 8 | ) ---------------------------------------------------------------------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / "A321033" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n n | \ 2 (3 n1 + 2) (3 n1 + 1) (3 n1 + 5) (3 n1 + 4) (3 n1 + 8) (3 n1 + 7) binomial(3 n1, n1) (5 n1 + 18)| {1, 8 , 27 , 8 | ) -------------------------------------------------------------------------------------------------------------|, | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / /n - 1 \ |----- n1 (-3 n1 - 3) | n | \ 2 3 (3 n1 + 1) (3 n1 + 2) (3 n1 + 4) (3 n1 + 5) (3 n1 + 7) (3 n1 + 8) (3 n1 + 11) binomial(3 n1, n1)| 27 | ) -----------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) (2 n1 + 1) (2 n1 + 3) (2 n1 + 5) (2 n1 + 7) | |----- | \n1 = 0 / "A321197" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A321200" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-3) } "A321204" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A321205" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321398" n n! 3 n! {----, -----} n n "A321425" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321427" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321574" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- (-n1 - 1) | |----- (-n1 - 1) | n n | \ 6 (LegendreP(n1 + 1, 3) - 3 LegendreP(n1, 3))| n | \ 6 (LegendreQ(n1 + 1, 3) - 3 LegendreQ(n1, 3))| {6 , 6 | ) ------------------------------------------------------|, 6 | ) ------------------------------------------------------|} | / n1 | | / n1 | |----- | |----- | \n1 = 0 / \n1 = 0 / "A321633" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A321798" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A321799" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A321837" LREtools/SearchTable: "SearchTable successful" n 3 ((3 n + 3) LaguerreL(n + 1, -1/3) + (-3 n - 4) LaguerreL(n, -1/3)) n! {------------------------------------------------------------------------} n "A321838" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (5 n + 11) { 2 binomial(n, n/2) (5 n + 6) { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { n + 2 {1, { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) (5 n + 11) { 2 2 (5 n + 6) { ------------------------------------- n::odd { ------------------------------------ n::odd { n + 3 { n (n + 2) binomial(n - 1, n/2 - 1/2) "A321839" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 3 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n | \ n1 6 5 4 3 2 {1, (-1) , (2 n + 7) | ) (-1) ((3 n1 - 235 n1 - 4108 n1 - 25469 n1 - 74519 n1 - 103932 n1 - 54972) hypergeom([1/2, -n1 - 1], [1], 4) | / |----- \n1 = 0 6 5 4 3 2 + (9 n1 + 486 n1 + 5994 n1 + 31254 n1 + 77661 n1 + 86232 n1 + 29268) hypergeom([1/2, -n1], [1], 4))/((n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) { n \ { 4 (5 n + 11) | { 1/2 -------------------------------- n::even | { (n + 1) (n + 3) binomial(n, n/2) (n1 + 2) (2 n1 + 9) (2 n1 + 7))|, 2 n + 7, { , | { (2 n - 2) | { 2 (n + 1) (5 n + 16) / { -------------------------------------------- n::odd { n (n + 2) (n + 4) binomial(n - 1, n/2 - 1/2) { 4 binomial(n, n/2) (5 n + 16) (n + 1) { ------------------------------------- n::even { (n + 2) (n + 4) { } { 2 binomial(n + 1, n/2 + 1/2) (5 n + 11) { --------------------------------------- n::odd { n + 3 "A321847" LREtools/SearchTable: "SearchTable successful" n 4 ((4 n + 4) LaguerreL(n + 1, -1/4) + (-4 n - 5) LaguerreL(n, -1/4)) n! {------------------------------------------------------------------------} n "A321848" LREtools/SearchTable: "SearchTable successful" n 5 ((5 n + 5) LaguerreL(n + 1, -1/5) + (-5 n - 6) LaguerreL(n, -1/5)) n! {------------------------------------------------------------------------} n "A321849" LREtools/SearchTable: "SearchTable successful" n 6 ((6 n + 6) LaguerreL(n + 1, -1/6) + (-6 n - 7) LaguerreL(n, -1/6)) n! {------------------------------------------------------------------------} n "A321850" LREtools/SearchTable: "SearchTable successful" n 7 ((7 n + 7) LaguerreL(n + 1, -1/7) + (-7 n - 8) LaguerreL(n, -1/7)) n! {------------------------------------------------------------------------} n "A321853" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | | \ (n1 + 1) n1! | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ------------------|} | / (n1 + 1)! (n1 + 3)| |----- | \n1 = 0 / "A321942" LREtools/SearchTable: "SearchTable successful" {(n + 1) n! (LaguerreL(n + 1, -1) - LaguerreL(n, -1))} "A321957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 binomial(3 n1, n1) (5 n1 + 11 n1 + 4)| {8 , 8 | ) ---------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A321965" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A322126" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A322239" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A322240" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(19*n^2+76*n+75)*E^2-15*(2*n+3)*(19*n^2+76*n+75)*E+3375*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-15*n-15 LREtools/SearchTable: "SearchTable successful" n 1/2 2 n 1/2 2 n 1/2 1/2 {(-15) LegendreP(n, 1/15 I 15 ) , (-15) LegendreQ(n, 1/15 I 15 ) , (-15) LegendreP(n, 1/15 I 15 ) LegendreQ(n, 1/15 I 15 )} "A322242" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 7 ) LegendreP(n, 3/7 I 7 ), (-I 7 ) LegendreQ(n, 3/7 I 7 )} "A322243" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(43*n^2+172*n+163)*E^2-7*(2*n+3)*(43*n^2+172*n+163)*E+343*(2*n+5)*(n+1)^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E-7*n-7 LREtools/SearchTable: "SearchTable successful" n 1/2 2 n 1/2 2 n 1/2 1/2 {(-7) LegendreP(n, 3/7 I 7 ) , (-7) LegendreQ(n, 3/7 I 7 ) , (-7) LegendreP(n, 3/7 I 7 ) LegendreQ(n, 3/7 I 7 )} "A322244" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 55 ) LegendreP(n, 3/55 I 55 ), (-I 55 ) LegendreQ(n, 3/55 I 55 )} "A322245" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(91*n^2+364*n+355)*E^2-55*(2*n+3)*(91*n^2+364*n+355)*E+166375*(2*n+5)*(n+1) ^2 "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E-55*n-55 LREtools/SearchTable: "SearchTable successful" n 1/2 2 n 1/2 2 n 1/2 1/2 {(-55) LegendreP(n, 3/55 I 55 ) , (-55) LegendreQ(n, 3/55 I 55 ) , (-55) LegendreP(n, 3/55 I 55 ) LegendreQ(n, 3/55 I 55 )} "A322246" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 11 ) LegendreP(n, 5/11 I 11 ), (-I 11 ) LegendreQ(n, 5/11 I 11 )} "A322247" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(111*n^2+444*n+419)*E^2-11*(2*n+3)*(111*n^2+444*n+419)*E+1331*(2*n+5)*(n+1) ^2 "to two: Symmetric square" (n+2)*E^2+(-10*n-15)*E-11*n-11 memory used=251323.3MB, alloc=3479.5MB, time=1936.06 LREtools/SearchTable: "SearchTable successful" n 1/2 2 n 1/2 2 n 1/2 1/2 {(-11) LegendreP(n, 5/11 I 11 ) , (-11) LegendreQ(n, 5/11 I 11 ) , (-11) LegendreP(n, 5/11 I 11 ) LegendreQ(n, 5/11 I 11 )} "A322248" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-I 39 ) LegendreP(n, 5/39 I 39 ), (-I 39 ) LegendreQ(n, 5/39 I 39 )} "A322249" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+3)^2*E^3-(2*n+5)*(139*n^2+556*n+531)*E^2-39*(2*n+3)*(139*n^2+556*n+531)*E+59319*(2*n+5)*(n+1 )^2 "to two: Symmetric square" (n+2)*E^2+(-10*n-15)*E-39*n-39 LREtools/SearchTable: "SearchTable successful" n 1/2 2 n 1/2 2 n 1/2 1/2 {(-39) LegendreP(n, 5/39 I 39 ) , (-39) LegendreQ(n, 5/39 I 39 ) , (-39) LegendreP(n, 5/39 I 39 ) LegendreQ(n, 5/39 I 39 )} "A322262" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 4 3 11 2 31 19\n {|1/120 RootOf(%1, index = 1) + 1/120 RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 2) + 1/120 RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 3) + 1/120 RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 4) + 1/120 RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + --- RootOf(%1, index = 4) + --| n!, \ 120 120 30/ / 4 3 11 2 31 19\n |1/120 RootOf(%1, index = 5) + 1/120 RootOf(%1, index = 5) + --- RootOf(%1, index = 5) + --- RootOf(%1, index = 5) + --| n!} \ 120 120 30/ 5 3 2 %1 := _Z + 10 _Z + 20 _Z + 45 _Z + 44 "A322282" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 6 5 11 4 23 3 {|1/5040 RootOf(%1, index = 1) + 1/5040 RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) \ 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) + ---| n!, |1/5040 RootOf(%1, index = 2) + 1/5040 RootOf(%1, index = 2) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 3) + 1/5040 RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) + ---| n!, |1/5040 RootOf(%1, index = 4) + 1/5040 RootOf(%1, index = 4) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---- RootOf(%1, index = 4) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 5) + 1/5040 RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) 2520 1260 407 2 1331 177\n / 6 5 + ---- RootOf(%1, index = 5) + ---- RootOf(%1, index = 5) + ---| n!, |1/5040 RootOf(%1, index = 6) + 1/5040 RootOf(%1, index = 6) 5040 5040 280/ \ 11 4 23 3 407 2 1331 177\n / + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---- RootOf(%1, index = 6) + ---| n!, | 2520 1260 5040 5040 280/ \ 6 5 11 4 23 3 1/5040 RootOf(%1, index = 7) + 1/5040 RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) 2520 1260 407 2 1331 177\n + ---- RootOf(%1, index = 7) + ---- RootOf(%1, index = 7) + ---| n!} 5040 5040 280/ 7 5 4 3 2 %1 := _Z + 21 _Z + 70 _Z + 315 _Z + 924 _Z + 1855 _Z + 1854 "A322295" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A322296" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A322543" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A322544" n (-1) n! (n + 2) n! (2 n + 5) (n + 2) {----------------, --------------------} n + 4 n + 4 "A322914" n n {12 , 3 binomial(2 n, n)} "A322938" (2 n + 1) binomial(2 n, n) {1, --------------------------} n + 2 "A323223" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)} / ----- n1 = 0 "A323229" binomial(2 n, n) n {1, ------------------} n + 1 "A323230" n binomial(2 n, n) {1, ------------------} 2 n - 1 "A323277" n n {12 , n 3 binomial(2 n, n)} "A323416" n {10 n!, n!} "A323499" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) 2 { 2 (n/2)! (n + 2) n::even { 2 (n + 1) (n + 2 n + 2) binomial(n, n/2) (n/2)! n::even {{ , { } { (n/2 + 1/2) 2 { (- n/2 + 1/2) { 2 (n/2 + 1/2)! (1/2 n + n + 1) n::odd { 2 2 n (n + 2) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A323618" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n | \ / (n1 + 1) n1! \| (-1) n! | ) |- ------------------|| | / \ (n1 - 1) (n1 + 1)!/| n |----- | (-1) n! \n1 = 0 / {---------, ----------------------------------------} (n - 1) n (n - 1) n "A323620" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 n 5 n 5 {(-1) GAMMA(n - 1/2 - ----), (-1) GAMMA(n - 1/2 + ----)} 2 2 "A323770" LREtools/SearchTable: "SearchTable successful" 2 ((n - 1) (n + 1) LaguerreL(n + 1, -1) + (-n + 2) LaguerreL(n, -1)) n! {----------------------------------------------------------------------} n "A323988" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (2 n + 2) n::even { binomial(n, n/2) {2 , { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A324167" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A324168" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 2 {3 n - 13 n - 2} "A324169" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n + 2 n - 2), /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n 2 | \ -I (-I 7 ) 7 ((119 n1 + 63) LegendreP(n1, 5/7 I 7 ) + 11 I 7 (n1 + 1) LegendreP(n1 + 1, 5/7 I 7 ))| (-1) (n + 2 n - 2) | ) ----------------------------------------------------------------------------------------------------------------|, | / (n1 + 1) 2 2 | |----- (-1) ((n1 + 1) + 2 n1) (n1 + 2 n1 - 2) | \n1 = 0 / /n - 1 \ |----- 1/2 n1 1/2 1/2 1/2 1/2 | n 2 | \ -I (-I 7 ) 7 ((119 n1 + 63) LegendreQ(n1, 5/7 I 7 ) + 11 I 7 (n1 + 1) LegendreQ(n1 + 1, 5/7 I 7 ))| (-1) (n + 2 n - 2) | ) ----------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (-1) ((n1 + 1) + 2 n1) (n1 + 2 n1 - 2) | \n1 = 0 / "A324352" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {n! (n + 5 n + 5), n! (n + 1) (n + 2), n! (n + 5 n + 5) | ) --------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 5 n1 + 10) (n1 + 5 n1 + 5)| \n1 = 0 / "A324353" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 3 2 | \ (-1) | {n! (n + 9 n + 23 n + 16), n! (n + 9 n + 23 n + 16) | ) ---------------------------------------------------------------------------|, | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 9 (n1 + 1) + 23 n1 + 39) (n1 + 9 n1 + 23 n1 + 16)| \n1 = 0 / (n + 3) (n + 2) (n + 1) n!} "A324354" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {n! (n + 14 n + 65 n + 116 n + 65), n! (n + 14 n + 65 n + 116 n + 65) /n - 1 \ |----- n1 | | \ (-1) | | ) --------------------------------------------------------------------------------------------------------|, | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 14 (n1 + 1) + 65 (n1 + 1) + 116 n1 + 181) (n1 + 14 n1 + 65 n1 + 116 n1 + 65)| \n1 = 0 / (n + 4) (n + 3) (n + 2) (n + 1) n!} "A324355" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 5 4 3 2 5 4 3 2 {n! (n + 20 n + 145 n + 470 n + 669 n + 326), n! (n + 20 n + 145 n + 470 n + 669 n + 326) /n - 1 \ |----- n1 | | \ (-1) | | ) -------------------------------------------------------------------------------------------------------------------------------------|, | / 5 4 3 2 5 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 20 (n1 + 1) + 145 (n1 + 1) + 470 (n1 + 1) + 669 n1 + 995) (n1 + 20 n1 + 145 n1 + 470 n1 + 669 n1 + 326)| \n1 = 0 / (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A324356" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 6 5 4 3 2 6 5 4 3 2 | \ n1 / {n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957), n! (n + 27 n + 280 n + 1415 n + 3634 n + 4429 n + 1957) | ) (-1) / ( | / / |----- \n1 = 0 6 5 4 3 2 (n1 + 1)! ((n1 + 1) + 27 (n1 + 1) + 280 (n1 + 1) + 1415 (n1 + 1) + 3634 (n1 + 1) + 4429 n1 + 6386) \ | 6 5 4 3 2 | (n1 + 27 n1 + 280 n1 + 1415 n1 + 3634 n1 + 4429 n1 + 1957))|, (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} | | / "A324357" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 7 6 5 4 3 2 {n! (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700), n! /n - 1 |----- 7 6 5 4 3 2 | \ n1 / (n + 35 n + 490 n + 3535 n + 14084 n + 30681 n + 33375 n + 13700) | ) (-1) / ((n1 + 1)! | / / |----- \n1 = 0 7 6 5 4 3 2 ((n1 + 1) + 35 (n1 + 1) + 490 (n1 + 1) + 3535 (n1 + 1) + 14084 (n1 + 1) + 30681 (n1 + 1) + 33375 n1 + 47075) \ | 7 6 5 4 3 2 | (n1 + 35 n1 + 490 n1 + 3535 n1 + 14084 n1 + 30681 n1 + 33375 n1 + 13700))|, (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} | | / "A324358" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 8 7 6 5 4 3 2 {n! (n + 44 n + 798 n + 7756 n + 43939 n + 147532 n + 284066 n + 283072 n + 109601), n! /n - 1 |----- 8 7 6 5 4 3 2 | \ n1 / (n + 44 n + 798 n + 7756 n + 43939 n + 147532 n + 284066 n + 283072 n + 109601) | ) (-1) / ((n1 + 1)! | / / |----- \n1 = 0 8 7 6 5 4 3 2 ((n1 + 1) + 44 (n1 + 1) + 798 (n1 + 1) + 7756 (n1 + 1) + 43939 (n1 + 1) + 147532 (n1 + 1) + 284066 (n1 + 1) + 283072 n1 + 392673) \ | 8 7 6 5 4 3 2 | (n1 + 44 n1 + 798 n1 + 7756 n1 + 43939 n1 + 147532 n1 + 284066 n1 + 283072 n1 + 109601))|, | | / (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A324359" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 9 8 7 6 5 4 3 2 {n! (n + 54 n + 1230 n + 15456 n + 117579 n + 558642 n + 1646714 n + 2878284 n + 2673321 n + 986410), n! /n - 1 |----- 9 8 7 6 5 4 3 2 | \ n1 / 9 (n + 54 n + 1230 n + 15456 n + 117579 n + 558642 n + 1646714 n + 2878284 n + 2673321 n + 986410) | ) (-1) / ((n1 + 1)! ((n1 + 1) | / / |----- \n1 = 0 8 7 6 5 4 3 2 + 54 (n1 + 1) + 1230 (n1 + 1) + 15456 (n1 + 1) + 117579 (n1 + 1) + 558642 (n1 + 1) + 1646714 (n1 + 1) + 2878284 (n1 + 1) + 2673321 n1 \ | 9 8 7 6 5 4 3 2 | + 3659731) (n1 + 54 n1 + 1230 n1 + 15456 n1 + 117579 n1 + 558642 n1 + 1646714 n1 + 2878284 n1 + 2673321 n1 + 986410))|, | | / (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A324360" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 10 9 8 7 6 5 4 3 2 {n! (n + 65 n + 1815 n + 28590 n + 280413 n + 1782207 n + 7396325 n + 19664350 n + 31777851 n + 27845293 n + 9864101), n! /n - 1 |----- 10 9 8 7 6 5 4 3 2 | \ n1 / (n + 65 n + 1815 n + 28590 n + 280413 n + 1782207 n + 7396325 n + 19664350 n + 31777851 n + 27845293 n + 9864101) | ) (-1) / ( | / / |----- \n1 = 0 10 9 8 7 6 5 4 (n1 + 1)! ((n1 + 1) + 65 (n1 + 1) + 1815 (n1 + 1) + 28590 (n1 + 1) + 280413 (n1 + 1) + 1782207 (n1 + 1) + 7396325 (n1 + 1) 3 2 + 19664350 (n1 + 1) + 31777851 (n1 + 1) + 27845293 n1 + 37709394) \ | 10 9 8 7 6 5 4 3 2 | (n1 + 65 n1 + 1815 n1 + 28590 n1 + 280413 n1 + 1782207 n1 + 7396325 n1 + 19664350 n1 + 31777851 n1 + 27845293 n1 + 9864101))|, | | / (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!} "A324361" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n + 1/2, 1/2), (-1) BesselK(n + 1/2, -1/2), n! binomial(2 n, n)} "A324445" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {(-1) (2 n BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) (2 n BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1)), (2 n + 1) (1/2) n! binomial(2 n, n) } "A324446" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n 2 {(-1) ((4 n + 6 n + 3) BesselI(n + 1/2, 1) + (-2 n - 2) BesselI(n - 1/2, 1)), n 2 (-1) ((4 n + 6 n + 3) BesselK(n + 1/2, -1) + (-2 n - 2) BesselK(n - 1/2, -1))} "A324568" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n n | \ (2 n1 + 1) binomial(2 n1, n1) (5 n1 + 8)| {1, 4 , (-1/2) , (-1/2) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-1/2) | \n1 = 0 / "A324591" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A324621" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(n + 1) n!, (-1) (n + 1) (n BesselI(n, 2) - BesselI(n - 1, 2)), (-1) (n + 1) (n BesselK(n, -2) - BesselK(n - 1, -2))} "A324622" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 7 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 n 2 {(-1) (n + 2) ((n + n + 1) BesselI(n, 2) + (-n - 1) BesselI(n - 1, 2)), (-1) (n + 2) ((n + n + 1) BesselK(n, -2) + (-n - 1) BesselK(n - 1, -2))} "A324804" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A325156" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A325157" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A325482" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {1, (-1/2 I 2 ) (2 HermiteH(n + 1, 2 I) + 2 (n + 1) HermiteH(n, 2 I) I) 2 I} "A325983" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 / 1/2\n / 1/2 \n { ------------------------ n::even n n | 5 | |5 | { (n + 1) binomial(n, n/2) { binomial(n, n/2) n::even {(-1) , 2 , |1/2 - ----| , |---- + 1/2| , 2 n + 5, { , { } \ 2 / \ 2 / { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A326290" LREtools/SearchTable: "SearchTable successful" n 2 4 ((n + 1) (17 n - 8) LegendreP(n + 1, 3) + (-99 n - 3 n + 24) LegendreP(n, 3)) {---------------------------------------------------------------------------------, n (n - 1) (n - 2) n 2 4 ((n + 1) (17 n - 8) LegendreQ(n + 1, 3) + (-99 n - 3 n + 24) LegendreQ(n, 3)) ---------------------------------------------------------------------------------} n (n - 1) (n - 2) "A326564" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A327370" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 { 0 n::even {(-1/2 I 2 ) (HermiteH(n + 1, 1/2 I 2 ) - 2 HermiteH(n, 1/2 I 2 ) I), { , { (n/2 - 1/2) { 2 (n/2 - 1/2)! n n::odd { (- n/2) { 1/2 2 n binomial(n, n/2) (n/2)! n::even} { { 0 n::odd "A327606" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) 2 | n n | \ 2 (n1 + 1) | {2 n! (2 n + 1), 2 n! (2 n + 1) | ) -------------------------------|} | / (2 n1 + 1) (2 n1 + 3) (n1 + 1)!| |----- | \n1 = 0 / "A327871" LREtools/SearchTable: "SearchTable successful" ((9 n + 6) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + (-8 n - 6) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) (n + 1) {-----------------------------------------------------------------------------------------------------------------------------------------} n (13 n + 9) "A327872" LREtools/SearchTable: "SearchTable successful" 2 {(3 (3 n + 2) (15 n + 1) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) + (-146 n - 116 n - 6) hypergeom([- n/2, - n/2 + 1/2], [n + 1], 4)) binomial(2 n, n) (n + 1)/(n (13 n + 9))} "A327904" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 4 { (n/2 + 1) ((n/2)!) n::even { (-n) 3 4 4 { { 4 16 (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ 4 , { } { 2 ((n/2 + 1/2)!) { (-4 n + 4) 4 4 4 { ----------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) (n + 2) n::odd { n + 1 "A327998" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { ----------------- n::even { 2 { 2 { binomial(n, n/2) { 16 binomial(n, n/2) n::even {{ , { } { (4 n - 4) { 2 2 { 16 2 { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------ n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) "A328002" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n/2 + 4) { ---------------- n::even { binomial(n, n/2) { binomial(n, n/2) (n + 6) (n + 1) n::even {{ , { } { (2 n - 2) { 1/2 (n + 8) (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { 2 (n + 1) (n + 6) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A328054" memory used=252752.0MB, alloc=3479.5MB, time=1946.05 LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n n! (1/2 - 1/2 I 3 ) n! (1/2 + 1/2 I 3 ) n! {----, ----------------------, ----------------------} n n n "A328141" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (- n/2) | 2 1/2 2 | 2 |HermiteH(n + 1, ----) - 2 HermiteH(n, ----)| \ 2 2 / {---------------------------------------------------------} n "A328286" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2 \n / 1/2 \n | 3 | |3 | {|- ---- + 1/2| n!, |---- + 1/2| n!} \ 2 / \ 2 / "A328345" memory used=253791.5MB, alloc=3479.5MB, time=1953.06 memory used=254689.5MB, alloc=3479.5MB, time=1958.80 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 2" LREtools/SearchTable: "SearchTable not successful" {} "A328378" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { 4 ((n/2)!) { ----------- n::even { (-n) 2 2 { 2 { 4 4 binomial(n, n/2) ((n/2)!) { n { ----------------------------------- n::even {{ , { n - 1 } { 2 { { 16 ((n/2 + 1/2)!) { (-2 n + 2) 2 2 { ------------------ n::odd { 2 binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { 2 { (n + 1) (n - 1) "A328426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A328494" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / /n1 - 1 \\ | |----- 1/2 n2 1/2 1/2 1/2 || | n1 | \ (-I 11 ) (11 LegendreP(n2 + 1, 1/11 I 11 ) I - 11 LegendreP(n2, 1/11 I 11 ))|| | (-3) | ) ---------------------------------------------------------------------------------------|| |n - 1 | / (n2 + 1) || |----- |----- (n2 + 2) (-3) || n n n | \ \n2 = 0 /| {(-3) , 5 , 5 | ) -------------------------------------------------------------------------------------------------------|, | / (n1 + 1) | |----- 5 | \n1 = 0 / / /n1 - 1 \\ | |----- / 1/2 n2 1/2 1/2 1/2 \|| | n1 | \ | (-I 11 ) (-11 LegendreQ(n2 + 1, 1/11 I 11 ) I + 11 LegendreQ(n2, 1/11 I 11 ))||| | (-3) | ) |- ----------------------------------------------------------------------------------------||| |n - 1 | / | (n2 + 1) ||| |----- |----- \ (n2 + 2) (-3) /|| n | \ \n2 = 0 /| 5 | ) ------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- 5 | \n1 = 0 / "A328713" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A328714" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A328715" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A328725" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A328874" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329024" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329475" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329478" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=255865.2MB, alloc=3481.8MB, time=1967.62 LREtools/SolveLRE: "Reduced the order of" (15*n+68)*(15*n+53)*(15*n+38)*(15*n+23)*(n+4)*(69*n^4+828*n^3+3695*n^2+7266*n+5315)*(n+5)^3*E^4+( 307395000*n^12+13105273500*n^11+253927774800*n^10+2956048994340*n^9+23020222689852*n^8+126296299200324*n^7+500352517423940*n^6+1441624296906228*n^5+ 2996502895184188*n^4+4379366545287776*n^3+4268686427741660*n^2+2489396043135272*n+656145552831200)*E^3+(8369527500*n^12+352636092000*n^11+ 6752018420100*n^10+77672735086680*n^9+597745474080384*n^8+3241078118818152*n^7+12692167519078876*n^6+36154969720761168*n^5+74320543613280436*n^4+ 107456465763946128*n^3+103661725835088064*n^2+59858370167405472*n+15630457858579840)*E^2+(-6147900000*n^12-255957570000*n^11-4840631766000*n^10-\ 54978274414800*n^9-417578394527040*n^8-2233970184582960*n^7-8629434084654160*n^6-24243495588077040*n^5-49144256606826560*n^4-70069273593815040*n^3-\ 66662744135254240*n^2-37970698893687360*n-9783708409555200)*E+1397250000*n^12+57473550000*n^11+1073075040000*n^10+12023430552000*n^9+90028746081600*n ^8+474516613272000*n^7+1804920660695600*n^6+4991294181979600*n^5+9958047823969600*n^4+13976450031568000*n^3+13097946492934400*n^2+7358258458630400*n+ 1873975351372800 "to two: Symmetric product" (n+2)^2*E^2+(-11*n^2-33*n-25)*E-(n+1)^2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 1 {-----, n + 1 1/2 n - 1 1/2 n1 2 5 ----- (-2 5 ) binomial(2 n1, n1) hypergeom([-n1 - 1, -n1 - 1, -n1 - 1], [1, -2 n1 - 2], 1) (2 n1 + 1) (15 n1 + 23) LegendreP(n1 + 1, ------) \ 5 ) ------------------------------------------------------------------------------------------------------------------------------------------ / n1 + 1 ----- n1 = 0 /(n + 1), 1/2 n - 1 1/2 n1 2 5 ----- (-2 5 ) binomial(2 n1, n1) hypergeom([-n1 - 1, -n1 - 1, -n1 - 1], [1, -2 n1 - 2], 1) (2 n1 + 1) (15 n1 + 23) LegendreQ(n1 + 1, ------) \ 5 ) ------------------------------------------------------------------------------------------------------------------------------------------ / n1 + 1 ----- n1 = 0 /(n + 1)} "A329521" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 2 3 2 { (-16) n (n + n - 9 n + 3) { 1/4 --------------------------------- n::even { binomial(n, n/2) {{ , { (n/2 - 1/2) 2 { (-16) (n + 1) (3 n - 1) n { ------------------------------------- n::odd { binomial(n - 1, n/2 - 1/2) { 3 (n/2) { 1/2 (n + 1) n (-1) binomial(n, n/2) (3 n - 1) n::even { } { (n/2 + 1/2) 2 3 2 { 1/8 (-1) binomial(n + 1, n/2 + 1/2) n (n + 1) (n + n - 9 n + 3) n::odd "A329550" n 2 (-1) n! (n - 2) n! (n - 2) (2 n - 1) {----------------, ---------------------} (n - 1) n (n - 1) n "A329664" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329665" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329666" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329667" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" 1/2 n 1/2 n {(1 - 2 ) , (1 + 2 ) } "A329668" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329669" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n 1/2 n 1/2 n {(-1) , (1 - 2 ) , (1 + 2 ) } "A329671" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329672" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1 - 3 ) , (1 + 3 ) } "A329673" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329674" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1 - 3 ) , (1 + 3 ) } "A329675" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329676" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(-1) } "A329688" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { /2 n\ { |---| { 2 n { \ 3 / { 12 binomial(---, n/3) { 2 (n + 5) GAMMA(n/3 + 7/6) { 3 { ------------------------------- irem(n, 3) = 0 { --------------------- irem(n, 3) = 0 { (2 n + 1) GAMMA(n/3 + 8/3) { n + 3 { { { /2 n \ { 2 n { |--- - 2/3| { 3 binomial(--- + 4/3, n/3 + 2/3) {{ \ 3 / , { 3 , { 2 2 GAMMA(n/3 + 5/6) { -------------------------------- irem(n, 3) = 1 { ------------------------------- irem(n, 3) = 1 { 2 n + 1 { GAMMA(n/3 + 7/3) { { { 2 n { /2 n \ { 6 binomial(--- + 2/3, n/3 + 1/3) { |--- - 4/3| { 3 { \ 3 / { -------------------------------- irem(n, 3) = 2 { 4 2 GAMMA(n/3 + 1/2) { n + 4 { ------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 2) { /2 n\ { |---| { \ 3 / { 2 2 GAMMA(n/3 + 5/6) { ------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 7/3) { { /2 n \ { |--- - 2/3| { \ 3 / } { 4 2 GAMMA(n/3 + 1/2) { ------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 2) { { /2 n \ { |--- + 2/3| { \ 3 / { 2 (n + 5) GAMMA(n/3 + 7/6) { ------------------------------------- irem(n, 3) = 2 { (2 n + 1) GAMMA(n/3 + 8/3) "A329689" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329690" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329691" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329692" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329693" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329694" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A329695" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A329696" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 2 binomial(n, n/2) { { ------------------ n::even {1, { (2 n - 2) , { n + 2 } { 2 { { ------------------------------------ n::odd { 0 n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A329698" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {n - 1} "A329699" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329700" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A329701" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A329702" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {n - 1} "A329703" memory used=257255.4MB, alloc=3479.5MB, time=1977.30 LREtools/SolveLRE: "Absolute Factorization reduced the order from 8 to 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A330016" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 / /n1 - 1 \\\ |----- | |----- | |----- ||| n n | \ n1 | n | \ | n1 | \ (n2 + 1) n2! ||| {(-1) , (-1) | ) (-(-1) (n1 + 1) n1!)|, (-1) | ) |-(-1) (n1 + 1) n1! | ) ------------------|||} | / | | / | | / (n2 + 2) (n2 + 1)!||| |----- | |----- | |----- ||| \n1 = 0 / \n1 = 0 \ \n2 = 0 /// "A330044" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) n!, (- 1/2 + 1/2 I 3 ) n!, n!} "A330045" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n {(-1) n!, (-I) n!, I n!, n!} "A330169" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 2 n n (-1) ((n + 16 n + 27) hypergeom([1/2, -n - 1], [1], 4) + 3 (n + 1) hypergeom([1/2, -n], [1], 4)) {(-1) , 3 , ---------------------------------------------------------------------------------------------------} n + 3 "A330793" LREtools/SearchTable: "SearchTable successful" ((20 n + 10) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (5 n - 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------} 3 n + 1 "A330796" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) (hypergeom([1/2, -n - 1], [1], 4) - hypergeom([1/2, -n], [1], 4))} "A330799" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-2) (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4))| {7 , 7 | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) 7 | \n1 = 0 / "A330800" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ 2 3 (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4))| {3 , 3 | ) ------------------------------------------------------------------------------------|} | / n1 + 2 | |----- | \n1 = 0 / "A330801" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-8 n - 5) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) (n + 1) {---------------------------------------------------------------------------------------------------------------------------------} n (10 n + 7) "A330802" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ 2 (3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3))| n | \ 2 (3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3))| {(-3) , (-3) | ) -----------------------------------------------|, (-3) | ) -----------------------------------------------|} | / (n1 + 1) | | / (n1 + 1) | |----- (n1 + 2) (-3) | |----- (n1 + 2) (-3) | \n1 = 0 / \n1 = 0 / "A330803" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | 3 6 (3 LegendreP(n1 + 1, 3) - LegendreP(n1, 3))|| {(-1/3) , (-1/3) | ) |- -------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / /n - 1 \ |----- / n1 \| n | \ | 3 6 (3 LegendreQ(n1 + 1, 3) - LegendreQ(n1, 3))|| (-1/3) | ) |- -------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A330966" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n 10 9 8 7 6 5 {(-1) (2 n + 25) (n + 10), 3 (2 n + 39) (n + 10), (-1) ((1093 n + 122707 n + 4729740 n + 93820110 n + 1108284969 n + 8306405571 n 4 3 2 + 40376458670 n + 126211592060 n + 243030970008 n + 259967364192 n + 116939064960) hypergeom([1/2, -n - 1], [1], 4) - 3 9 8 7 6 5 4 3 2 (365 n + 40406 n + 1517574 n + 29023176 n + 326116329 n + 2280212394 n + 10031801476 n + 26946731704 n + 40244569056 n + 25511345280) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A331007" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 | \ (n1 - 1) (n1 - 2) (-1) n1 (n1 + 1) | n! (n - 3 n + 1) | ) -------------------------------------------------| | / 2 2 | 2 |----- (n1 + 1)! ((n1 + 1) - 3 n1 - 2) (n1 - 3 n1 + 1)| n! (n - 3 n + 1) \n1 = 0 / {-----------------, ----------------------------------------------------------------------------} (n - 1) n (n - 1) n "A331323" LREtools/SearchTable: "SearchTable successful" n n {2 (n + 1) (LegendreP(n + 1, 2) + LegendreP(n, 2)), 2 (n + 1) (LegendreQ(n + 1, 2) + LegendreQ(n, 2))} "A331325" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -1), n! LaguerreL(n, 1)} "A331326" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" {n! LaguerreL(n, -1), n! LaguerreL(n, 1)} "A331328" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ 2 15 (LegendreP(n1 + 1, 3) - 3 LegendreP(n1, 3))| {15 , 15 | ) -----------------------------------------------------------|, | / n1 | |----- | \n1 = 0 / /n - 1 \ |----- n1 (-n1 - 1) | n | \ 2 15 (LegendreQ(n1 + 1, 3) - 3 LegendreQ(n1, 3))| 15 | ) -----------------------------------------------------------|} | / n1 | |----- | \n1 = 0 / "A331329" LREtools/SearchTable: "SearchTable successful" 2 2 {(16 (4 n + 7) (4 n + 3) (7 n + 4) (2 n + 3) (4 n + 5) hypergeom([-n - 1, 4 n + 8], [3 n + 6], -1) 5 4 3 2 - 3 (10156 n + 56652 n + 122500 n + 127770 n + 63949 n + 12228) (3 n + 5) (3 n + 4) hypergeom([-n, 4 n + 4], [3 n + 3], -1)) / 3 2 binomial(4 n, n) (2 n + 1) / ((n + 1) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (170 n + 720 n + 1009 n + 468))} / "A331334" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, 1)} "A331396" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {2 , 2 LegendreP(n + 1, 3), 2 LegendreQ(n + 1, 3)} "A331397" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n n {2 , 2 LegendreP(n, 3), 2 LegendreQ(n, 3)} "A331403" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 (-n1) \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1) n1!|| {(-1) n!, (-1) n! | ) |- -----------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A331473" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- -----------------------------------------------||} | / \ (n1 + 3) (n1 + 1) /| |----- | \n1 = 0 / "A331515" LREtools/SearchTable: "SearchTable successful" n n {2 (n + 1) (2 LegendreP(n + 1, 2) - LegendreP(n, 2)), 2 (n + 1) (2 LegendreQ(n + 1, 2) - LegendreQ(n, 2))} "A331516" LREtools/SearchTable: "SearchTable successful" {((8 n + 3) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 7) hypergeom([-1/2, -n], [1], -8)) (n + 1)} "A331551" LREtools/SearchTable: "SearchTable successful" n n {(-3) (n + 1) (LegendreP(n + 1, 1/3) + LegendreP(n, 1/3)), (-3) (n + 1) (LegendreQ(n + 1, 1/3) + LegendreQ(n, 1/3))} "A331552" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (-16) { (n/2) { 1/2 ---------------- n::even { (-1) binomial(n, n/2) (-2 n - 2) n::even { binomial(n, n/2) {{ , { } { (n/2 + 1/2) { (n/2 - 1/2) { (n + 1) (-1) binomial(n + 1, n/2 + 1/2) n::odd { (-16) (n + 1) { - ---------------------------- n::odd { binomial(n - 1, n/2 - 1/2) n "A331688" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1) (LaguerreL(n1 + 1, 1) - LaguerreL(n1, 1)) n1!| {2 n!, 2 n! | ) -----------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A331689" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 (n1 + 1) (LaguerreL(n1 + 1, -1) - LaguerreL(n1, -1)) n1!| {2 n!, 2 n! | ) -------------------------------------------------------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A331725" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 1) (LaguerreL(n1 + 1, -1) - LaguerreL(n1, -1)) n1!|| {(-1) n!, (-1) n! | ) |- ---------------------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A331792" LREtools/SearchTable: "SearchTable successful" n n 2 (n + 1) (2 LegendreP(n + 1, 2) - LegendreP(n, 2)) 2 (n + 1) (2 LegendreQ(n + 1, 2) - LegendreQ(n, 2)) {----------------------------------------------------, ----------------------------------------------------} n + 2 n + 2 "A331793" LREtools/SearchTable: "SearchTable successful" ((8 n + 3) hypergeom([-1/2, -n - 1], [1], -8) + (-8 n - 7) hypergeom([-1/2, -n], [1], -8)) (n + 1) {--------------------------------------------------------------------------------------------------} n + 2 "A331817" LREtools/SearchTable: "SearchTable successful" {((2 n + 2) hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 3) hypergeom([-1/2, -n], [1], -2)) n!} "A331836" LREtools/SearchTable: "SearchTable successful" 2 (2 n + 5) (n + 2) hypergeom([-n - 1, -n - 2, -n - 3/2], [1, 3/2], -1) + (-10 n - 32 n - 26) hypergeom([-n, -n - 1, -n - 1/2], [1, 3/2], -1) {--------------------------------------------------------------------------------------------------------------------------------------------} (n + 2) (2 n + 1) "A331951" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) {----------------} n + 1 "A332051" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A332710" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 3 2 4 3 2 {1, ) (-1) ((-n1 - 7 n1 - 16 n1 - 13) BesselK(n1 - 1, -2) + (n1 + 7 n1 + 17 n1 + 19 n1 + 9) BesselK(n1, -2)), / ----- n1 = 0 n - 1 ----- \ n1 4 3 2 3 2 ) (-1) ((n1 + 7 n1 + 17 n1 + 19 n1 + 9) BesselI(n1, 2) + (-n1 - 7 n1 - 16 n1 - 13) BesselI(n1 - 1, 2))} / ----- n1 = 0 "A332754" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / /{ n1 \\\ | | |{ 4 ||| | | |{ 1/2 --------------------------- n1::even||| |n - 1 | |{ n1 ||| |----- | |{ (n1 + 1) binomial(n1, ----) ||| n n | \ | n1 |{ 2 ||| {(-1) , (-1) | ) |-(-1) |{ |||, | / | |{ (2 n1 - 2) ||| |----- | |{ 2 (n1 + 1) ||| |n1 = 0 | |{ ---------------------------------------- n1::odd ||| | | |{ n1 ||| | | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) ||| \ \ \{ 2 /// / / /{ n1 \\\ |n - 1 | |{ 4 binomial(n1, ----) (n1 + 1) ||| |----- | |{ 2 ||| n | \ | n1 |{ ----------------------------- n1::even||| (-1) | ) |-(-1) |{ n1 + 2 |||} | / | |{ ||| |----- | |{ n1 ||| |n1 = 0 | |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd ||| \ \ \{ 2 /// "A333017" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 n (-1) ((n + 4) (2 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-3 n - 12 n - 6) hypergeom([1/2, -n], [1], 4)) {3 , ------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) "A333156" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A333472" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A333473" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A333564" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (3 n1 + 5) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A333565" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (3 n1 + 2) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A333592" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- 2 3 2 \ binomial(2 n1, n1) (21 n1 + 36 n1 + 21 n1 + 4) binomial(2 n1 + 2, n1 + 1) ) ---------------------------------------------------------------------------- / (n1 + 1) (4 n1 + 2) ----- 1 n1 = 0 {------------------, -----------------------------------------------------------------------------------} n binomial(2 n, n) n binomial(2 n, n) "A333593" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { 0 n::even { (n/2) { { 2 (-1) {{ /-1\(n/2 - 1/2) , { ------------------ n::even} { |--| binomial(n - 1, n/2 - 1/2) n::odd { n binomial(n, n/2) { \16/ { { 0 n::odd "A333715" LREtools/SearchTable: "SearchTable successful" 3 2 3 (n + 1) (3 n + 4) (3 n + 2) hypergeom([-n, -3 n - 3], [2], 2) + (-293 n - 429 n - 190 n - 24) hypergeom([-3 n, -n + 1], [2], 2) {-----------------------------------------------------------------------------------------------------------------------------------} 2 35 n + 42 n + 12 "A333902" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A334000" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 (n1 + 1) (2 n1 + 1) binomial(2 n1, n1) n1! \| {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A334066" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 (n1 + 2) (2 n1 + 1) binomial(2 n1, n1) n1! \| {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A334155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) n1! (2 n1 + 1)} / ----- n1 = 0 "A334551" 2 {3 n + 2 n + 7, binomial(2 n, n)} "A334562" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A334564" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A334565" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A334569" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A334570" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A334571" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A334578" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { 0 n::even { (n/2) {{ , { 2 (n/2)! n::even} { (- n/2 + 1/2) { { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { 0 n::odd "A334670" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / 2 (2 n1 + 1) binomial(2 n1, n1) n1! \| {(2 n + 1) (1/2) n! binomial(2 n, n), (2 n + 1) (1/2) n! binomial(2 n, n) | ) |-----------------------------------------------||} | / \(2 n1 + 3) binomial(2 n1 + 2, n1 + 1) (n1 + 1)!/| |----- | \n1 = 0 / "A334716" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) | {n! n, n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A335026" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1!| {n! (n + 3) (n + 2) (n + 1), n! (n + 3) (n + 2) (n + 1) | ) -------------------|} | / (n1 + 1)! (n1 + 4) | |----- | \n1 = 0 / "A335111" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-2) (n1 + 1)| {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A335344" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A335349" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 memory used=258633.5MB, alloc=3479.5MB, time=1986.97 LREtools/SearchTable: "SearchTable successful" n n n 6 5 4 3 2 {(-1) (6 n + 47), 3 (2 n + 19), (-1) ((121 n + 4997 n + 64631 n + 384895 n + 1161576 n + 1711908 n + 968112) hypergeom([1/2, -n - 1], [1], 4) 5 4 3 2 - 3 (41 n + 1624 n + 19327 n + 100436 n + 239076 n + 212976) (n + 1) hypergeom([1/2, -n], [1], 4)) (n + 7)/((n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A335355" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SearchTable: "SearchTable successful" n n n 10 9 8 7 6 5 {(-1) (2 n + 25) (n + 10), 3 (2 n + 39) (n + 10), (-1) ((1093 n + 122707 n + 4729740 n + 93820110 n + 1108284969 n + 8306405571 n 4 3 2 + 40376458670 n + 126211592060 n + 243030970008 n + 259967364192 n + 116939064960) hypergeom([1/2, -n - 1], [1], 4) - 3 9 8 7 6 5 4 3 2 (365 n + 40406 n + 1517574 n + 29023176 n + 326116329 n + 2280212394 n + 10031801476 n + 26946731704 n + 40244569056 n + 25511345280) (n + 1) hypergeom([1/2, -n], [1], 4))/((n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2))} "A335537" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A335595" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ (-1) (2 HermiteH(n1 + 1, 1) + (-n1 - 1) HermiteH(n1, 1))| {n! (n + 5), n! (n + 5) | ) ----------------------------------------------------------|} | / (n1 + 1)! (n1 + 6) (n1 + 5) | |----- | \n1 = 0 / "A335693" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-1) ((-n - 3 n - 3) BesselK(n - 1, -2) + (n + 2 n + 2) (n + 1) BesselK(n, -2)), n 2 2 (-1) ((n + 2 n + 2) (n + 1) BesselI(n, 2) + (-n - 3 n - 3) BesselI(n - 1, 2))} "A335700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A335819" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 6 ) HermiteH(n, 1/2 I 6 )} "A335827" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n!} "A335873" n (-1) n! n! (4 n - 1) {--------, ------------} n n "A335974" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 8 7 6 5 4 3 2 { 4 (n + 49 n + 2443 n + 10045 n + 358729 n + 177331 n + 5584137 n + 135135 n + 4054050) { ---------------------------------------------------------------------------------------------- n::even { n (n + 1) (n + 3) (n + 5) (n + 7) (n + 9) (n + 11) (n + 13) (n + 15) binomial(n, n/2) {{ , { (2 n - 2) { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 2 binomial(n, n/2) n::even { { 8 7 6 5 4 3 2 } { binomial(n + 1, n/2 + 1/2) (n + 49 n + 2443 n + 10045 n + 358729 n + 177331 n + 5584137 n + 135135 n + 4054050) { ---------------------------------------------------------------------------------------------------------------------- n::odd { (n + 15) (n + 13) (n + 11) (n + 9) (n + 7) (n + 5) (n + 3) n "A336114" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n + 1/2, 1/2) + BesselI(n - 1/2, 1/2)), (-1) (BesselK(n + 1/2, -1/2) + BesselK(n - 1/2, -1/2))} "A336165" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 2} "A336170" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A336174" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A336181" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A336182" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A336283" LREtools/SearchTable: "SearchTable successful" n n 2 ((n + 5) LegendreP(n + 1, 3) + (-11 n - 15) LegendreP(n, 3)) 2 ((n + 5) LegendreQ(n + 1, 3) + (-11 n - 15) LegendreQ(n, 3)) {---------------------------------------------------------------, ---------------------------------------------------------------} (n - 1) n (n - 1) n "A336286" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n + 1/2, 1/2) + 2 BesselI(n - 1/2, 1/2)), (-1) (BesselK(n + 1/2, -1/2) + 2 BesselK(n - 1/2, -1/2))} "A336400" LREtools/SearchTable: "SearchTable successful" n n {(-1) (BesselI(n + 1/2, 1) - BesselI(n - 1/2, 1)), (-1) (BesselK(n + 1/2, -1) - BesselK(n - 1/2, -1))} "A336538" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (3 n + 2) hypergeom([-n - 1, -3 n - 3], [1], 3) + (-60 n - 72 n - 20) hypergeom([-n, -3 n], [1], 3) {------------------------------------------------------------------------------------------------------------} 2 (3 n + 2) n "A336539" LREtools/SearchTable: "SearchTable successful" binomial(3 n, n) hypergeom([-n, 3 n + 1], [2 n + 2], -2) {--------------------------------------------------------} 2 n + 1 "A336540" LREtools/SearchTable: "SearchTable successful" n 2 binomial(4 n, n) hypergeom([-n, 4 n + 1], [3 n + 2], -1/2) {-------------------------------------------------------------} 3 n + 1 "A336572" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A336614" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A336634" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A336729" LREtools/SearchTable: "SearchTable successful" n n (-4) (2 LegendreP(n + 1, 1/2) - LegendreP(n, 1/2)) (-4) (2 LegendreQ(n + 1, 1/2) - LegendreQ(n, 1/2)) {---------------------------------------------------, ---------------------------------------------------} n n "A336804" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n 2 n 2 | \ 2 | {2 (n!) , 2 (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A336805" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n 2 n 2 | \ 3 | {3 (n!) , 3 (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A336807" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-2 n1 - 2)| n 2 n 2 | \ 2 | {4 (n!) , 4 (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A336808" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n 2 n 2 | \ 5 | {5 (n!) , 5 (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A336981" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=259888.2MB, alloc=3479.5MB, time=1995.86 LREtools/SolveLRE: "Reduced the order of" (2*n+5)*(4290*n+17527)*(4290*n+13237)*(4290*n+8947)*(4290*n+4657)*(n+5)^2*E^4+(-644905547526240000*n^7-\ 14731055785253808000*n^6-139978158596101425600*n^5-714705582479936567520*n^4-2108850978019874926544*n^3-3577534685373676821416*n^2-\ 3207966351979266169236*n-1159845804977066816490)*E^3+(211323083363346240000*n^7+4827082262942747808000*n^6+45941088120856893993600*n^5+ 235299625352905239846720*n^4+697292977612730414577504*n^3+1188770478140357526223152*n^2+1071242015472701470917048*n+388994425727462299076220)*E^2+(-\ 27859919653133568000000*n^7-636381609922964505600000*n^6-6070149662140985541120000*n^5-31250699254896388210944000*n^4-93425731366150142412364800*n^3-\ 161318211708390654023769600*n^2-147718875159027452071008000*n-54530772121604247482160000)*E+1866240000*(n+2)*(677421793620000*n^6+14118954506514000*n ^5+119809604201086800*n^4+528847849249107960*n^3+1278572570710585442*n^2+1603584117254705053*n+815158627859311218) "to two: Symmetric product" (4 *n+8)*E^2+(-14*n-15)*E+12*n+12 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" /n - 1 n |----- 3136 n | \ n1 {1/2 --------------------------, 1/2 3136 | ) 108 (n + 1/2) binomial(2 n, n) | / |----- \n1 = 0 ((2 n1 + 1) hypergeom([-1/2, -n1 - 1], [1], -16/9) + (-2 n1 - 2) hypergeom([-1/2, -n1], [1], -16/9)) 3 2 2 ((4 n1 - 78 n1 + 200 n1 + 162) hypergeom([-5/2, -n1 - 1], [1], -1/3) - (n1 + 1) (4 n1 - 72 n1 + 167) hypergeom([-5/2, -n1], [1], -1/3)) \ | /286000 n1 \ / (n1 + 1) | |--------- + 931400/9| (n1 + 3/2) binomial(2 n1 + 2, n1 + 1) / (3136 (2 n1 + 3))|/((n + 1/2) binomial(2 n, n))} \ 3 / / | | / "A336982" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=261237.5MB, alloc=3479.5MB, time=2005.39 LREtools/SolveLRE: "Reduced the order of" (2*n+7)*(540*n+2837)*(540*n+2297)*(540*n+1757)*(540*n+1217)*(n+6)^2*E^4+(-19046845440000*n^7-\ 581210961408000*n^6-7480985197593600*n^5-52599565557342720*n^4-217964831035083104*n^3-531739026663865328*n^2-706248210085883080*n-393365110197412276) *E^3+(-56944625909760000*n^7-1737654714335232000*n^6-22383516418655846400*n^5-157623785262383431680*n^4-654620395187723974656*n^3-\ 1601386598444337017856*n^2-2133544960900213613056*n-1192206166485839721216)*E^2+(-780158789222400000*n^7-23806400979271680000*n^6-\ 306805454134050816000*n^5-2162841009817072435200*n^4-8998697557503648727040*n^3-22071384428996666982400*n^2-29507257039497475194880*n-\ 16555744458575589048320)*E+1677721600*(n+3)*(170061120000*n^6+4679200224000*n^5+52925081572800*n^4+314766064056960*n^3+1037540859412802*n^2+ 1796334870598697*n+1275960097173993) "to two: Symmetric product" (4*n+8)*E^2+(-14*n-21)*E-8*n-8 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" /n - 1 n |----- (n + 1) 3136 n | \ / 1/2 n1 {------------------------------------, (n + 1) 3136 | ) |- (-64 10 ) (2 n + 3) (2 n + 1) binomial(2 n, n) | / \ |----- \n1 = 0 1/2 1/2 1/2 ((-20 n1 - 20) LegendreP(n1, 1/20 I 5 ) + 5 (2 n1 + 3) LegendreP(n1 + 1, 1/20 I 5 ) I) 1/2 1/2 1/2 ((-8 n1 - 8) LegendreP(n1, 7/8 I 2 ) + 7 I 2 (2 n1 + 3) LegendreP(n1 + 1, 7/8 I 2 )) (540 n1 + 1217) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) \ /n - 1 | |----- / 3 (n1 + 1) \| n | \ / 1/2 n1 / ((n1 + 2) 3136 )||/((2 n + 3) (2 n + 1) binomial(2 n, n)), (n + 1) 3136 | ) |- (-64 10 ) / /| | / \ | |----- / \n1 = 0 1/2 1/2 1/2 ((-20 n1 - 20) LegendreP(n1, 1/20 I 5 ) + 5 (2 n1 + 3) LegendreP(n1 + 1, 1/20 I 5 ) I) 1/2 1/2 1/2 ((-8 n1 - 8) LegendreQ(n1, 7/8 I 2 ) + 7 I 2 (2 n1 + 3) LegendreQ(n1 + 1, 7/8 I 2 )) (540 n1 + 1217) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) \ /n - 1 | |----- / 3 (n1 + 1) \| n | \ / 1/2 n1 / ((n1 + 2) 3136 )||/((2 n + 3) (2 n + 1) binomial(2 n, n)), (n + 1) 3136 | ) |- (-64 10 ) / /| | / \ | |----- / \n1 = 0 1/2 1/2 1/2 ((-20 n1 - 20) LegendreQ(n1, 1/20 I 5 ) + 5 (2 n1 + 3) LegendreQ(n1 + 1, 1/20 I 5 ) I) 1/2 1/2 1/2 ((-8 n1 - 8) LegendreQ(n1, 7/8 I 2 ) + 7 I 2 (2 n1 + 3) LegendreQ(n1 + 1, 7/8 I 2 )) (540 n1 + 1217) (2 n1 + 3) binomial(2 n1 + 2, n1 + 1) \ /n - 1 | |----- / 3 (n1 + 1) \| n | \ / 1/2 n1 / ((n1 + 2) 3136 )||/((2 n + 3) (2 n + 1) binomial(2 n, n)), (n + 1) 3136 | ) |- (-64 10 ) / /| | / \ | |----- / \n1 = 0 1/2 1/2 1/2 ((-8 n1 - 8) LegendreP(n1, 7/8 I 2 ) + 7 I 2 (2 n1 + 3) LegendreP(n1 + 1, 7/8 I 2 )) 1/2 1/2 1/2 ((-20 n1 - 20) LegendreQ(n1, 1/20 I 5 ) + 5 (2 n1 + 3) LegendreQ(n1 + 1, 1/20 I 5 ) I) (540 n1 + 1217) (2 n1 + 3) \ | / 3 (n1 + 1) \| binomial(2 n1 + 2, n1 + 1) / ((n1 + 2) 3136 )||/((2 n + 3) (2 n + 1) binomial(2 n, n))} / /| | / "A337001" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A337002" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 4| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A337152" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n 2 n 2 | \ (-1) 2 | {2 (n!) , 2 (n!) | ) -----------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A337153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n 2 n 2 | \ (-1) 3 | {3 (n!) , 3 (n!) | ) -----------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A337154" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2)| n 2 n 2 | \ (-1) 2 | {4 (n!) , 4 (n!) | ) -------------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A337155" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1)| n 2 n 2 | \ (-1) 5 | {5 (n!) , 5 (n!) | ) -----------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A337167" LREtools/SearchTable: "SearchTable successful" {hypergeom([-1/2, -n - 1], [1], -12) - hypergeom([-1/2, -n], [1], -12)} "A337168" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 7 ) 7 (7 LegendreP(n + 1, 3/7 I 7 ) I + 7 LegendreP(n, 3/7 I 7 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreQ(n + 1, 3/7 I 7 ) I + 7 LegendreQ(n, 3/7 I 7 ))} "A337169" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 11 ) 11 (11 LegendreP(n + 1, 5/11 I 11 ) I + 11 LegendreP(n, 5/11 I 11 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 11 ) 11 (11 LegendreQ(n + 1, 5/11 I 11 ) I + 11 LegendreQ(n, 5/11 I 11 ))} "A337247" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n (n + 1) 160 {------------------------------------} (2 n + 3) (2 n + 1) binomial(2 n, n) "A337332" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A337370" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n n (-2) ((2 n + 2) LegendreP(2 n + 2, I) + (2 n + 1) LegendreP(2 n, I)) (-2) ((2 n + 2) LegendreQ(2 n + 2, I) + (2 n + 1) LegendreQ(2 n, I)) {---------------------------------------------------------------------, ---------------------------------------------------------------------} 4 n + 3 4 n + 3 "A337390" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n n {(-2) LegendreP(2 n, I), (-2) LegendreQ(2 n, I)} "A337393" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 n 5 n 5 {(-5) LegendreP(2 n, ----), (-5) LegendreQ(2 n, ----)} 5 5 "A337394" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ n | 5 5 | (-5) |(2 n + 1) LegendreP(2 n, ----) + (2 n + 2) LegendreP(2 n + 2, ----)| \ 5 5 / {---------------------------------------------------------------------------, 4 n + 3 / 1/2 1/2 \ n | 5 5 | (-5) |(2 n + 1) LegendreQ(2 n, ----) + (2 n + 2) LegendreQ(2 n + 2, ----)| \ 5 5 / ---------------------------------------------------------------------------} 4 n + 3 "A337396" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 n 2 n 2 {(-8) LegendreP(2 n, ----), (-8) LegendreQ(2 n, ----)} 2 2 "A337397" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ n | 2 2 | (-8) |(2 n + 2) LegendreP(2 n + 2, ----) + (2 n + 1) LegendreP(2 n, ----)| \ 2 2 / {---------------------------------------------------------------------------, 4 n + 3 / 1/2 1/2 \ n | 2 2 | (-8) |(2 n + 2) LegendreQ(2 n + 2, ----) + (2 n + 1) LegendreQ(2 n, ----)| \ 2 2 / ---------------------------------------------------------------------------} 4 n + 3 "A337421" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 n 3 n 3 {(-6) LegendreP(2 n, ----), (-6) LegendreQ(2 n, ----)} 3 3 "A337422" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" 1/2 1/2 n 21 n 21 {(-7) LegendreP(2 n, -----), (-7) LegendreQ(2 n, -----)} 7 7 "A337466" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ n | 3 3 | (-6) |(2 n + 1) LegendreP(2 n, ----) + (2 n + 2) LegendreP(2 n + 2, ----)| \ 3 3 / {---------------------------------------------------------------------------, 4 n + 3 / 1/2 1/2 \ n | 3 3 | (-6) |(2 n + 1) LegendreQ(2 n, ----) + (2 n + 2) LegendreQ(2 n + 2, ----)| \ 3 3 / ---------------------------------------------------------------------------} 4 n + 3 "A337467" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" / 1/2 1/2 \ n | 21 21 | (-7) |(2 n + 1) LegendreP(2 n, -----) + (2 n + 2) LegendreP(2 n + 2, -----)| \ 7 7 / {-----------------------------------------------------------------------------, 4 n + 3 / 1/2 1/2 \ n | 21 21 | (-7) |(2 n + 1) LegendreQ(2 n, -----) + (2 n + 2) LegendreQ(2 n + 2, -----)| \ 7 7 / -----------------------------------------------------------------------------} 4 n + 3 "A337499" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { 2 binomial(n, n/2) (3 n + 2) { ------------------------ n::even { ---------------------------- n::even { (n + 1) binomial(n, n/2) { n + 2 {{ , { } { (2 n + 2) { 8 binomial(n - 1, n/2 - 1/2) n { 2 (3 n + 2) { ------------------------------ n::odd { ------------------------------------------ n::odd { n + 1 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) "A337500" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { binomial(n, n/2) (3 n + 2) { 2 4 { -------------------------- n::even { ------------------------ n::even { n + 2 { (n + 3) binomial(n, n/2) {{ , { } { 2 binomial(n + 1, n/2 + 1/2) (n + 1) { (2 n - 2) { ------------------------------------ n::odd { 2 (3 n + 2) { n + 3 { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A337512" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A337589" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A337749" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(-I) n!, I n!} "A337750" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n 1/2 n 1/2 n {(-1) n!, (1/2 - 1/2 I 3 ) n!, (1/2 + 1/2 I 3 ) n!} "A337751" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 n 4 n 4 n 4 n {RootOf(_Z + 1, index = 1) n!, RootOf(_Z + 1, index = 2) n!, RootOf(_Z + 1, index = 3) n!, RootOf(_Z + 1, index = 4) n!} "A337905" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A337992" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- | |----- | n n n | \ (-n1 - 1) | n | \ (-n1 - 1) | {2 , 6 , 6 | ) 6 (LegendreP(n1 + 1, 3) - 3 LegendreP(n1, 3))|, 6 | ) 6 (LegendreQ(n1 + 1, 3) - 3 LegendreQ(n1, 3))|} | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / "A338187" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A338188" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A338193" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A338726" n binomial(2 n, n) {2 , ----------------, n + 1} n + 1 "A339034" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 | | \ n1! (n1 + n1 + 1)| {1, n, n | ) ------------------|} | / 2 | |----- n1 (n1 + 1) | \n1 = 0 / "A339240" n {n binomial(2 n, n), 4 n} "A339390" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A339516" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 n1 (2 n1 + 1) | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A339565" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A339654" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 (7 n + 10)} "A339710" memory used=262733.2MB, alloc=3479.5MB, time=2015.59 LREtools/SearchTable: "SearchTable successful" 3 (2 n + 3) (n + 1) hypergeom([-n, 2 n + 4], [2], -2) - 2 (17 n + 11) n hypergeom([2 n + 2, -n + 1], [2], -2) {-------------------------------------------------------------------------------------------------------------} 13 n + 9 "A339987" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A340567" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 3) { 1/2 ------------------------ n::even n { binomial(n, n/2) (2 n - 6) n::even { (n + 1) binomial(n, n/2) {2 , { , { } { binomial(n + 1, n/2 + 1/2) (n - 3) n::odd { (2 n - 2) { 2 (n - 3) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A340568" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { 4 (n + 1) { n binomial(n, n/2) n::even { -------------------------- n::even n { { n (n + 1) binomial(n, n/2) {2 , { 2 , { } { binomial(n + 1, n/2 + 1/2) (n + 1) { (2 n - 2) { 1/2 ----------------------------------- n::odd { 2 2 { n { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A340569" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (2 n + 1) { -------------------------- n::even { 4 binomial(n, n/2) n::even n { n (n + 1) binomial(n, n/2) { {2 , { , { (2 n + 1) binomial(n + 1, n/2 + 1/2) } { (2 n - 2) { ------------------------------------ n::odd { 4 2 { n { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A340766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / 1/2\n / 1/2\n / 1/2\n | 5 | | 5 | | 5 | {|3/2 - ----| , |3/2 + ----| , |3/2 - ----| \ 2 / \ 2 / \ 2 / / / / /{ 3 n2 n2 2 \\\\ | | | |{ binomial(----, ----) (n2 - 6 n2 - 4) |||| |n - 1 | |n1 - 1 |{ 2 2 |||| |----- | |----- / 1/2\(-n2 - 1) |{ ------------------------------------- n2::even|||| | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 5 | |{ (n2 + 1) (n2 + 2) |||| | ) |2 (3 + 5 ) (3 - 5 ) | ) |3/2 + ----| |{ ||||, | / | | / \ 2 / |{ 3 n2 n2 |||| |----- | |----- |{ 3 binomial(---- - 3/2, ---- - 1/2) (3 n2 - 1) |||| |n1 = 0 | |n2 = 0 |{ 2 2 |||| | | | |{ - --------------------------------------------- n2::odd |||| \ \ \ \{ n2 + 1 //// / / / | | | | | | | | | |n - 1 | |n1 - 1 / 1/2\n |----- | |----- | 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ |3/2 - ----| | ) |2 (3 + 5 ) (3 - 5 ) | ) \ 2 / | / | | / |----- | |----- |n1 = 0 | |n2 = 0 | | | | | | | | | \ \ \ /{ (-n2) 3 n2 3 n2 n2 \\\\ |{ 4 n2 binomial(3 n2, ----) binomial(----, ----) |||| |{ 2 2 2 |||| |{ -3/2 --------------------------------------------------- n2::even|||| |{ n2 |||| / 1/2\(-n2 - 1) |{ (n2 + 1) binomial(n2, ----) |||| | 5 | |{ 2 |||| |3/2 + ----| |{ ||||} \ 2 / |{ (-2 n2 - 2) 2 3 n2 3 n2 n2 |||| |{ 2 (n2 - 6 n2 - 4) binomial(3 n2 + 3, ---- + 3/2) binomial(---- + 3/2, ---- + 1/2) |||| |{ 2 2 2 |||| |{ 1/2 --------------------------------------------------------------------------------------------- n2::odd |||| |{ n2 |||| |{ (n2 + 2) (3 n2 + 2) binomial(n2 + 1, ---- + 1/2) |||| \{ 2 //// "A340789" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n!) , (n!) | ) ------------|} | / 2| |----- ((n1 + 1)!) | \n1 = 0 / "A340973" LREtools/SearchTable: "SearchTable successful" {(2 n + 2) hypergeom([-1/2, -n - 1], [1], -12) + (-2 n - 3) hypergeom([-1/2, -n], [1], -12)} "A341266" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341300" 2 {n, n! (n + 2) (n + 5 n + 5)} "A341302" {n! (n + 2), n - 1} "A341303" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 1, 1, 2 / 1/2\n / 1/2 \n n | 5 | |5 | (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {2 , |1/2 - ----| , |---- + 1/2| , (n + 3) (n + 2) (n + 1) n!, ----------------------------------------------, n - 2 n - 5} \ 2 / \ 2 / (n + 4) (n + 3) (n + 2) (n + 1) "A341344" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) { 8 2 ((n + 2) LegendreP(n/2 + 1, 3) + (-6 n - 6) LegendreP(n/2, 3)) { - ----------------------------------------------------------------------- n::even { n {{ , { (n/2 + 1/2) { 2 2 ((n + 3) (7 n - 1) LegendreP(n/2 + 3/2, 3) + %1 LegendreP(n/2 + 1/2, 3)) { - --------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { (n/2) { 8 2 ((n + 2) LegendreQ(n/2 + 1, 3) + (-6 n - 6) LegendreQ(n/2, 3)) { - ----------------------------------------------------------------------- n::even { n { , { (n/2 + 1/2) { 2 2 ((n + 3) (7 n - 1) LegendreQ(n/2 + 3/2, 3) + %1 LegendreQ(n/2 + 1/2, 3)) { - --------------------------------------------------------------------------------------- n::odd { (n - 1) (n + 1) { (n/2) { 2 ((n + 3) (7 n - 1) LegendreP(n/2 + 3/2, 3) + %1 LegendreP(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { , { (n/2 - 1/2) { 4 2 ((n + 2) LegendreP(n/2 + 1, 3) + (-6 n - 6) LegendreP(n/2, 3)) { - ----------------------------------------------------------------------------- n::odd { n { (n/2) { 2 ((n + 3) (7 n - 1) LegendreQ(n/2 + 3/2, 3) + %1 LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------------- n::even { (n - 1) (n + 1) { } { (n/2 - 1/2) { 4 2 ((n + 2) LegendreQ(n/2 + 1, 3) + (-6 n - 6) LegendreQ(n/2, 3)) { - ----------------------------------------------------------------------------- n::odd { n 2 %1 := -41 n - 76 n + 13 "A341491" LREtools/SearchTable: "SearchTable successful" /46656\n |-----| GAMMA(n + 1/3) GAMMA(1/6 + n) GAMMA(n + 2/3) GAMMA(n + 1/2) GAMMA(n + 5/6) hypergeom([-n, -5 n], [-6 n], -1) \3125 / {---------------------------------------------------------------------------------------------------------------------} GAMMA(n + 3/5) GAMMA(n + 1/5) GAMMA(n + 1) GAMMA(n + 4/5) GAMMA(n + 2/5) "A341500" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 6" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A341546" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n n | \ (3 n1 + 3) LegendreP(n1 + 1, 3) + (-13 n1 - 9) LegendreP(n1, 3)| {(-1) , (-3/2) , (-3/2) | ) ---------------------------------------------------------------|, | / (n1 + 1) | |----- n1 (-3/2) | \n1 = 0 / /n - 1 \ |----- | n | \ (3 n1 + 3) LegendreQ(n1 + 1, 3) + (-13 n1 - 9) LegendreQ(n1, 3)| (-3/2) | ) ---------------------------------------------------------------|} | / (n1 + 1) | |----- n1 (-3/2) | \n1 = 0 / "A341732" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 24 binomial(n, n/2) (n + 1) { 2 4 (n + 2) { --------------------------- n::even { ---------------------------------------- n::even n { (n + 2) (n + 6) { (n + 1) (n + 3) (n + 5) binomial(n, n/2) {1, (-1) , { , { } { 32 binomial(n - 1, n/2 - 1/2) (n + 2) n { (2 n + 2) { --------------------------------------- n::odd { 2 { (n + 5) (n + 3) (n + 1) { 3/2 ------------------------------------------ n::odd { (n + 2) (n + 6) binomial(n + 1, n/2 + 1/2) "A341900" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- | n \ n1 | \ n2 | {1, (-1) , ) (-1) | ) (-(-1) (n2 + 2) (n2 + 1) n2!)|} / | / | ----- |----- | n1 = 0 \n2 = 0 / "A341922" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(-1/4) (24 n + 31), /n - 1 \ |----- 2 | n | \ (3 n1 + 5) (3 n1 + 2) (3 n1 + 4) (3 n1 + 1) binomial(3 n1, n1) (93 n1 + 371 n1 + 358) | (-1/4) (24 n + 31) | ) ----------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- (2 n1 + 7) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) (n1 + 3) (n1 + 2) (n1 + 1) (-1/4) (24 n1 + 55) (96 n1 + 124)| \n1 = 0 / "A341961" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341962" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341963" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341964" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341965" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341966" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341967" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A341968" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A342600" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\ / 1/2\ 1/2 n | 3 2 | 1/2 n | 3 2 | {(1 - 2 ) |n - 3 + ------|, (1 + 2 ) |n - 3 - ------|} \ 2 / \ 2 / "A342906" n (2 n + 1) binomial(2 n, n) {4 , --------------------------} (n + 1) (n + 2) "A342912" LREtools/SearchTable: "SearchTable successful" n (-1) (n hypergeom([1/2, -n - 1], [1], 4) + (3 n + 12) hypergeom([1/2, -n], [1], 4)) (n + 1) {--------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A343089" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 \\\ |----- | |----- ||| n n n | \ | n1 n1 | \ (3 n2 + 5) (3 n2 + 2) (3 n2 + 4) (3 n2 + 1) binomial(3 n2, n2) (11 n2 + 35) (7 n2 + 17)||| {(-1/4) , (27/4) , (-1/4) | ) |-4 27 (-1) | ) ---------------------------------------------------------------------------------------||| | / | | / (n2 + 1) ||| |----- | |----- (2 n2 + 7) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) (n2 + 3) (n2 + 2) (n2 + 1) (27/4) ||| \n1 = 0 \ \n2 = 0 /// } "A343091" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=263852.9MB, alloc=3479.5MB, time=2023.83 LREtools/SearchTable: "SearchTable not successful" n /196\n {(-4) , |---| } \27 / "A343093" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | /256\n /256\n | \ (4 n1 + 5) (4 n1 + 1) (2 n1 + 3) (2 n1 + 1) (4 n1 + 7) (4 n1 + 3) binomial(4 n1, n1) (56 n1 + 287 n1 + 366)| {|---| , |---| | ) ------------------------------------------------------------------------------------------------------------|} \27 / \27 / | / /256\(n1 + 1) | |----- (n1 + 3) (n1 + 2) (n1 + 1) (3 n1 + 7) (3 n1 + 4) (3 n1 + 1) (3 n1 + 8) (3 n1 + 5) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A343276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ (n1 + 1) (n1 + 2) | {n! (n - 2 n + 2), n! (n - 2 n + 2) | ) ---------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) - 2 n1) (n1 - 2 n1 + 2)| \n1 = 0 / "A343304" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A343305" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A343386" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2) | 1/2 5 5 | (n/2) | 1/2 5 5 | 5 |5 LegendreP(n + 1, ----) - 5 LegendreP(n, ----)| 5 |5 LegendreQ(n + 1, ----) - 5 LegendreQ(n, ----)| \ 5 5 / \ 5 5 / {-----------------------------------------------------------, -----------------------------------------------------------, n + 2 n + 2 n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) -------------------------------------------------------------------------} n + 2 "A343445" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { {{ (n - 1) 2 3 n , { 2 ((n/2 - 1/2)!) binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 { (-n) 2 3 n 3 n { 2 ((n/2)!) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------------- n::even} { 3 n - 1 { { 0 n::odd "A343580" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { n { 4 (n!) { 8 (n/2)! { --------- n::even { ------------------------ n::even { 3 { (n + 1) binomial(n, n/2) { ((n/2)!) {{ , { } { (3 n - 3) { 2 { 4 2 (n/2 - 1/2)! { ((n + 1)!) { ---------------------------- n::odd { --------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 3 { ((n/2 + 1/2)!) "A343581" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { n { 4 (n!) { 8 (n - 1) (n/2)! { --------- n::even { -------------------------- n::even { 3 { n (n + 1) binomial(n, n/2) { ((n/2)!) {{ , { } { (3 n - 3) { 2 { 4 2 (n/2 - 1/2)! { ((n + 1)!) (n - 1) { ---------------------------- n::odd { ------------------- n::odd { n binomial(n - 1, n/2 - 1/2) { 3 { ((n/2 + 1/2)!) n "A343582" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 3 | {(-2) n!, (-2) n! | ) ----------------------|} | / (n1 + 1) | |----- (-2) (n1 + 1)!| \n1 = 0 / "A343688" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n + 2) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((4 n + 2) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A343689" LREtools/SearchTable: "SearchTable successful" n n {(-1) ((4 n + 2) BesselI(n + 1/2, 1/2) - BesselI(n - 1/2, 1/2)), (-1) ((4 n + 2) BesselK(n + 1/2, -1/2) - BesselK(n - 1/2, -1/2))} "A343773" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ / 1/2 1/2 \ (n/2) | 1/2 5 5 | (n/2) | 1/2 5 5 | 5 |5 LegendreP(n + 1, ----) - 5 LegendreP(n, ----)| 5 |5 LegendreQ(n + 1, ----) - 5 LegendreQ(n, ----)| \ 5 5 / \ 5 5 / {-----------------------------------------------------------, -----------------------------------------------------------} n + 2 n + 2 "A343832" LREtools/SearchTable: "SearchTable successful" ((n - 1) hypergeom([-n - 1], [n + 1], -1) - n hypergeom([-n], [n], -1)) binomial(2 n, n) n! (2 n + 1) {-----------------------------------------------------------------------------------------------------} n + 2 "A343884" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A343896" LREtools/SearchTable: "SearchTable successful" (2 n + 1) n! binomial(2 n, n) (2 hypergeom([-n - 1], [n + 2], 1) - hypergeom([-n], [n + 1], 1)) {-----------------------------------------------------------------------------------------------} n "A343898" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A344014" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n 2 n 2 {54000 (11979 n + 27819 n + 15872), 54000 (11979 n + 27819 n + 15872) /n - 1 \ |----- n1 2 | | \ (3 n1 + 1) (3 n1 + 2) 256 binomial(3 n1, n1) (4158 n1 + 12063 n1 - 31) | | ) ----------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) 54000 (11979 (n1 + 1) + 27819 n1 + 43691) (11979 n1 + 27819 n1 + 15872)| \n1 = 0 / "A344055" LREtools/SearchTable: "SearchTable successful" n {2 ((2 n - 1) hypergeom([-1/2, -n - 1], [1], -2) - 2 n hypergeom([-1/2, -n], [1], -2))} "A344216" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1! n1 | {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) ---------------------------|} | / (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A344229" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- n1 | |----- n1 | |----- n1 | | \ 3 n1 + 4 | | \ (-1) n1| | \ (-I) n1| | \ I n1 | {n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / \n1 = 0 / "A344262" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1 + 2 | | \ (-1) n1| {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A344317" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1 + 2 | | \ (-1) n1| {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A344396" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) hypergeom([- n/2, - n/2 - 1/2], [n + 2], 4) {----------------------------------------------------------------------} n + 1 "A344418" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1 + 2 | | \ (-1) n1| {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A344419" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- n1 | | \ n1 + 2 | | \ (-1) n1| {n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A344500" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A344504" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {3 , (-1) ((4 n + 7) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 5) hypergeom([1/2, -n], [1], 4))} "A344506" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4))| {(13/3) , (13/3) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (13/3) | \n1 = 0 / "A344507" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | n n | \ (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - 3 hypergeom([1/2, -n1], [1], 4))| {(-3/2) , (-3/2) | ) ----------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (-3/2) | \n1 = 0 / "A344553" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (n + 1) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------} (10 n + 7) (n + 1) "A344558" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n n | \ | 15 (-1/6 I) (-15 LegendreP(n1 + 1, 1/15 I 15 ) I + 15 LegendreP(n1, 1/15 I 15 ))|| {6 , 6 | ) |-1/6 -----------------------------------------------------------------------------------------------||, | / \ n1 + 2 /| |----- | \n1 = 0 / / / / n1 \ \\ |n - 1 | |----| || |----- | \ 2 / n1 1/2 1/2 1/2 || n | \ | 15 (-1/6 I) (-15 LegendreQ(n1 + 1, 1/15 I 15 ) I + 15 LegendreQ(n1, 1/15 I 15 ))|| 6 | ) |-1/6 -----------------------------------------------------------------------------------------------||} | / \ n1 + 2 /| |----- | \n1 = 0 / "A344559" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A344560" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A344576" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2\n | 3 5 | | 3 5 | {|7/2 - ------| , |7/2 + ------| , \ 2 / \ 2 / /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- / 1/2\(-n2 - 1) ||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 3 5 | ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) |7/2 + ------| (LegendreP(n2 + 1, 3) + LegendreP(n2, 3))|||, \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- / 1/2\(-n2 - 1) ||| | 3 5 | | \ | 1/2 n1 1/2 (-n1 - 1) | \ | 3 5 | ||| |7/2 - ------| | ) |2 (7 + 3 5 ) (7 - 3 5 ) | ) |7/2 + ------| (LegendreQ(n2 + 1, 3) + LegendreQ(n2, 3))|||} \ 2 / | / | | / \ 2 / ||| |----- | |----- ||| \n1 = 0 \ \n2 = 0 /// "A344623" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A344935" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-I) (n1 + 1 - I)| | \ I (n1 + 1 + I)| {n! | ) -------------------|, n! | ) ----------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A345132" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A345189" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 binomial(2 n, n) {-------------------} n + 1 "A345190" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 3 binomial(2 n, n) {-------------------} n + 1 "A345340" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {2 , (-2) hypergeom([1/2, -n], [1], 4)} "A345367" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1) {1, ) ---------------------------------------------------} / (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) ----- n1 = 0 "A345368" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /3125\n1 ----- |----| GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) \ \256 / {1, ) ------------------------------------------------------------------------} / GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 9/4) ----- n1 = 0 "A345646" LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) binomial(4 n, 2 n) hypergeom([-n, -n, -n, -n], [1, 1, 1], 1)} "A345887" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A345889" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 3) (n1 + 2) (n1 + 1) n1!} / ----- n1 = 0 "A345908" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346047" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A346048" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A346049" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A346065" memory used=264772.1MB, alloc=3479.5MB, time=2031.57 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /46656\n1 ----- |-----| GAMMA(n1 + 3/2) GAMMA(n1 + 11/6) GAMMA(n1 + 5/3) GAMMA(n1 + 4/3) GAMMA(n1 + 7/6) \ \3125 / {1, ) ------------------------------------------------------------------------------------------} / GAMMA(n1 + 11/5) GAMMA(n1 + 2) GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) ----- n1 = 0 "A346073" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A346074" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A346075" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A346076" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A346224" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) n! HermiteH(n, I)} "A346258" LREtools/SearchTable: "SearchTable successful" n {(-3) ((3 n + 3) LaguerreL(n + 1, -n - 1/3, 1/3) + LaguerreL(n, -n + 2/3, 1/3)) n!} "A346380" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) n n {------------------, (-1) (n + 1) (2 n - 1) 2 (4 (2 n + 1) hypergeom([-n - 1, n/2 + 1, n/2 + 1/2], [n + 2, n + 1], 4) + (n + 1) n hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4)) binomial(2 n, n)/((n + 1) (5 n + 3) n)} "A346394" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 /n1 - 1 \\ |----- | |----- |----- n2 || n n | \ (-n1 - 1) | n | \ (-n1 - 1) | \ 2 || {2 , 2 | ) 2 n1!|, 2 | ) 2 n1! | ) ---------||} | / | | / | / (n2 + 1)!|| |----- | |----- |----- || \n1 = 0 / \n1 = 0 \n2 = 0 // "A346395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 /n1 - 1 \\ |----- | |----- |----- n2 || n n | \ (-n1 - 1) | n | \ (-n1 - 1) | \ 3 || {3 , 3 | ) 3 n1!|, 3 | ) 3 n1! | ) ---------||} | / | | / | / (n2 + 1)!|| |----- | |----- |----- || \n1 = 0 / \n1 = 0 \n2 = 0 // "A346396" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 /n1 - 1 \\ |----- | |----- |----- n2 || n n | \ (-2 n1 - 2) | n | \ (-2 n1 - 2) | \ 4 || {4 , 4 | ) 2 n1!|, 4 | ) 2 n1! | ) ---------||} | / | | / | / (n2 + 1)!|| |----- | |----- |----- || \n1 = 0 / \n1 = 0 \n2 = 0 // "A346397" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- n2 || | | \ (-2) || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || n n | \ n1! | n | \ \n2 = 0 /| {(-2) , (-2) | ) ------------|, (-2) | ) ----------------------|} | / (n1 + 1)| | / (n1 + 1) | |----- (-2) | |----- (-2) | \n1 = 0 / \n1 = 0 / "A346398" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- n2 || | | \ (-3) || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || n n | \ n1! | n | \ \n2 = 0 /| {(-3) , (-3) | ) ------------|, (-3) | ) ----------------------|} | / (n1 + 1)| | / (n1 + 1) | |----- (-3) | |----- (-3) | \n1 = 0 / \n1 = 0 / "A346432" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n {(2 - 2 ) n!, (2 + 2 ) n!} "A346503" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346504" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A346505" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346506" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346550" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346626" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346627" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346628" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346646" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346647" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A346648" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A346649" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A346650" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A346664" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346665" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A346666" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A346667" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A346668" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A346671" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /823543\n1 ----- |------| GAMMA(n1 + 12/7) GAMMA(n1 + 8/7) GAMMA(n1 + 10/7) GAMMA(n1 + 11/7) GAMMA(n1 + 9/7) GAMMA(n1 + 13/7) \ \46656 / {1, ) --------------------------------------------------------------------------------------------------------------} / GAMMA(n1 + 3/2) GAMMA(n1 + 13/6) GAMMA(n1 + 2) GAMMA(n1 + 11/6) GAMMA(n1 + 5/3) GAMMA(n1 + 4/3) ----- n1 = 0 "A346672" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /16777216\n1 ----- |--------| GAMMA(n1 + 3/2) GAMMA(n1 + 11/8) GAMMA(n1 + 7/4) GAMMA(n1 + 15/8) GAMMA(n1 + 9/8) GAMMA(n1 + 13/8) GAMMA(n1 + 5/4) \ \ 823543 / {1, ) -------------------------------------------------------------------------------------------------------------------------------} / GAMMA(n1 + 12/7) GAMMA(n1 + 10/7) GAMMA(n1 + 2) GAMMA(n1 + 11/7) GAMMA(n1 + 9/7) GAMMA(n1 + 15/7) GAMMA(n1 + 13/7) ----- n1 = 0 "A346680" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) /| |----- | \n1 = 0 / "A346681" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /3125\n1 \ |----- |----| GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) | n n | \ \256 / | {(-1) , (-1) | ) --------------------------------------------------------------------------|} | / (n1 + 1)| |----- GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 9/4) (-1) | \n1 = 0 / "A346682" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /46656\n1 \ |----- |-----| GAMMA(n1 + 3/2) GAMMA(n1 + 11/6) GAMMA(n1 + 5/3) GAMMA(n1 + 4/3) GAMMA(n1 + 7/6) | n n | \ \3125 / | {(-1) , (-1) | ) -------------------------------------------------------------------------------------------|} | / (n1 + 1)| |----- GAMMA(n1 + 11/5) GAMMA(n1 + 2) GAMMA(n1 + 9/5) GAMMA(n1 + 8/5) GAMMA(n1 + 7/5) (-1) | \n1 = 0 / "A346683" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /823543\n1 \ |----- |------| GAMMA(n1 + 12/7) GAMMA(n1 + 8/7) GAMMA(n1 + 10/7) GAMMA(n1 + 11/7) GAMMA(n1 + 9/7) GAMMA(n1 + 13/7)| n n | \ \46656 / | {(-1) , (-1) | ) --------------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- GAMMA(n1 + 3/2) GAMMA(n1 + 13/6) GAMMA(n1 + 2) GAMMA(n1 + 11/6) GAMMA(n1 + 5/3) GAMMA(n1 + 4/3) (-1) | \n1 = 0 / "A346684" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n {(-1) , /n - 1 /16777216\n1 \ |----- |--------| GAMMA(n1 + 3/2) GAMMA(n1 + 11/8) GAMMA(n1 + 7/4) GAMMA(n1 + 15/8) GAMMA(n1 + 9/8) GAMMA(n1 + 13/8) GAMMA(n1 + 5/4)| n | \ \ 823543 / | (-1) | ) -------------------------------------------------------------------------------------------------------------------------------|} | / (n1 + 1)| |----- GAMMA(n1 + 12/7) GAMMA(n1 + 10/7) GAMMA(n1 + 2) GAMMA(n1 + 11/7) GAMMA(n1 + 9/7) GAMMA(n1 + 15/7) GAMMA(n1 + 13/7) (-1) | \n1 = 0 / "A346762" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346763" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A346845" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | {(n + 1) n! (4 n + 9), (n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A346846" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 {(n + 1) n! (6 n + 33 n + 43), (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------------------------------------------|} | / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A346847" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 {(n + 1) n! (16 n + 156 n + 484 n + 475), (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n!, /n - 1 \ |----- n1 | | \ (n1 + 1) (-1) n1! | (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------------------------|} | / (n1 + 6) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A346943" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 2 { (-n) 2 2 2 { 2 ((n/2)!) (n/4 + 1/2) n::even { 2 (n + 1) binomial(n, n/2) ((n/2)!) n::even {{ , { } { (n - 1) 2 2 { (-n - 1) 2 2 { 1/4 2 (n + 1) ((n/2 - 1/2)!) n::odd { 2 binomial(n + 1, n/2 + 1/2) ((n/2 + 1/2)!) (n + 2) n::odd "A346960" LREtools/SearchTable: "SearchTable not successful" {} "A347012" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, - 1/4 - n, 1/4)} "A347013" LREtools/SearchTable: "SearchTable successful" n {(-5) n! LaguerreL(n, - 1/5 - n, 1/5)} "A347014" LREtools/SearchTable: "SearchTable successful" n {(-6) n! LaguerreL(n, - 1/6 - n, 1/6)} "A347036" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 8 binomial(n1, ----) (n1 + 1) (2 n1 - 7) | | { 2 | | { ---------------------------------------- n1::even| | { n1 + 2 | | { | | { n1 2 | | { 2 binomial(n1 + 1, ---- + 1/2) (2 n1 + 3 n1 - 40) | |n - 1 { 2 | |----- { -------------------------------------------------- n1::odd | | \ { n1 + 3 | {1, (6 n - 13) | ) --------------------------------------------------------------------|, | / (6 n1 - 7) (6 n1 - 13) | |----- | \n1 = 0 / / { n1 2 \ | { 4 (2 n1 + 3 n1 - 40) | | { 1/2 ------------------------------------ n1::even| | { n1 | | { (n1 + 1) (n1 + 3) binomial(n1, ----) | | { 2 | | { | | { (2 n1 - 2) | | { 2 2 (n1 + 1) (2 n1 - 7) | | { ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | | \ { 2 | (6 n - 13) | ) ----------------------------------------------------------|, 6 n - 13} | / (6 n1 - 7) (6 n1 - 13) | |----- | \n1 = 0 / "A347051" LREtools/SearchTable: "SearchTable not successful" {} "A347106" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | | \ (-1) | | \ (-1) HermiteH(n1 + 1, 1/2)| {n! | ) ---------|, n! | ) ----------------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A347210" n! n {----, 2 (n - 1) (n - 4)} n "A347304" memory used=266254.5MB, alloc=3511.5MB, time=2041.56 LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 1 (Liouvillian solutions)" { 2 (n/6) { (n + 5) 432 GAMMA(n/6 + 1) GAMMA(n/6 + 5/3) { ------------------------------------------------- irem(n, 6) = 0 { 2 { (n + 4) GAMMA(n/6 + 11/6) { { 2 (n/6 - 1/6) { 2 (n + 4) 432 GAMMA(n/6 + 5/6) GAMMA(n/6 + 3/2) { ----------------------------------------------------------- irem(n, 6) = 1 { 2 { (n + 3) GAMMA(n/6 + 5/3) { { 2 (n/6 - 1/3) { 6 (n + 3) 432 GAMMA(n/6 + 2/3) GAMMA(n/6 + 4/3) { ----------------------------------------------------------- irem(n, 6) = 2 { 2 { (n + 2) GAMMA(n/6 + 3/2) {{ , { 2 (n/6 - 1/2) { 12 (n + 2) 432 GAMMA(n/6 + 1/2) GAMMA(7/6 + n/6) { ------------------------------------------------------------ irem(n, 6) = 3 { 2 { (n + 1) GAMMA(n/6 + 4/3) { { 2 (n/6 - 2/3) { 12 (n + 1) 432 GAMMA(n/6 + 1/3) GAMMA(n/6 + 1) { ---------------------------------------------------------- irem(n, 6) = 4 { 2 { GAMMA(7/6 + n/6) { { (n/6 - 5/6) { 432 432 GAMMA(n/6 + 1/6) GAMMA(n/6 + 5/6) { ---------------------------------------------------- irem(n, 6) = 5 { 2 { GAMMA(n/6 + 1) { 432 binomial(n/2, n/6) binomial(n, n/2) irem(n, 6) = 0 { { 2 { (n + 5) binomial(n/2 + 5/2, n/6 + 5/6) binomial(n + 5, n/2 + 5/2) { ------------------------------------------------------------------ irem(n, 6) = 1 { n + 4 { { 2 { 2 (n + 4) binomial(n/2 + 2, n/6 + 2/3) binomial(n + 4, n/2 + 2) { ---------------------------------------------------------------- irem(n, 6) = 2 { n + 3 { , { 2 { 6 (n + 3) binomial(n/2 + 3/2, n/6 + 1/2) binomial(n + 3, n/2 + 3/2) { -------------------------------------------------------------------- irem(n, 6) = 3 { n + 2 { { 2 { 12 (n + 2) binomial(n/2 + 1, n/6 + 1/3) binomial(n + 2, n/2 + 1) { ----------------------------------------------------------------- irem(n, 6) = 4 { n + 1 { { 2 { 12 (n + 1) binomial(n/2 + 1/2, n/6 + 1/6) binomial(n + 1, n/2 + 1/2) irem(n, 6) = 5 { /4 n\ { |---| { \ 3 / { 27 2 binomial(n/2, n/6) { ---------------------------- irem(n, 6) = 0 { binomial(n/3, n/6) { { /4 n \ { |--- - 4/3| { \ 3 / { 54 2 binomial(n/2 - 1/2, n/6 - 1/6) { ---------------------------------------------- irem(n, 6) = 1 { binomial(n/3 - 1/3, n/6 - 1/6) { { /4 n \ { |--- - 8/3| { \ 3 / { 54 2 n binomial(n/2 - 1, n/6 - 1/3) { ---------------------------------------------- irem(n, 6) = 2 { binomial(n/3 - 2/3, n/6 - 1/3) { , { /4 n \ { |--- - 4| { \ 3 / { 1944 2 (n - 1) binomial(n/2 - 3/2, n/6 - 1/2) { ------------------------------------------------------ irem(n, 6) = 3 { 2 { n binomial(n/3 - 1, n/6 - 1/2) { { /4 n \ { |--- + 8/3| { \ 3 / { 2 binomial(n/2 + 1, n/6 + 1/3) { 9/2 ----------------------------------------- irem(n, 6) = 4 { binomial(n/3 + 2/3, n/6 + 1/3) { { /4 n \ { |--- + 4/3| { \ 3 / { 9 2 binomial(n/2 + 1/2, n/6 + 1/6) { --------------------------------------------- irem(n, 6) = 5 { binomial(n/3 + 1/3, n/6 + 1/6) { 2 (n/6) { 12 (n + 1) 432 GAMMA(n/6 + 1/3) GAMMA(n/6 + 1) { ---------------------------------------------------- irem(n, 6) = 0 { 2 { GAMMA(7/6 + n/6) { { (n/6 - 1/6) { 432 432 GAMMA(n/6 + 1/6) GAMMA(n/6 + 5/6) { ---------------------------------------------------- irem(n, 6) = 1 { 2 { GAMMA(n/6 + 1) { { 2 (n/6 + 2/3) { (n + 5) 432 GAMMA(n/6 + 1) GAMMA(n/6 + 5/3) { ------------------------------------------------------- irem(n, 6) = 2 { 2 { (n + 4) GAMMA(n/6 + 11/6) { , { 2 (n/6 + 1/2) { 2 (n + 4) 432 GAMMA(n/6 + 5/6) GAMMA(n/6 + 3/2) { ----------------------------------------------------------- irem(n, 6) = 3 { 2 { (n + 3) GAMMA(n/6 + 5/3) { { 2 (n/6 + 1/3) { 6 (n + 3) 432 GAMMA(n/6 + 2/3) GAMMA(n/6 + 4/3) { ----------------------------------------------------------- irem(n, 6) = 4 { 2 { (n + 2) GAMMA(n/6 + 3/2) { { 2 (n/6 + 1/6) { 12 (n + 2) 432 GAMMA(n/6 + 1/2) GAMMA(7/6 + n/6) { ------------------------------------------------------------ irem(n, 6) = 5 { 2 { (n + 1) GAMMA(n/6 + 4/3) { 2 (n/6) { 12 (n + 2) 432 GAMMA(n/6 + 1/2) GAMMA(7/6 + n/6) { ------------------------------------------------------ irem(n, 6) = 0 { 2 { (n + 1) GAMMA(n/6 + 4/3) { { 2 (n/6 - 1/6) { 12 (n + 1) 432 GAMMA(n/6 + 1/3) GAMMA(n/6 + 1) { ---------------------------------------------------------- irem(n, 6) = 1 { 2 { GAMMA(7/6 + n/6) { { (n/6 - 1/3) { 432 432 GAMMA(n/6 + 1/6) GAMMA(n/6 + 5/6) { ---------------------------------------------------- irem(n, 6) = 2 { 2 { GAMMA(n/6 + 1) { , { 2 (n/6 + 1/2) { (n + 5) 432 GAMMA(n/6 + 1) GAMMA(n/6 + 5/3) { ------------------------------------------------------- irem(n, 6) = 3 { 2 { (n + 4) GAMMA(n/6 + 11/6) { { 2 (n/6 + 1/3) { 2 (n + 4) 432 GAMMA(n/6 + 5/6) GAMMA(n/6 + 3/2) { ----------------------------------------------------------- irem(n, 6) = 4 { 2 { (n + 3) GAMMA(n/6 + 5/3) { { 2 (n/6 + 1/6) { 6 (n + 3) 432 GAMMA(n/6 + 2/3) GAMMA(n/6 + 4/3) { ----------------------------------------------------------- irem(n, 6) = 5 { 2 { (n + 2) GAMMA(n/6 + 3/2) { 2 (n/6) { 2 (n + 4) 432 GAMMA(n/6 + 3/2) GAMMA(n/6 + 5/6) { ----------------------------------------------------- irem(n, 6) = 0 { 2 { (n + 3) GAMMA(n/6 + 5/3) { { 2 (n/6 - 1/6) { 6 (n + 3) 432 GAMMA(n/6 + 4/3) GAMMA(n/6 + 2/3) { ----------------------------------------------------------- irem(n, 6) = 1 { 2 { (n + 2) GAMMA(n/6 + 3/2) { { 2 (n/6 - 1/3) { 12 (n + 2) 432 GAMMA(7/6 + n/6) GAMMA(n/6 + 1/2) { ------------------------------------------------------------ irem(n, 6) = 2 { 2 { (n + 1) GAMMA(n/6 + 4/3) { } { 2 (n/6 - 1/2) { 12 (n + 1) 432 GAMMA(n/6 + 1) GAMMA(n/6 + 1/3) { ---------------------------------------------------------- irem(n, 6) = 3 { 2 { GAMMA(7/6 + n/6) { { (n/6 - 2/3) { 432 432 GAMMA(n/6 + 5/6) GAMMA(n/6 + 1/6) { ---------------------------------------------------- irem(n, 6) = 4 { 2 { GAMMA(n/6 + 1) { { 2 (n/6 + 1/6) { (n + 5) 432 GAMMA(n/6 + 5/3) GAMMA(n/6 + 1) { ------------------------------------------------------- irem(n, 6) = 5 { 2 { (n + 4) GAMMA(n/6 + 11/6) "A347429" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- | |----- | n n | \ (n1 + 1) | n | \ n1 (n1 + 1) | {(1/2) , (1/2) | ) 2 (n1 + 1) n1!|, (1/2) | ) (-1) 2 (n1 + 1) n1!|, | / | | / | |----- | |----- | \n1 = 0 / \n1 = 0 / /n - 1 /n1 - 1 \ \ |----- |----- / n2 \| | n | \ n1 (n1 + 1) | \ | (-1) || | (1/2) | ) (-1) 2 (n1 + 1) | ) |- ------------------|| n1!|} | / | / \ (n2 + 2) (n2 + 1)!/| | |----- |----- | | \n1 = 0 \n2 = 0 / / "A347431" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) /n - 1 /n1 - 1 \\ |----- |----- /{ 0 n2::even\|| | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ ||| | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) |{ / n2 \ / n2 \ |||, | / | / |{ |---- + 1/2| |---- - 1/2|! n2::odd ||| |----- |----- \{ \ 2 / \ 2 / /|| \n1 = 0 \n2 = 0 // /n - 1 |----- 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- /{ (-n2) n2 / n2 \ \|| | \ 1/2 (-n2 - 1) |{ 2 (n2 + 1) binomial(n2, ----) |----|! n2::even||| | ) (1/2 + 1/2 I 3 ) |{ 2 \ 2 / |||, | / |{ ||| |----- \{ 0 n2::odd /|| \n2 = 0 // /n - 1 /n1 - 1 \\ |----- |----- || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) || (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (n2 + 2) (n2 + 1) n2!||} | / | / || |----- |----- || \n1 = 0 \n2 = 0 // "A347497" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 5, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / 1/2\n / 1/2\n / 1/2\n |----- | 5 | | 5 | | 5 | | \ {|- 1/2 - ----| , |- 1/2 + ----| , |- 1/2 - ----| | ) \ 2 / \ 2 / \ 2 / | / |----- \n1 = 0 / /n1 - 1 /{ 0 n2::even\\\\ | |----- / 1/2\(-n2 - 1) |{ |||| | n1 1/2 (-n1 - 1) 1/2 n1 | \ | 5 | |{ / n2 \ |||| |-2 (-1) (5 + 1) (5 - 1) | ) |- 1/2 + ----| |{ |---- - 1/2| ||||, | | / \ 2 / |{ / n2 \ \ 2 / / n2 \ |||| | |----- |{ |---- + 1/2| (-1) |---- - 1/2|! n2::odd |||| \ \n2 = 0 \{ \ 2 / \ 2 / //// /n - 1 / / 1/2\n |----- | | 5 | | \ | |- 1/2 - ----| | ) | \ 2 / | / | |----- | \n1 = 0 \ /n1 - 1 /{ / n2 \ \\\ |----- / 1/2\(-n2 - 1) |{ |----| ||| n1 1/2 (-n1 - 1) 1/2 n1 | \ | 5 | |{ \ 2 / / n2 \ n2 ||| -2 (-1) (5 + 1) (5 - 1) | ) |- 1/2 + ----| |{ (n2 + 1) (-1/4) |----|! binomial(n2, ----) n2::even||| | / \ 2 / |{ \ 2 / 2 ||| |----- |{ ||| \n2 = 0 \{ 0 n2::odd /// \ /n - 1 / /n1 - 1 \\\ | / 1/2\n |----- | |----- / 1/2\(-n2 - 1) ||| | | 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ | 5 | ||| |, |- 1/2 - ----| | ) |-2 (-1) (5 + 1) (5 - 1) | ) |- 1/2 + ----| (n2 + 2) (n2 + 1) n2!|||} | \ 2 / | / | | / \ 2 / ||| | |----- | |----- ||| / \n1 = 0 \ \n2 = 0 /// "A347947" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n 3 2 n 3 2 n 3 2 n {(-1) , RootOf(_Z - 2 _Z - 5 _Z - 3, index = 1) , RootOf(_Z - 2 _Z - 5 _Z - 3, index = 2) , RootOf(_Z - 2 _Z - 5 _Z - 3, index = 3) } "A347953" LREtools/SearchTable: "SearchTable not successful" {} "A348012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 2 _Z - _Z - 2, index = 1) , RootOf(_Z - 2 _Z - _Z - 2, index = 2) , RootOf(_Z - 2 _Z - _Z - 2, index = 3) } "A348063" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- || | n1 | \ / (n2 + 1) n2! \|| | (n1 + 1) (-1) n1! | ) |- ------------------||| /n - 1 \ |n - 1 | / \ (n2 + 3) (n2 + 1)!/|| |----- n1 | |----- |----- || | \ (n1 + 1) (-1) n1!| | \ \n2 = 0 /| {(n + 2) (n + 1) n!, (n + 2) (n + 1) n! | ) -------------------|, (n + 2) (n + 1) n! | ) ---------------------------------------------------| | / (n1 + 3) (n1 + 1)! | | / (n1 + 3) (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / } "A348189" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(-1 - 1/3 I 3 ) , (-1 + 1/3 I 3 ) , (-1 - 1/3 I 3 ) | ) (-1 + 1/3 I 3 ) (-1 - 1/3 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- n2 1/2 (-n2 - 1) || | \ (-1) (-1 + 1/3 I 3 ) (hypergeom([1/2, -n2 - 1], [1], 4) - hypergeom([1/2, -n2], [1], 4))|| | ) -----------------------------------------------------------------------------------------------------||} | / n2 + 3 || |----- || \n2 = 0 // "A348202" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A348311" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 - 1) (n1 + 1)| {n! | ) ------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A348312" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 3 (n1 + 1)| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A348314" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 4 (n1 + 1)| {n! | ) ------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A348410" LREtools/SearchTable: "SearchTable successful" {(24 (3 n + 2) (3 n + 4) (7 n + 1) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) 3 2 / 2 + (-2797 n - 6213 n - 3602 n - 528) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1) / ((2 n + 1) (4 n + 1) (17 n + 24))} / "A348457" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) n { 0 n::even { 2 (-64) {(-8) , { , { ------------------ n::even} { (n/2 - 1/2) { n binomial(n, n/2) { (-4) binomial(n - 1, n/2 - 1/2) n::odd { { 0 n::odd "A348474" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A348476" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A348597" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ |----- n1 | |----- 1/2 n1| |----- 1/2 n1| | \ (-1) | | \ (1/2 - 1/2 I 3 ) | | \ (1/2 + 1/2 I 3 ) | {n! | ) ---------|, n! | ) --------------------|, n! | ) --------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / "A348618" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { n 3 n { 3 n 3 n { 4 binomial(---, n/2) n::even {{ (3 n - 2) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) , { 2 } { 2 2 { { --------------------------------------------------------------------- n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A348662" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) n { 0 n::even { 2 (-16) {(-4) , { , { ------------------ n::even} { (n/2 - 1/2) { n binomial(n, n/2) { (-1) binomial(n - 1, n/2 - 1/2) n::odd { { 0 n::odd "A348700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 |----- n1 (-6 n1 - 6) 2 | |----- n n | \ 9 2 (13 n1 - 15 n1 - 10) binomial(3 n1, n1)| n | \ n1 (-6 n1 - 6) 2 {64 , 64 | ) ---------------------------------------------------------|, 64 | ) 9 2 (13 n1 - 15 n1 - 10) | / (n1 + 1) (2 n1 + 1) | | / |----- | |----- \n1 = 0 / \n1 = 0 /n1 - 1 \ |----- n2 (-2 n2 - 2) 5 4 3 2 | | \ 2 3 (3 n2 + 1) (3 n2 + 2) (65 n2 - 114 n2 - 643 n2 - 464 n2 - 220 n2 - 400) binomial(2 n2, n2) binomial(3 n2, n2)| | ) ----------------------------------------------------------------------------------------------------------------------------------| | / 2 2 2 | |----- (n2 + 1) (n2 + 2) (13 (n2 + 1) - 15 n2 - 25) (13 n2 - 15 n2 - 10) binomial(3 n2 + 3, n2 + 1) | \n2 = 0 / \ | | binomial(3 n1, n1)/((n1 + 1) (2 n1 + 1))|} | | / "A348793" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A348813" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A348815" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A348818" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A348864" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ---------------- n::even n { binomial(n, n/2) (2 n + 2) n::even { binomial(n, n/2) {2 (n + 1), (n + 1) n, { , { } { (n + 1) binomial(n + 1, n/2 + 1/2) n::odd { (2 n - 2) { 2 (n + 1) { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A348912" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349014" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {1} "A349015" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A349017" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349018" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A349047" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349048" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349087" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ |----- 4 n1| |----- 4 n1| |----- 4 n1| | \ RootOf(_Z + 1, index = 1) | | \ RootOf(_Z + 1, index = 2) | | \ RootOf(_Z + 1, index = 3) | {n! | ) ----------------------------|, n! | ) ----------------------------|, n! | ) ----------------------------|, | / (n1 + 1)! | | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / /n - 1 \ |----- 4 n1| | \ RootOf(_Z + 1, index = 4) | n! | ) ----------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A349088" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- 1/2 n1| |----- 1/2 n1| | \ 1 | | \ (- 1/2 - 1/2 I 3 ) | | \ (- 1/2 + 1/2 I 3 ) | {n! | ) ---------|, n! | ) ----------------------|, n! | ) ----------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / "A349089" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- n1 | |----- n1 | |----- n1 | | \ 1 | | \ (-1) | | \ (-I) | | \ I | {n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / \n1 = 0 / "A349185" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349186" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349253" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349254" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349255" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349256" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349270" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349289" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A349290" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A349299" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A349300" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A349310" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349311" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A349312" memory used=267678.3MB, alloc=3543.5MB, time=2051.86 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A349318" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349331" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349332" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A349333" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A349334" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A349335" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A349361" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A349362" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A349363" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A349364" memory used=269140.0MB, alloc=3543.5MB, time=2061.02 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A349458" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A349513" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) binomial(2 n1, n1)| {n! | ) -----------------------------|, n!} | / (n1 + 1) (n1 + 1)! | |----- | \n1 = 0 / "A349514" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349531" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349532" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349533" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349534" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349535" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349540" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 35 ) 35 (35 LegendreP(n + 1, 1/35 I 35 ) I - 5 LegendreP(n, 1/35 I 35 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 35 ) 35 (35 LegendreQ(n + 1, 1/35 I 35 ) I - 5 LegendreQ(n, 1/35 I 35 ))} "A349541" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-3 I 7 ) 7 (3 I 7 LegendreP(n + 1, 1/21 I 7 ) - 7 LegendreP(n, 1/21 I 7 )), 1/2 n 1/2 1/2 1/2 1/2 -I (-3 I 7 ) 7 (3 I 7 LegendreQ(n + 1, 1/21 I 7 ) - 7 LegendreQ(n, 1/21 I 7 ))} "A349554" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (2 n1 + 5) (2 n1 + 3) (2 n1 + 1) binomial(2 n1, n1)|| {(-1) , (-1) | ) |- ----------------------------------------------------------||} | / \ (n1 + 3) (n1 + 2) (n1 + 1) /| |----- | \n1 = 0 / "A349640" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1 n {-----, (-4) n! LaguerreL(n + 1, -n - 1/2, 1/4)} n + 1 "A349648" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n { 0 n::even { n binomial(2 n, n) (-1) binomial(2 n, n) { { 2 8 {----------------, ----------------------, { (n - 1) , { -------------------------- n::even} n + 1 n + 1 { 2 2 binomial(n - 1, n/2 - 1/2) { n (n + 1) binomial(n, n/2) { ------------------------------------- n::odd { { n + 1 { 0 n::odd "A349713" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A349768" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+4)*(n+3)*E^3-(2*n+5)*(19*n^2+77*n+72)*E^2+(2*n+3)*(19*n^2+75*n+68)*E-(n+1)*(2*n+5)*n "to two: Symmetric square" (n+2)*E^2+(-10*n-15)*E+5*n+5 LREtools/SearchTable: "SearchTable successful" 1/2 2 1/2 1/2 2 (n + 5) LegendreP(n, 5 ) - 2 5 (n + 1) LegendreP(n, 5 ) %2 + (n + 1) %2 {--------------------------------------------------------------------------------, n 1/2 2 1/2 1/2 2 (n + 5) LegendreQ(n, 5 ) - 2 5 (n + 1) LegendreQ(n, 5 ) %1 + (n + 1) %1 --------------------------------------------------------------------------------, n 1/2 1/2 1/2 1/2 1/2 1/2 (n + 5) LegendreP(n, 5 ) LegendreQ(n, 5 ) - 5 (n + 1) LegendreQ(n, 5 ) %2 - 5 (n + 1) %1 LegendreP(n, 5 ) + (n + 1) %2 %1 ---------------------------------------------------------------------------------------------------------------------------------------} n 1/2 %1 := LegendreQ(n + 1, 5 ) 1/2 %2 := LegendreP(n + 1, 5 ) "A349781" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) n! {-------------------------------------------------------------, n!} n "A349818" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { / 3 n 3 n {{ 512 |3/2 (3 n + 5) (3 n + 1) (n + 2) hypergeom([-n - 2, - --- - 3], [- --- - 5/2], 1/2) { \ 2 2 2 3 n 3 n \ 3 n - 1/8 (3 n + 2) (9 n + 22 n + 12) hypergeom([-n, - ---], [- --- + 1/2], 1/2)| binomial(3 n, ---)/(n (3 n + 2) (5 n + 8)) , n::even 2 2 / 2 / 3 n 3 n - 64 |3/2 (3 n + 4) (3 n + 1) (3 n + 8) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) \ 2 2 3 2 3 n 3 n \ 3 n + (-81/8 n - 359/8 n - 443/8 n - 141/8) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| binomial(3 n + 3, --- + 3/2)/(n (n + 1) (5 n + 13) 2 2 / 2 { n / 3 n 3 n ) , n::odd, { - 4 64 (3 n + 2) |3/2 (3 n + 4) (3 n + 1) (3 n + 8) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) { \ 2 2 3 2 3 n 3 n \ / / 2 + (-81/8 n - 359/8 n - 443/8 n - 141/8) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| / |n (n + 1) (3 n + 1) (5 n + 13) 2 2 / / \ 3 n \ binomial(3 n, ---)| , n::even 2 / (6 n - 6) / 3 n 3 n 32 2 (3 n - 1) |3/2 (3 n + 5) (3 n + 1) (n + 2) hypergeom([-n - 2, - --- - 3], [- --- - 5/2], 1/2) \ 2 2 2 3 n 3 n \ / / 2 - 1/8 (3 n + 2) (9 n + 22 n + 12) hypergeom([-n, - ---], [- --- + 1/2], 1/2)| / |n (3 n - 2) (3 n + 2) (5 n + 8) 2 2 / / \ 3 n \ binomial(3 n - 3, --- - 3/2)| , n::odd} 2 / "A349819" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { {{ - 4096 { / 3 n 3 n 3 n 3 n \ |3 (3 n + 5) (3 n + 1) hypergeom([-n - 2, - --- - 3], [- --- - 5/2], 1/2) - 1/2 (2 n + 3) (3 n + 2) hypergeom([-n, - ---], [- --- + 1/2], 1/2)| \ 2 2 2 2 / 3 n binomial(3 n, ---)/(n (3 n + 2) (5 n + 8)) , n::even 2 / 3 n 3 n - 256 |3 (3 n + 8) (3 n + 4) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) \ 2 2 2 3 n 3 n \ 3 n { + (-17/4 n - 17 n - 63/4) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| binomial(3 n + 3, --- + 3/2)/(n (n + 1) (5 n + 13)) , n::odd, { - 2 2 / 2 { n / 3 n 3 n 16 64 (3 n + 2) |3 (3 n + 8) (3 n + 4) hypergeom([-n - 3, - --- - 9/2], [- --- - 4], 1/2) \ 2 2 2 3 n 3 n \ / / 2 3 n \ + (-17/4 n - 17 n - 63/4) hypergeom([-n - 1, - --- - 3/2], [- --- - 1], 1/2)| / |n (n + 1) (3 n + 1) (5 n + 13) binomial(3 n, ---)| , 2 2 / / \ 2 / n::even (6 n - 6) - 256 2 (3 n - 1) / 3 n 3 n 3 n 3 n \ |3 (3 n + 5) (3 n + 1) hypergeom([-n - 2, - --- - 3], [- --- - 5/2], 1/2) - 1/2 (2 n + 3) (3 n + 2) hypergeom([-n, - ---], [- --- + 1/2], 1/2)| \ 2 2 2 2 / / / 2 3 n \ / |n (3 n - 2) (3 n + 2) (5 n + 8) binomial(3 n - 3, --- - 3/2)| , n::odd} / \ 2 / "A349834" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ (-1) 2 binomial(2 n1, n1)| {4 , 4 | ) --------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A349844" LREtools/SearchTable: "SearchTable successful" n 8 ((4 n + 4) hypergeom([-1/4, -n - 3/4], [5/4], -1) + (-4 n - 3) hypergeom([-1/4, -n + 1/4], [5/4], -1)) {---------------------------------------------------------------------------------------------------------} 4 n - 1 "A349845" LREtools/SearchTable: "SearchTable successful" n (-8) ((2 n + 2) hypergeom([3/4, -n - 3/4], [5/4], 3) + (4 n + 1) hypergeom([3/4, -n + 1/4], [5/4], 3)) {-------------------------------------------------------------------------------------------------------} 4 n - 1 "A349936" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A350114" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A350265" LREtools/SearchTable: "SearchTable successful" ((3 n + 4) hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) - 8 hypergeom([-n, -n, -n], [1, 1], -1)) (n + 1) {-----------------------------------------------------------------------------------------------------------} 2 (n + 3) (n + 2) n "A350267" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 1 {-----, n! (LaguerreL(n + 1, -1) - LaguerreL(n, -1))} n + 1 "A350290" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A350309" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ {1, ) (n1 + 2) (n1 + 1) n1! (2 n1 + 3)} / ----- n1 = 0 "A350383" LREtools/SearchTable: "SearchTable successful" n {(-3) GAMMA(n - 1/3) GAMMA(n + 1/3) ((18 n + 15) hypergeom([n - 1/3, -n - 4/3], [-1/3], 1/9) + (-30 n - 17) hypergeom([n - 4/3, -n - 1/3], [-1/3], 1/9))/(GAMMA(n + 1) GAMMA(n + 4/3) n ), (-3) GAMMA(n - 1/3) GAMMA(n + 1/3) (n + 1) n (3 (-1) (6 n + 5) hypergeom([n - 1/3, -n - 4/3], [-1/3], 8/9) - (-1) (30 n + 17) hypergeom([n - 4/3, -n - 1/3], [-1/3], 8/9))/( GAMMA(n + 1) GAMMA(n + 4/3))} "A350406" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A350407" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A350461" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -1) + (-n - 2) LaguerreL(n, -1)) binomial(2 n, n) n! {------------------------------------------------------------------------------} n "A350599" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A350645" LREtools/SearchTable: "SearchTable successful" ((2 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4) + (-2 n - 2) hypergeom([n, -n], [-n + 1/2], 1/4)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------} 2 n - 1 "A350653" (2 n + 1) binomial(2 n, n) {--------------------------, n + 1} n + 2 "A350851" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" /{ n1 \ |{ 4 | |{ 1/2 --------------------------- n1::even| /{ n1 \ n - 1 |{ n1 | n - 1 |{ 8 binomial(n1, ----) (n1 + 1) | ----- |{ (n1 + 1) binomial(n1, ----) | ----- |{ 2 | n \ |{ 2 | \ |{ ----------------------------- n1::even| {1, 2 , ) |{ |, ) |{ n1 + 2 |} / |{ (2 n1 - 2) | / |{ | ----- |{ 2 2 (n1 + 1) | ----- |{ n1 | n1 = 0 |{ ---------------------------------------- n1::odd | n1 = 0 |{ 2 binomial(n1 + 1, ---- + 1/2) n1::odd | |{ n1 | \{ 2 / |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | \{ 2 / "A350855" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1 - 1 | {(n + 1) n!, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A351046" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (- n/2 + 1/2) n | 2 1/2 2 | {-2 (-1) (n + 1) (n + 2) |2 hypergeom([-n - 1, 1 + ----], [2], 2) - 2 (n + 1) hypergeom([-n, 1 + ----], [2], 2)| n!} \ 2 2 / "A351531" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 {(-1/2 I 6 ) HermiteH(n + 1, 1/6 I 6 )} "A351767" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A351825" LREtools/SearchTable: "SearchTable successful" 2 3 2 ((n + 1) (n + 2 n - 1) LaguerreL(n + 1, -1) + (-n - 6 n - 2 n + 2) LaguerreL(n, -1)) n! {------------------------------------------------------------------------------------------} n (n - 1) (n - 2) "A351858" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A351929" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A351930" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A351932" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A351990" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (- n/2) { (n/2) { 2 n (n + 1) binomial(n, n/2) (n/2)! n::even { 2 2 (n/2)! n::even {{ , { } { (- n/2 + 1/2) { (n/2 + 1/2) { 2 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd { n 2 (n/2 + 1/2)! n::odd "A352027" (2 n + 1) binomial(2 n, n) 2 {--------------------------, n + n + 1} n + 1 "A352275" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ /3125\n2 {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) |----| | / | / \108 / |----- |----- \n1 = 0 \n2 = 0 GAMMA(n2 + 3/5) GAMMA(n2 + 4/5) GAMMA(n2 + 2/5) GAMMA(n2 + 6/5) 7 6 5 4 3 2 / (496831 n2 + 3959499 n2 + 13088161 n2 + 23161029 n2 + 23584900 n2 + 13750380 n2 + 4229280 n2 + 527040) / (GAMMA(n2 + 3) GAMMA(n2 + 7/3) / \\ || 1/2 (n2 + 1) || GAMMA(n2 + 5/2) GAMMA(n2 + 8/3) (1/2 + 1/2 I 3 ) )||} || || // "A352276" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ /823543\n2 {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) |------| | / | / \ 6912 / |----- |----- \n1 = 0 \n2 = 0 11 10 GAMMA(n2 + 3/7) GAMMA(n2 + 2/7) GAMMA(n2 + 8/7) GAMMA(n2 + 6/7) GAMMA(n2 + 4/7) GAMMA(n2 + 5/7) (18177797821 n2 + 217964994617 n2 9 8 7 6 5 4 3 + 1160421778812 n2 + 3614184401658 n2 + 7302284510985 n2 + 10027664867169 n2 + 9527316464198 n2 + 6246597580012 n2 + 2761962880824 n2 2 / + 781914821184 n2 + 127120129920 n2 + 8955878400) / (GAMMA(n2 + 3) GAMMA(n2 + 9/4) GAMMA(n2 + 7/3) GAMMA(n2 + 11/4) GAMMA(n2 + 5/2) / \\ || 1/2 (n2 + 1) || GAMMA(n2 + 8/3) (1/2 + 1/2 I 3 ) )||} || || // "A352373" LREtools/SearchTable: "SearchTable not successful" {} "A352385" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A352526" n n (-2) n! 2 n! (2 n + 1) {--------, ---------------} n n "A352654" LREtools/SearchTable: "SearchTable successful" 2 {(6 (n + 1) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-34 n - 33 n - 9) hypergeom([-n, -n, -n], [1, -2 n], 1)) / 2 binomial(2 n, n) / n } / "A352655" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (n + 1) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------} n + 1 "A352659" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- 1/2 n1| |----- 1/2 n1| | \ 1 | | \ (- 1/2 - 1/2 I 3 ) | | \ (- 1/2 + 1/2 I 3 ) | {n! | ) ---------|, n! | ) ----------------------|, n! | ) ----------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / "A352660" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ /n - 1 \ /n - 1 \ |----- | |----- n1 | |----- n1 | |----- n1 | | \ 1 | | \ (-1) | | \ (-I) | | \ I | {n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n! | ) ---------|, n!} | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| | / (n1 + 1)!| |----- | |----- | |----- | |----- | \n1 = 0 / \n1 = 0 / \n1 = 0 / \n1 = 0 / "A352916" memory used=270689.5MB, alloc=3543.5MB, time=2071.20 LREtools/SearchTable: "SearchTable successful" n 3 2 4 3 2 {(-1) ((n + 1) (41 n - 27 n - 164 n + 120) hypergeom([1/2, -n - 1], [1], 4) + (121 n - 34 n - 463 n + 76 n + 120) hypergeom([1/2, -n], [1], 4)) /(n (n - 1) (n - 2))} "A352972" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 6 } "A353133" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 (-1) ((8 n + 27 n + 16 n - 6) hypergeom([1/2, -n - 1], [1], 4) + (-8 n - 23 n - 8 n - 6) hypergeom([1/2, -n], [1], 4)) {---------------------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n "A353182" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ n1 \\ | |{ 12 || | |{ 1/2 --------------------------- n1::even|| |n - 1 |{ n1 || |----- |{ (n1 + 1) binomial(n1, ----) || n n | \ (-n1 - 1) |{ 2 || {10 , 10 | ) 10 |{ ||, | / |{ (n1 - 1) || |----- |{ 9 12 (n1 + 1) || |n1 = 0 |{ - ---------------------------------------- n1::odd || | |{ n1 || | |{ n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // / /{ n1 n1 \\ |n - 1 |{ 36 3 (n1 + 1) binomial(n1, ----) || |----- |{ 2 || n | \ (-n1 - 1) |{ - ---------------------------------- n1::even|| 10 | ) 10 |{ n1 + 2 ||} | / |{ || |----- |{ (n1 + 1) n1 || |n1 = 0 |{ 2 3 binomial(n1 + 1, ---- + 1/2) n1::odd || \ \{ 2 // "A353183" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ n2 n2 \\ | |{ 4 3 (n2 + 1) binomial(n2, ----) || n - 1 |n1 - 1 |{ 2 || ----- |----- |{ - --------------------------------- n2::even|| n \ n1 | \ (-n2 - 1) |{ n2 + 2 || {1, 10 , ) 10 | ) 10 |{ ||, / | / |{ (n2 + 1) n2 || ----- |----- |{ 2 3 binomial(n2 + 1, ---- + 1/2) (n2 + 2) || n1 = 0 |n2 = 0 |{ 2 || | |{ ------------------------------------------------- n2::odd || \ \{ n2 + 3 // / /{ n2 \\ | |{ 12 (n2 + 2) || | |{ 1/2 ------------------------------------ n2::even|| n - 1 |n1 - 1 |{ n2 || ----- |----- |{ (n2 + 1) (n2 + 3) binomial(n2, ----) || \ n1 | \ (-n2 - 1) |{ 2 || ) 10 | ) 10 |{ ||} / | / |{ (n2 - 1) || ----- |----- |{ 12 (n2 + 1) || n1 = 0 |n2 = 0 |{ - ---------------------------------------- n2::odd || | |{ n2 || | |{ n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) || \ \{ 2 // "A353230" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (2 n1 + 1) binomial(2 n1, n1)| {9 , 9 | ) ----------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A353546" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- (-n2 - 1)| \ n1 \ n1 | \ 2 | {1, ) 2 n1!, ) 2 n1! | ) ----------|} / / | / (n2 + 1)! | ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A353547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- (-n2 - 1)| \ n1 \ n1 | \ 3 | {1, ) 3 n1!, ) 3 n1! | ) ----------|} / / | / (n2 + 1)! | ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A353548" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- (-2 n2 - 2)| \ n1 \ n1 | \ 2 | {1, ) 4 n1!, ) 4 n1! | ) ------------|} / / | / (n2 + 1)! | ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A353549" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ n1 \ n1 | \ 1 | {1, ) (-3) n1!, ) (-3) n1! | ) ----------------------|} / / | / (n2 + 1) | ----- ----- |----- (-3) (n2 + 1)!| n1 = 0 n1 = 0 \n2 = 0 / "A354019" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) 3 n 3 n { { (-3) binomial(3 n, ---) binomial(---, n/2) { (n/2 - 1/2) 3 n { 2 2 {{ (-48) binomial(--- - 3/2, n/2 - 1/2) , { ----------------------------------------------- n::even} { 2 { binomial(n, n/2) (3 n - 1) { ----------------------------------------------- n::odd { { n { 0 n::odd "A354152" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {1, (-1) n BesselI(n, 2), (-1) n BesselK(n, -2)} "A354419" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ n1 \ n1 | \ 1 | {1, ) (-4) n1!, ) (-4) n1! | ) ----------------------|} / / | / (n2 + 1) | ----- ----- |----- (-4) (n2 + 1)!| n1 = 0 n1 = 0 \n2 = 0 / "A354540" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" / / | | |n - 1 |n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) {(1 - 2 ) , (1 + 2 ) , (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) | / | / |----- |----- |n1 = 0 |n2 = 0 | | \ \ /{ / n2 n2 \ \\\ |{ 2 |(2 n2 - 1) hypergeom([-1/2, - ---- - 1], [1], -4) + (-2 n2 - 1) hypergeom([-1/2, - ----], [1], -4)| ||| |{ \ 2 2 / ||| |{ ------------------------------------------------------------------------------------------------------ n2::even||| |{ n2 ||| 1/2 n |{ |||, (1 - 2 ) |{ / n2 n2 \ ||| |{ 4 |(2 n2 + 1) hypergeom([-1/2, - ---- - 3/2], [1], -4) + (-2 n2 - 3) hypergeom([-1/2, - ---- - 1/2], [1], -4)| ||| |{ \ 2 2 / ||| |{ -------------------------------------------------------------------------------------------------------------- n2::odd ||| \{ n2 + 1 /// /n - 1 /n1 - 1 /{ |----- |----- |{ | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) |{ | / | / |{ |----- |----- |{ \n1 = 0 \n2 = 0 \{ / / n2 \ / n2 \ \ | |---- + 1| |----| | | \ 2 / n2 \ 2 / n2 | 2 |5 (2 n2 - 1) hypergeom([3/2, - ---- - 1], [1], 4/5) - 5 (2 n2 + 1) hypergeom([3/2, - ----], [1], 4/5)| \ 2 2 / ------------------------------------------------------------------------------------------------------------------------- , n2::even n2 / / n2 \ / n2 \ \ | |---- + 3/2| |---- + 1/2| | | \ 2 / n2 \ 2 / n2 | 4 |5 (2 n2 + 1) hypergeom([3/2, - ---- - 3/2], [1], 4/5) - 5 (2 n2 + 3) hypergeom([3/2, - ---- - 1/2], [1], 4/5)| \ 2 2 / ----------------------------------------------------------------------------------------------------------------------------------------- , n2 + 1 / / | | \\\ |n - 1 |n1 - 1 ||| |----- |----- ||| 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |||, (1 - 2 ) | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) ||| | / | / n2::odd||| |----- |----- /// |n1 = 0 |n2 = 0 | | \ \ /{ / n2 n2 \ \\\ |{ 2 |(2 n2 + 1) hypergeom([-1/2, - ---- - 3/2], [1], -4) + (-2 n2 - 3) hypergeom([-1/2, - ---- - 1/2], [1], -4)| ||| |{ \ 2 2 / ||| |{ -------------------------------------------------------------------------------------------------------------- n2::even||| |{ n2 + 1 ||| 1/2 n |{ |||, (1 - 2 ) |{ n2 n2 ||| |{ (2 n2 - 1) hypergeom([-1/2, - ---- - 1], [1], -4) + (-2 n2 - 1) hypergeom([-1/2, - ----], [1], -4) ||| |{ 2 2 ||| |{ -------------------------------------------------------------------------------------------------- n2::odd ||| \{ n2 /// /n - 1 /n1 - 1 /{ |----- |----- |{ | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) |{ | ) (1 + 2 ) (1 - 2 ) | ) (1 + 2 ) |{ | / | / |{ |----- |----- |{ \n1 = 0 \n2 = 0 \{ / / n2 \ / n2 \ \ | |---- + 3/2| |---- + 1/2| | | \ 2 / n2 \ 2 / n2 | 2 |5 (2 n2 + 1) hypergeom([3/2, - ---- - 3/2], [1], 4/5) - 5 (2 n2 + 3) hypergeom([3/2, - ---- - 1/2], [1], 4/5)| \ 2 2 / ----------------------------------------------------------------------------------------------------------------------------------------- , n2 + 1 n2::even / n2 \ / n2 \ \\\ |---- + 1| |----| ||| \ 2 / n2 \ 2 / n2 ||| 5 (2 n2 - 1) hypergeom([3/2, - ---- - 1], [1], 4/5) - 5 (2 n2 + 1) hypergeom([3/2, - ----], [1], 4/5) |||} 2 2 ||| --------------------------------------------------------------------------------------------------------------------- , n2::odd||| n2 /// "A354588" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A354690" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" n 1/2 n 1/2 n {2 n!, (- 1/7 - 3/7 I 3 ) n!, (- 1/7 + 3/7 I 3 ) n!} "A354733" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A354734" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A354735" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A354736" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A355127" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 3 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 1/2 n 1/2 1/2 1/2 {8 (n + 2) , (-4 I 2 ) ((-2 n - 1) LegendreP(n, 1/4 I 2 ) + 2 I 2 (n + 1) LegendreP(n + 1, 1/4 I 2 )) (n + 2), 1/2 n 1/2 1/2 1/2 (-4 I 2 ) ((-2 n - 1) LegendreQ(n, 1/4 I 2 ) + 2 I 2 (n + 1) LegendreQ(n + 1, 1/4 I 2 )) (n + 2)} "A355171" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- | | \ 2 n1! | | \ n1! (n1 + 3) | {(n + 1) n!, (n + 1) n! | ) ------------------|, (n + 1) n! | ) ------------------|} | / (n1 + 2) (n1 + 1)!| | / (n1 + 2) (n1 + 1)!| |----- | |----- | \n1 = 0 / \n1 = 0 / "A355258" n n! 2 n! (n - 2) {---------, -------------} (n - 1) n (n - 1) n "A355268" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { n1 / n1 \ \ | { | | { 2 |----|! n1::even| |n - 1 { / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) ---------------------------------------------------------------|, n! | ) -----------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A355293" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 2 3 n {(5/6 + 1/12 RootOf(_Z + 9 _Z + 34, index = 1) - 1/12 RootOf(_Z + 9 _Z + 34, index = 1)) n!, 3 2 3 n (5/6 + 1/12 RootOf(_Z + 9 _Z + 34, index = 2) - 1/12 RootOf(_Z + 9 _Z + 34, index = 2)) n!, 3 2 3 n (5/6 + 1/12 RootOf(_Z + 9 _Z + 34, index = 3) - 1/12 RootOf(_Z + 9 _Z + 34, index = 3)) n!} "A355372" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- 2 | | \ 2 n1! | | \ n1! (n1 + 7 n1 + 14) | {n! (n + 1) (n + 2), n! (n + 1) (n + 2) | ) ---------------------------|, n! (n + 1) (n + 2) | ) ---------------------------|} | / (n1 + 1)! (n1 + 2) (n1 + 3)| | / (n1 + 1)! (n1 + 2) (n1 + 3)| |----- | |----- | \n1 = 0 / \n1 = 0 / "A355407" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 n1! | {(n + 3) (n + 2) (n + 1) n!, (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|, | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / /n - 1 \ |----- 2 | | \ n1! (n1 + 5) (n1 + 7 n1 + 18) | (n + 3) (n + 2) (n + 1) n! | ) ------------------------------------|} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A355414" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 2 n1! | {(n + 4) (n + 3) (n + 2) (n + 1) n!, (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------------------------------------------|, | / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / /n - 1 \ |----- 4 3 2 | | \ n1! (n1 + 18 n1 + 131 n1 + 474 n1 + 744) | (n + 4) (n + 3) (n + 2) (n + 1) n! | ) ---------------------------------------------|} | / (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)!| |----- | \n1 = 0 / "A355481" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 binomial(2 n, n) (n + 1) hypergeom([-n - 1, -n - 1, -n - 1, -n - 1], [1, 1, 1], 1) + (-16 n + 8) hypergeom([-n, -n, -n, -n], [1, 1, 1], 1) {-----------------, -------------------------------------------------------------------------------------------------------------------------} 2 n + 1 (n + 1) "A355989" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 (n/2 + 1) (n/2)! n::even { binomial(n, n/2) (n/2)! (n + 1) n::even {{ , { } { (n + 1) { (n + 2) n (n/2 - 1/2)! binomial(n - 1, n/2 - 1/2) n::odd { 1/2 2 (n/2 + 1/2)! n::odd "A356012" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 2 1/2 n 1/2 1/2 n 1/2 {n! (3 n + 12 n + 11), (- 1/2 - 1/2 I 3 ) n! (3 I + 3 n + 6), (- 1/2 + 1/2 I 3 ) n! (-3 I + 3 n + 6)} "A356118" binomial(2 n, n) {----------------, n!} n + 1 "A356258" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A356521" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A356684" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 | \ n1! (n1 + 1) (n1 + 2) | 2 {(n + n - 1) | ) -------------------------------|, n + n - 1} | / 2 2 | |----- ((n1 + 1) + n1) (n1 + n1 - 1)| \n1 = 0 / "A356858" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { n 13 17 { 5 GAMMA(n/4 + 1/20) GAMMA(n/4 + 9/20) GAMMA(n/4 + --) GAMMA(n/4 + --) irem(n, 4) = 0 { 20 20 { { n { 8000 5 GAMMA(n/4 + 4/5) GAMMA(n/4 + 6/5) GAMMA(n/4 + 7/5) GAMMA(n/4 + 8/5) { --------------------------------------------------------------------------- irem(n, 4) = 1 { (5 n + 4) (5 n + 8) (5 n + 12) { {{ n 11 19 23 27 , { 400 5 GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) { 20 20 20 20 { ---------------------------------------------------------------------- irem(n, 4) = 2 { (5 n + 3) (5 n + 7) { { (n + 1) 11 { 4 5 GAMMA(n/4 + 3/10) GAMMA(n/4 + 7/10) GAMMA(n/4 + 9/10) GAMMA(n/4 + --) { 10 { -------------------------------------------------------------------------------- irem(n, 4) = 3 { 5 n + 2 { n 11 { 4 5 GAMMA(n/4 + 3/10) GAMMA(n/4 + 7/10) GAMMA(n/4 + 9/10) GAMMA(n/4 + --) { 10 { -------------------------------------------------------------------------- irem(n, 4) = 0 { 5 n + 2 { { (n - 1) 13 17 { 5 GAMMA(n/4 + 1/20) GAMMA(n/4 + 9/20) GAMMA(n/4 + --) GAMMA(n/4 + --) irem(n, 4) = 1 { 20 20 { , { n { 1600 5 GAMMA(n/4 + 4/5) GAMMA(n/4 + 6/5) GAMMA(n/4 + 7/5) GAMMA(n/4 + 8/5) { --------------------------------------------------------------------------- irem(n, 4) = 2 { (5 n + 4) (5 n + 8) (5 n + 12) { { (n + 1) 11 19 23 27 { 16 5 GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) { 20 20 20 20 { --------------------------------------------------------------------------- irem(n, 4) = 3 { (5 n + 3) (5 n + 7) { n 11 19 23 27 { 16 5 GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) { 20 20 20 20 { --------------------------------------------------------------------- irem(n, 4) = 0 { (5 n + 3) (5 n + 7) { { (n - 1) 11 { 4 5 GAMMA(n/4 + 3/10) GAMMA(n/4 + 7/10) GAMMA(n/4 + 9/10) GAMMA(n/4 + --) { 10 { -------------------------------------------------------------------------------- irem(n, 4) = 1, { 5 n + 2 { { (n - 2) 13 17 { 5 GAMMA(n/4 + 1/20) GAMMA(n/4 + 9/20) GAMMA(n/4 + --) GAMMA(n/4 + --) irem(n, 4) = 2 { 20 20 { { (n + 1) { 64 5 GAMMA(n/4 + 4/5) GAMMA(n/4 + 6/5) GAMMA(n/4 + 7/5) GAMMA(n/4 + 8/5) { ------------------------------------------------------------------------------- irem(n, 4) = 3 { (5 n + 4) (5 n + 8) (5 n + 12) { n { 64 5 GAMMA(n/4 + 4/5) GAMMA(n/4 + 6/5) GAMMA(n/4 + 7/5) GAMMA(n/4 + 8/5) { ------------------------------------------------------------------------- irem(n, 4) = 0 { (5 n + 4) (5 n + 8) (5 n + 12) { { (n - 1) 11 19 23 27 { 16 5 GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) GAMMA(n/4 + --) { 20 20 20 20 { --------------------------------------------------------------------------- irem(n, 4) = 1 { (5 n + 3) (5 n + 7) } { { (n - 2) 11 { 4 5 GAMMA(n/4 + 3/10) GAMMA(n/4 + 7/10) GAMMA(n/4 + 9/10) GAMMA(n/4 + --) { 10 { -------------------------------------------------------------------------------- irem(n, 4) = 2 { 5 n + 2 { { (n - 3) 13 17 { 5 GAMMA(n/4 + 1/20) GAMMA(n/4 + 9/20) GAMMA(n/4 + --) GAMMA(n/4 + --) irem(n, 4) = 3 { 20 20 "A357307" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A357308" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A357479" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) (n1 - 1) n1| {n! | ) --------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A357480" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) n1 (n1 - 1) (n1 - 2)| {n! | ) -----------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A357509" {binomial(2 n, n), binomial(3 n, n)} "A357532" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A357533" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A357558" LREtools/SearchTable: "SearchTable successful" {((4 n + 2) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-3 n - 3) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n)} "A357568" 2 {binomial(2 n, n) , binomial(3 n, n)} "A357570" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A357571" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A357641" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { (n/2) binomial(2 n, n) n { { 2 (-16) {------------------, { (n/2 - 1/2) , { -------------------------- n::even} n + 1 { 2 (-1) binomial(n - 1, n/2 - 1/2) { (n + 1) n binomial(n, n/2) { -------------------------------------------- n::odd { { n + 1 { 0 n::odd "A357642" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 2 { 0 n::even { (n/2) binomial(2 n, n) (n - n + 1) { { 2 (-16) {-----------------------------, { (n/2 - 1/2) , { -------------------------- n::even} (n + 1) (2 n - 1) { 2 (-1) binomial(n - 1, n/2 - 1/2) { (n + 1) n binomial(n, n/2) { -------------------------------------------- n::odd { { n + 1 { 0 n::odd "A357652" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 binomial(2 n, n) (n + 1) hypergeom([-n - 1, -n - 1, -n - 1, -n - 1], [1, 1, 1], 1) + (-16 n + 8) hypergeom([-n, -n, -n, -n], [1, 1, 1], 1) {-----------------, -------------------------------------------------------------------------------------------------------------------------} 2 n + 1 (n + 1) "A357654" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / / |n - 1 | / 1/2\n / 1/2 \n / 1/2\n |----- | | 5 | |5 | | 5 | | \ | {|1/2 - ----| , |---- + 1/2| , |1/2 - ----| | ) |- \ 2 / \ 2 / \ 2 / | / | |----- | |n1 = 0 | \ \ / /{ 0 irem(n2, 3) = 0\\ |n1 - 1 |{ || |----- / 1/2 \(-n2 - 1) |{ 0 irem(n2, 3) = 1|| n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ || 2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 2 n2 n2 || | / \ 2 / |{ 3 binomial(---- - 4/3, ---- - 2/3) (2 n2 - 1) || |----- |{ 3 3 || |n2 = 0 |{ --------------------------------------------- irem(n2, 3) = 2|| \ \{ (n2 + 1) (n2 + 4) // \\ || || / 1/2\n || | 5 | ||, |1/2 - ----| || \ 2 / || || // / / / /{ 0 irem(n2, 3) = 0\\\\ | | | |{ |||| | | | |{ /2 n2 \ |||| |n - 1 | |n1 - 1 |{ |---- - 2/3| |||| |----- | |----- / 1/2 \(-n2 - 1) |{ \ 3 / n2 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ 2 GAMMA(---- + 5/6) |||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ 3 ||||, | / | | / \ 2 / |{ ------------------------------- irem(n2, 3) = 1|||| |----- | |----- |{ n2 |||| |n1 = 0 | |n2 = 0 |{ GAMMA(---- + 7/3) |||| | | | |{ 3 |||| | | | |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// / 1/2\n | 5 | |1/2 - ----| \ 2 / / / / /{ /2 n2\ \\\\ | | | |{ |----| |||| | | | |{ \ 3 / n2 |||| |n - 1 | |n1 - 1 |{ 2 GAMMA(---- + 5/6) |||| |----- | |----- / 1/2 \(-n2 - 1) |{ 3 |||| | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ |5 | |{ ------------------------- irem(n2, 3) = 0|||| | ) |-2 (-1) (5 - 1) (5 + 1) | ) |---- + 1/2| |{ n2 ||||} | / | | / \ 2 / |{ GAMMA(---- + 7/3) |||| |----- | |----- |{ 3 |||| |n1 = 0 | |n2 = 0 |{ |||| | | | |{ 0 irem(n2, 3) = 1|||| | | | |{ |||| \ \ \ \{ 0 irem(n2, 3) = 2//// "A357655" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / / | | | | | | | | | | | | | | |n - 1 | / 1/2\n / 1/2\ / 1/2 \n / 1/2 \ / 1/2\n |----- | | 5 | | 3 5 | |5 | | 3 5 | | 5 | 1/2 | \ | n1 1/2 (-n1 - 1) {|1/2 - ----| |n + 7/2 - ------|, |---- + 1/2| |n + ------ + 7/2|, |1/2 - ----| (10 n + 35 - 3 5 ) | ) |- 2 (-1) (5 - 1) \ 2 / \ 10 / \ 2 / \ 10 / \ 2 / | / | |----- | \n1 = 0 \ 1/2 n1 (5 + 1) (n1 + 3) (n1 + 5) / /{ 2 n2 n2 /4 n2 \ \\ | |{ binomial(----, ----) |---- + 2| irem(n2, 3) = 0|| | |{ 3 3 \ 3 / || | / 1/2 \(-n2 - 1) / 1/2\ |{ || | |5 | | 3 5 | |{ /2 n2 \ 2 n2 n2 || | |---- + 1/2| |n2 + 9/2 - ------| |{ |---- + 7/3| binomial(---- + 4/3, ---- + 2/3) irem(n2, 3) = 1|| | \ 2 / \ 10 / |{ \ 3 / 3 3 || | |{ || |n1 - 1 |{ 2 n2 n2 || |----- |{ -1/3 binomial(---- + 2/3, ---- + 1/3) (n2 + 4) (2 n2 + 5) irem(n2, 3) = 2|| / | \ \{ 3 3 /| / | | ) ------------------------------------------------------------------------------------------------------------------------------| / | | / (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 6) | / \ |----- | \n2 = 0 / / / | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \\ | | || | | || | | || | | || | | || | | || | | || | | || |n - 1 | / 1/2\ \|| / 1/2\n |----- | | 3 5 | 1/2 ||| | 5 | 1/2 | \ | n1 1/2 (-n1 - 1) 1/2 n1 |n1 + 7/2 - ------| (10 n1 + 45 - 3 5 )|||, |1/2 - ----| (10 n + 35 - 3 5 ) | ) |- 2 (-1) (5 - 1) (5 + 1) (n1 + 3) \ 10 / /|| \ 2 / | / | || |----- | // \n1 = 0 \ / /{ /2 n2\ \\ | |{ |----| || | |{ \ 3 / n2 || | |{ 2 GAMMA(---- + 13/6) || | |{ 3 || | |{ -------------------------- irem(n2, 3) = 0|| | |{ n2 || | |{ GAMMA(5/3 + ----) || | |{ 3 || | |{ || | |{ /2 n2 \ || | |{ |---- - 2/3| || | / 1/2 \(-n2 - 1) / 1/2\ |{ \ 3 / n2 || | |5 | | 3 5 | |{ 2 (-n2 - 4) GAMMA(---- + 11/6) || | |---- + 1/2| |n2 + 9/2 - ------| |{ 3 || | \ 2 / \ 10 / |{ ------------------------------------------ irem(n2, 3) = 1|| | |{ n2 || | |{ GAMMA(---- + 4/3) || | |{ 3 || | |{ || | |{ /2 n2 \ || | |{ |---- - 4/3| || | |{ \ 3 / n2 || | |{ 2 2 GAMMA(---- + 3/2) || | |{ 3 || | |{ --------------------------------- irem(n2, 3) = 2|| |n1 - 1 |{ n2 || |----- |{ GAMMA(---- + 1) || / | \ \{ 3 /| / | (n1 + 5) | ) ---------------------------------------------------------------------------------------------------------------| / | | / (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 6) | / \ |----- | \n2 = 0 / \\ / / || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || | | || |n - 1 | / 1/2\ \|| / 1/2\n |----- | | 3 5 | 1/2 ||| | 5 | 1/2 | \ | n1 1/2 (-n1 - 1) 1/2 n1 |n1 + 7/2 - ------| (10 n1 + 45 - 3 5 )|||, |1/2 - ----| (10 n + 35 - 3 5 ) | ) |- 2 (-1) (5 - 1) (5 + 1) (n1 + 3) \ 10 / /|| \ 2 / | / | || |----- | // \n1 = 0 \ / /{ /2 n2\ \\ | |{ |----| || | |{ \ 3 / n2 || | |{ 2 (-n2 - 4) GAMMA(---- + 11/6) || | |{ 3 || | |{ ------------------------------------ irem(n2, 3) = 0|| | |{ n2 || | |{ GAMMA(---- + 4/3) || | |{ 3 || | |{ || | |{ /2 n2 \ || | |{ |---- - 2/3| || | / 1/2 \(-n2 - 1) / 1/2\ |{ \ 3 / n2 || | |5 | | 3 5 | |{ 2 2 GAMMA(---- + 3/2) || | |---- + 1/2| |n2 + 9/2 - ------| |{ 3 || | \ 2 / \ 10 / |{ --------------------------------- irem(n2, 3) = 1|| | |{ n2 || | |{ GAMMA(---- + 1) || | |{ 3 || | |{ || | |{ /2 n2 \ || | |{ |---- + 2/3| || | |{ \ 3 / n2 || | |{ 2 GAMMA(---- + 13/6) || | |{ 3 || | |{ -------------------------------- irem(n2, 3) = 2|| |n1 - 1 |{ n2 || |----- |{ GAMMA(5/3 + ----) || / | \ \{ 3 /| / | (n1 + 5) | ) ---------------------------------------------------------------------------------------------------------| / | | / (n2 + 3) (n2 + 4) (n2 + 5) (n2 + 6) | / \ |----- | \n2 = 0 / \\ || || || || || || || || || || || || || || || || || || || || || || || || || || / 1/2\ \|| | 3 5 | 1/2 ||| |n1 + 7/2 - ------| (10 n1 + 45 - 3 5 )|||} \ 10 / /|| || // "A357693" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n 1/2 n 1/2 {(-1) GAMMA(n - 2 I), (-1) GAMMA(n + 2 I)} "A357703" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(n - 3 ), GAMMA(n + 3 )} "A357718" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n 1/2 n 1/2 {(-1) GAMMA(n - 3 I), (-1) GAMMA(n + 3 I)} "A357719" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-1) GAMMA(n - 2 I), (-1) GAMMA(n + 2 I)} "A357770" LREtools/SearchTable: "SearchTable successful" n {2 hypergeom([1/2, -n, -n], [1, 1], 4)} "A357771" LREtools/SearchTable: "SearchTable successful" n {2 hypergeom([1/2, -n, -n], [1, 1], 4)} "A357828" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(- 1/2 - 1/2 I 3 + n), GAMMA(n - 1/2 + 1/2 I 3 ), n!} "A357829" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(- 1/2 - 1/2 I 3 + n), GAMMA(n - 1/2 + 1/2 I 3 ), n!} "A357830" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 1/2 {GAMMA(- 1/2 - 1/2 I 3 + n), GAMMA(n - 1/2 + 1/2 I 3 ), n!} "A357831" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 3 3 {GAMMA(n - RootOf(_Z + 2, index = 1)), GAMMA(n - RootOf(_Z + 2, index = 2)), GAMMA(n - RootOf(_Z + 2, index = 3))} "A357832" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 3 3 {GAMMA(n - RootOf(_Z + 2, index = 1)), GAMMA(n - RootOf(_Z + 2, index = 2)), GAMMA(n - RootOf(_Z + 2, index = 3))} "A357833" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 3 3 {GAMMA(n - RootOf(_Z + 2, index = 1)), GAMMA(n - RootOf(_Z + 2, index = 2)), GAMMA(n - RootOf(_Z + 2, index = 3))} "A357834" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n 1/2 n 1/2 {(-1) GAMMA(1/2 - 1/2 I 3 + n), (-1) GAMMA(n + 1/2 + 1/2 I 3 )} "A357835" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n 1/2 n 1/2 {(-1) GAMMA(1/2 - 1/2 I 3 + n), (-1) GAMMA(n + 1/2 + 1/2 I 3 )} "A357836" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n 1/2 n 1/2 {(-1) GAMMA(1/2 - 1/2 I 3 + n), (-1) GAMMA(n + 1/2 + 1/2 I 3 )} "A357847" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A357955" {binomial(2 n, n), binomial(3 n, n), binomial(4 n, n)} "A358072" LREtools/SearchTable: "SearchTable successful" n 2 {(-1/2) (n!) ((3 n + 3) LaguerreL(n + 1, - 5/3 - n, -2/3) - 2 LaguerreL(n, - 2/3 - n, -2/3)) (n + 1)} "A358092" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / | 1/2 n 1/2 n 1/2 n | {(1 - 5 ) , (5 + 1) , (1 - 5 ) | | | \ n - 1 /n1 - 1 \ ----- |----- n2 1/2 (-n2 - 1) | \ 1/2 n1 1/2 (-n1 - 1) | \ (-1) (5 + 1) (hypergeom([1/2, -n2 - 1], [1], 4) - 3 hypergeom([1/2, -n2], [1], 4))| ) (5 + 1) (1 - 5 ) | ) ------------------------------------------------------------------------------------------------| / | / n2 + 2 | ----- |----- | n1 = 0 \n2 = 0 / \ | | |} | | / "A358108" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358109" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358113" LREtools/SearchTable: "SearchTable successful" n {16 ((3 n + 3) hypergeom([1/2, 1/2, -n - 1], [1, 1], 1) + (-3 n - 2) hypergeom([1/2, 1/2, -n], [1, 1], 1)) (n + 1)} "A358114" LREtools/SearchTable: "SearchTable successful" n {16 ((2 n + 2) hypergeom([-1/2, -n - 1], [1], -1) + (-2 n - 3) hypergeom([-1/2, -n], [1], -1))} "A358119" LREtools/SolveLRE: "Reduced the order of" (2*n+3)*(n+6)*(n+4)*E^3-(2*n+5)*(13*n^2+80*n+96)*E^2-(2*n+3)*(13*n^2+24*n-16)*E+(2*n+5)*(n-2)*n "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" n 1/2 2 1/2 2 1/2 2 2 2 (-1) (%3 LegendreP(n, 3 I) + 2 I 3 (n + 1) (n + 4) (n + n - 1) LegendreP(n, 3 I) %2 - (n + n + 4) (n + n - 1) %2 ) {- --------------------------------------------------------------------------------------------------------------------------------, 2 (n - 2) (n - 1) n (n + 2) (n + 3) n 1/2 2 1/2 2 1/2 2 2 2 (-1) (%3 LegendreQ(n, 3 I) + 2 I 3 (n + 1) (n + 4) (n + n - 1) LegendreQ(n, 3 I) %1 - (n + n + 4) (n + n - 1) %1 ) n - --------------------------------------------------------------------------------------------------------------------------------, - (-1) ( 2 (n - 2) (n - 1) n (n + 2) (n + 3) 1/2 1/2 1/2 2 1/2 %3 LegendreP(n, 3 I) LegendreQ(n, 3 I) + 3 (n + 1) (n + 4) (n + n - 1) LegendreP(n, 3 I) %1 I 1/2 2 1/2 2 2 / 2 + 3 (n + 1) (n + 4) (n + n - 1) %2 LegendreQ(n, 3 I) I - (n + n + 4) (n + n - 1) %2 %1) / ((n - 2) (n - 1) n (n + 2) (n + 3))} / 1/2 %1 := LegendreQ(n + 1, 3 I) 1/2 %2 := LegendreP(n + 1, 3 I) 4 3 2 %3 := 7 n + 34 n + 36 n - 15 n - 12 "A358362" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=272170.3MB, alloc=3543.5MB, time=2082.04 /n - 1 \ |----- n1 (-4 n1 - 4) 2 2| n n | \ (-1) 2 (2 n1 + 1) binomial(2 n1, n1) | {16 , 16 | ) ---------------------------------------------------|} | / 2 | |----- (n1 + 1) | \n1 = 0 / "A358363" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) 2| n n | \ (-1) 2 binomial(2 n1, n1) | {16 , 16 | ) ---------------------------------------|} | / 2 | |----- (n1 + 1) | \n1 = 0 / "A358365" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 2| n n | \ 2 (2 n1 + 1) binomial(2 n1, n1) | {16 , 16 | ) --------------------------------------------|} | / 2 | |----- (n1 + 1) | \n1 = 0 / "A358367" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { n 3 n { (n - 1) 3 n 3 n { 8 binomial(---, n/2) n::even {{ 2 (3 n - 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) , { 2 } { 2 2 { { ------------------------------------------------------------------------------ n::odd { 0 n::odd { n binomial(n - 1, n/2 - 1/2) "A358436" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | | \ (2 n1 + 1) binomial(2 n1, n1) | binomial(2 n, n) | ) ---------------------------------------------| | / 2 | |----- (n1 + 2) (n1 + 1) binomial(2 n1 + 2, n1 + 1)| binomial(2 n, n) \n1 = 0 / {----------------, -----------------------------------------------------------------------} n + 1 n + 1 "A358437" LREtools/SearchTable: "SearchTable successful" binomial(2 n, n) (hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4)) {--------------------------------------------------------------------------------------} n + 1 "A358446" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 5 | 1/2 | 5 | 1/2 {|- 1/2 - ----| n! (5 n + 5 - 5 ), |- 1/2 + ----| n! (5 n + 5 + 5 )} \ 2 / \ 2 / "A358493" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358498" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n 1/2 n {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , (- 1/2 - 1/2 I 3 ) /n - 1 /n1 - 1 /n2 - 1 \\\ |----- |----- |----- ||| | \ 1/2 n1 1/2 (-n1 - 1) | \ 1/2 (-n2 - 1) | \ ||| | ) (- 1/2 + 1/2 I 3 ) (- 1/2 - 1/2 I 3 ) | ) (- 1/2 + 1/2 I 3 ) | ) (n3 + 3) (n3 + 2) (n3 + 1) n3!|||} | / | / | / ||| |----- |----- |----- ||| \n1 = 0 \n2 = 0 \n3 = 0 /// "A358499" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" / / n2 - 1 /n3 - 1 \\ \ | | ----- |----- || | | | \ n3 | \ n4 || | | | ) (-1) | ) (-(-1) (n4 + 4) (n4 + 3) (n4 + 2) (n4 + 1) n4!)|| | |n - 1 |n1 - 1 / | / || | |----- |----- ----- |----- || | n n n n | \ n1 | \ n3 = 0 \n4 = 0 /| | {1, (-1) , (-I) , I , (-I) | ) (-1) | ) ------------------------------------------------------------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A358500" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {1, RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z + _Z + _Z + _Z + 1 "A358518" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {2 n + 3} "A358547" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358560" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358585" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n { { 2 16 binomial(2 n, n) { 2 { ---------------------------- n::even {----------------, { 4 binomial(n - 1, n/2 - 1/2) n , { 2 2 } n + 1 { ------------------------------- n::odd { n (n + 1) binomial(n, n/2) { 2 { { (n + 1) { 0 n::odd "A358586" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n { { 2 16 binomial(2 n, n) { 2 { ---------------------------- n::even {----------------, { 4 binomial(n - 1, n/2 - 1/2) n , { 2 2 } n + 1 { ------------------------------- n::odd { n (n + 1) binomial(n, n/2) { 2 { { (n + 1) { 0 n::odd "A358591" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(2 n1, n1) (9 n1 + n1 - 2)| {(-1/2) , (-1/2) | ) -----------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-1/2) | \n1 = 0 / "A358603" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" (-n) {2 HermiteH(n + 1, 1/2)} "A358604" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358605" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358606" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A358607" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | n n n | \ n1 | \ (n2 + 1) (n2 + 2) n2!| | {(-I) , I , (-I) | ) (-1) | ) ---------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A358608" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- n 1/2 n 1/2 n 1/2 n | \ {(-1) , (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) | / |----- \n1 = 0 /n1 - 1 /n2 - 1 \\\ |----- |----- ||| 1/2 n1 1/2 (-n1 - 1) | \ n2 1/2 (-n2 - 1) | \ n3 ||| (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) (-1) (1/2 + 1/2 I 3 ) | ) (-(-1) (n3 + 3) (n3 + 2) (n3 + 1) n3!)|||} | / | / ||| |----- |----- ||| \n2 = 0 \n3 = 0 /// "A358609" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 n 4 n 4 n 4 n {RootOf(_Z + 1, index = 1) , RootOf(_Z + 1, index = 2) , RootOf(_Z + 1, index = 3) , RootOf(_Z + 1, index = 4) } "A358611" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n {(-1) , RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - _Z + _Z - _Z + 1 "A358613" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A358791" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n n | \ 2 (n1 + 1) (3 n1 + 7) n1!| (n + 1) (-1) n! {(n + 1) (1/2) n!, (n + 1) (1/2) n! | ) ---------------------------------|, ----------------} | / (n1 + 2) (n1 + 3) (n1 + 1)! | n + 2 |----- | \n1 = 0 / "A358987" n {1, (1/2) n! binomial(2 n, n)} "A359039" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { (- n/2) { 2 (n/2)! n::even { 2 (n + 1) binomial(n, n/2) (n/2)! n::even {{ , { } { (n/2 - 1/2) { (- n/2 - 1/2) { (n/2 + 1/2) 2 (n/2 - 1/2)! n::odd { 2 2 binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd "A359066" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / / { / n2 \ \ \ | | { |----| | | | | { \ 2 / n2 | | | | { 24 2 binomial(n2, ----) (n2 + 1) | | | | { 2 | | | | { -------------------------------------- n2::even| | | | { n2 + 2 | | | | { | | | | { / n2 \ | | | | { |---- + 1/2| | | | | { \ 2 / n2 | | | | { 2 2 binomial(n2 + 1, ---- + 1/2) (3 n2 + 5) | | |n - 1 |n1 - 1 { 2 | | |----- |----- { ------------------------------------------------------- n2::odd | | n n n | \ n1 | \ { n2 + 3 | | {(-I) , I , (-I) | ) (-1) | ) -------------------------------------------------------------------------| I|, | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / / / { /5 n2\ \ \ | | { |----| | | | | { \ 2 / | | | | { 2 (3 n2 + 5) | | | | { 1/2 ------------------------------------ n2::even| | | | { n2 | | | | { (n2 + 1) (n2 + 3) binomial(n2, ----) | | | | { 2 | | | | { | | | | { /5 n2 \ | | | | { |---- - 5/2| | | | | { \ 2 / | | | | { 6 2 (n2 + 1) | | | | { ---------------------------------------- n2::odd | | |n - 1 |n1 - 1 { n2 | | |----- |----- { n2 (n2 + 2) binomial(n2 - 1, ---- - 1/2) | | n | \ n1 | \ { 2 | | (-I) | ) (-1) | ) ----------------------------------------------------------| I|} | / | / (n2 + 1) | | |----- |----- I | | \n1 = 0 \n2 = 0 / / "A359087" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 1) hypergeom([1/2, -n], [1], 4)), 3 (n + 1)} "A359140" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A359175" (2 n + 3) (2 n + 1) binomial(2 n, n) {------------------------------------, n + 2} (n + 3) (n + 1) "A359176" {n + 1, binomial(2 n, n)} "A359435" (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {----------------------------------------------, n + 6 n + 10} (n + 3) (n + 2) (n + 1) "A359489" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A359643" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A359646" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, 5 n + 1], [1], -1)} "A359758" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A359984" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 {(-1) (3 n + 7 n - 2)} "A360024" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360025" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360026" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A360027" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360045" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360046" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 3} "A360047" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360048" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 5 1/2 5 | ----- 5 |(5 n1 + 5) LegendreQ(n1, ----) + 5 (3 n1 + 7) LegendreQ(n1 + 1, ----)| \ \ 5 5 / {1, ) ----------------------------------------------------------------------------------------, / (n1 + 2) (n1 + 3) ----- n1 = 0 / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 (3 n1 + 7) LegendreP(n1 + 1, ----) + (5 n1 + 5) LegendreP(n1, ----)| \ \ 5 5 / ) ----------------------------------------------------------------------------------------} / (n1 + 2) (n1 + 3) ----- n1 = 0 "A360049" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360050" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 3} "A360051" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360057" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360058" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360059" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {2 n + 3} "A360060" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360076" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(n + 2) (n + 3) (-1) } "A360082" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(n + 4) (n + 3) (n + 2) (-1) } "A360083" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(n + 5) (n + 4) (n + 3) (n + 2) (-1) } "A360084" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(n + 2) (n + 3) (-1) } "A360085" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(n + 4) (n + 3) (n + 2) (-1) } "A360086" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(n + 5) (n + 4) (n + 3) (n + 2) (-1) } "A360100" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360102" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360103" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360133" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360143" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A360144" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 6 _Z + 5 _Z - 1 "A360149" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A360150" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A360151" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - _Z + 2 _Z + 1 "A360153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , (- 1/2 - 1/2 I 3 ) | ) (- 1/2 + 1/2 I 3 ) (- 1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 /n2 - 1 \\\ |----- |----- ||| | \ 1/2 (-n2 - 1) | \ (2 n3 + 5) (2 n3 + 3) (2 n3 + 1) binomial(2 n3, n3)||| | ) (- 1/2 + 1/2 I 3 ) | ) ---------------------------------------------------|||} | / | / (n3 + 3) (n3 + 2) (n3 + 1) ||| |----- |----- ||| \n2 = 0 \n3 = 0 /// "A360168" n {1, 4 , binomial(2 n, n)} "A360185" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | n n n | \ n1 | \ (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)| | {(-I) , I , (-I) | ) (-1) | ) ----------------------------------------| I|} | / | / (n2 + 1) | | |----- |----- (n2 + 1) (n2 + 2) I | | \n1 = 0 \n2 = 0 / / "A360186" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- n 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(-1) , (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 /n2 - 1 \\\ |----- |----- / n3 \||| | \ n2 1/2 (-n2 - 1) | \ | (-1) (2 n3 + 5) (2 n3 + 3) (2 n3 + 1) binomial(2 n3, n3)|||| | ) (-1) (1/2 + 1/2 I 3 ) | ) |- ----------------------------------------------------------||||} | / | / \ (n3 + 3) (n3 + 2) (n3 + 1) /||| |----- |----- ||| \n2 = 0 \n3 = 0 /// "A360211" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z + _Z + 1, index = 1) , RootOf(_Z + _Z + 1, index = 2) , RootOf(_Z + _Z + 1, index = 3) } "A360219" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360266" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360267" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360271" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A360272" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n 1/2 n 1/2 n {(-1) , (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A360273" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 /n1 - 1 \ ----- |----- / n2 \| n \ n1 | \ | (-1) (2 n2 + 3) (2 n2 + 1) binomial(2 n2, n2)|| {1, (-1) , ) (-1) | ) |- -----------------------------------------------||} / | / \ (n2 + 3) (n2 + 2) (n2 + 1) /| ----- |----- | n1 = 0 \n2 = 0 / "A360274" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) , (- 1/2 - 1/2 I 3 ) | ) (- 1/2 + 1/2 I 3 ) (- 1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 /n2 - 1 \\\ |----- |----- ||| | \ 1/2 (-n2 - 1) | \ (2 n3 + 5) (2 n3 + 3) (2 n3 + 1) binomial(2 n3, n3)||| | ) (- 1/2 + 1/2 I 3 ) | ) ---------------------------------------------------|||} | / | / (n3 + 4) (n3 + 3) (n3 + 2) (n3 + 1) ||| |----- |----- ||| \n2 = 0 \n3 = 0 /// "A360290" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360291" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360292" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A360293" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360294" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A360295" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A360309" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360310" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A360313" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360314" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360315" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A360317" LREtools/SearchTable: "SearchTable successful" n 2 ((n + 1) (4 n - 1) hypergeom([-1/2, -n - 1], [1], -2) - n (4 n + 5) hypergeom([-1/2, -n], [1], -2)) {------------------------------------------------------------------------------------------------------} n "A360318" LREtools/SearchTable: "SearchTable successful" n 2 3 ((n + 1) (8 n - 3) hypergeom([-1/2, -n - 1], [1], -4/3) + (-8 n - 9 n + 1) hypergeom([-1/2, -n], [1], -4/3)) {----------------------------------------------------------------------------------------------------------------} n "A360319" LREtools/SearchTable: "SearchTable successful" n 2 4 (2 (n + 1) (2 n - 1) hypergeom([-1/2, -n - 1], [1], -1) + (-4 n - 4 n + 1) hypergeom([-1/2, -n], [1], -1)) {--------------------------------------------------------------------------------------------------------------} n "A360321" LREtools/SearchTable: "SearchTable successful" n 2 3 2 5 ((n + 1) (64 n - 232 n + 75) hypergeom([-3/2, -n - 1], [1], -4/5) + (-64 n + 72 n + 181 n + 15) hypergeom([-3/2, -n], [1], -4/5)) {---------------------------------------------------------------------------------------------------------------------------------------} n "A360322" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-n + 1) hypergeom([-1/2, -n], [1], -4)) {--------------------------------------------------------------------------------------------} n "A360852" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 2 2 | \ (n1 + 1) (n1 + 2)| {(n + 1) (n!) , (n + 1) (n!) | ) -----------------|} | / 2 | |----- ((n1 + 1)!) | \n1 = 0 / "A360854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n2 - 1 \\ | |----- 6 5 4 3 2 || | 2 2 | \ 8 n3 + 74 n3 + 242 n3 + 271 n3 - 161 n3 - 510 n3 - 218|| | (n2 + 1) (n2!) | ) -----------------------------------------------------------|| n - 1 n - 1 |n1 - 1 | / 2 2 || ----- ----- |----- |----- (n3 + 3) (n3 + 2) ((n3 + 1)!) || \ 2 2 \ 2 2 | \ \n3 = 0 /| {1, ) (n1 + 1) (n1!) , ) (n1 + 1) (n1!) | ) -------------------------------------------------------------------------------------|, / / | / 2 | ----- ----- |----- ((n2 + 1)!) | n1 = 0 n1 = 0 \n2 = 0 / n - 1 ----- \ 2 2 ) (n1 + 2) (n1 + 1) (n1!) } / ----- n1 = 0 "A360856" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A360861" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 { -------------------------- n::even { 2 2 { 2 (2 n + 1) binomial(2 n, n) { (n + 1) binomial(n, n/2) { 4 binomial(n, n/2) n::even {--------------------------, { , { } n + 1 { (4 n - 4) { 2 { 4 2 { binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------ n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) "A360878" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361229" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361244" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361245" LREtools/SearchTable: "SearchTable not successful" {} "A361278" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361279" memory used=273610.0MB, alloc=3575.5MB, time=2092.73 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361280" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361283" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361305" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361306" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361313" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A361375" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/3, -n - 1], [1], -9) + (-n - 4) hypergeom([1/3, -n], [1], -9) {----------------------------------------------------------------------------------} n "A361488" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361528" LREtools/SearchTable: "SearchTable successful" (n + 2) (n + 1) n! (LaguerreL(n + 1, -3) - 4 LaguerreL(n, -3)) {--------------------------------------------------------------} n "A361532" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361571" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361596" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361626" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361636" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361637" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361649" LREtools/SearchTable: "SearchTable successful" ((n + 1) LaguerreL(n + 1, -2) + (-n - 3) LaguerreL(n, -2)) n! (n + 1) {---------------------------------------------------------------------} n "A361657" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361673" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361675" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A361699" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361700" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361701" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A361703" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361710" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 3 n 3 n { (-1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------- n::even { 2 { binomial(n, n/2) n {{ , { (n/2 - 1/2) 3 n 3 n { (3 n - 2) (-1) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { - ------------------------------------------------------------------------------------- n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) { (n/2) 3 n { -(-1) binomial(n, n/2) binomial(---, n/2) n::even { 2 { { (n/2 + 1/2) 3 n 2 } { (-1) binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------------------------------------------------- n::odd { (3 n + 1) n "A361711" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 3 n { (-1) binomial(n, n/2) binomial(---, n/2) (3 n + 2) { 2 { 3/2 ------------------------------------------------------- n::even { n + 1 {{ , { (n/2 + 1/2) 3 n { (n - 2) (-1) binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) { 2 { 1/2 --------------------------------------------------------------------------------- n::odd { n { (n/2) 3 n 3 n { 2 (-1) binomial(3 n, ---) binomial(---, n/2) (3 n + 1) (n - 2) { 2 2 { ------------------------------------------------------------------- n::even { 2 2 { binomial(n, n/2) (n + 1) n { } { (n/2 - 1/2) 3 n 3 n { 6 (-1) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) (3 n - 2) (3 n + 2) { 2 2 { ------------------------------------------------------------------------------------------------- n::odd { 2 2 { binomial(n - 1, n/2 - 1/2) (n + 1) n "A361712" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) , 3 3 2 (n + 1) hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + (-41 n - 51 n - 27 n - 5) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1) ----------------------------------------------------------------------------------------------------------------------------------------------} 3 n "A361713" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) , 3 3 2 (n + 1) hypergeom([n + 2, n + 2, -n - 1, -n - 1], [1, 1, 1], 1) + (-35 n - 51 n - 27 n - 5) hypergeom([-n, -n, n + 1, n + 1], [1, 1, 1], 1) ----------------------------------------------------------------------------------------------------------------------------------------------} 3 n "A361714" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" 2 {binomial(2 n, n) , 2 (6 (n + 1) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-34 n - 33 n - 9) hypergeom([-n, -n, -n], [1, -2 n], 1)) / 2 binomial(2 n, n) / n } / "A361716" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) 3 n 3 n { (-1) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------- n::even { 2 { binomial(n, n/2) n {{ , { (n/2 - 1/2) 3 n 3 n { 3 (3 n - 2) (-1) binomial(--- - 3/2, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 2 { - --------------------------------------------------------------------------------------- n::odd { 2 2 { n binomial(n - 1, n/2 - 1/2) { (n/2) 3 n { -3 (-1) binomial(n, n/2) binomial(---, n/2) n::even { 2 { { (n/2 + 1/2) 3 n 2 } { (-1) binomial(n + 1, n/2 + 1/2) binomial(--- + 3/2, n/2 + 1/2) (n + 1) { 2 { ---------------------------------------------------------------------------------- n::odd { (3 n + 1) n "A361717" LREtools/SearchTable: "SearchTable successful" 2 {(6 (n + 1) (2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-34 n - 33 n - 9) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n)/(n (n - 1))} "A361719" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (n/2) { n (n + 1) (-16) { -------------------- n::even { 3 (n/2) { binomial(n, n/2) { 1/4 n (-1) binomial(n, n/2) n::even {{ , { } { (n/2 - 1/2) 2 { 2 (n/2 + 1/2) { 2 (-16) n { 1/8 n (n + 1) (-1) binomial(n + 1, n/2 + 1/2) n::odd { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) "A361726" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361727" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361729" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361730" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A361738" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361739" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361743" LREtools/SearchTable: "SearchTable successful" {(n + 1) LegendreP(n + 1, 3) + (-7 n - 3) LegendreP(n, 3), (n + 1) LegendreQ(n + 1, 3) + (-7 n - 3) LegendreQ(n, 3)} "A361752" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361753" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A361790" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361791" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361792" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361801" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" n - 1 /{ 0 n1::even\ n - 1 /{ n1 \ ----- |{ | ----- |{ 4 | \ |{ n1 | \ |{ --------------------------- n1::even| {1, ) |{ 2 binomial(n1 - 1, ---- - 1/2) n1 |, ) |{ n1 |} / |{ 2 | / |{ (n1 + 1) binomial(n1, ----) | ----- |{ --------------------------------- n1::odd | ----- |{ 2 | n1 = 0 \{ n1 + 1 / n1 = 0 |{ | \{ 0 n1::odd / "A361812" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361813" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361814" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A361815" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361816" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361817" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361841" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361842" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361843" LREtools/SearchTable: "SearchTable successful" n 3 GAMMA(n - 1/3) (3 hypergeom([-n - 1, n - 1/3], [-1/6], -1/4) - 2 hypergeom([-n, n - 4/3], [-1/6], -1/4)) {-----------------------------------------------------------------------------------------------------------} GAMMA(n + 1) "A361844" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361845" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361880" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361881" LREtools/SearchTable: "SearchTable successful" n (-1) ((n + 1) hypergeom([1/3, -n - 1], [1], 9) + (-n + 2) hypergeom([1/3, -n], [1], 9)) {----------------------------------------------------------------------------------------} n "A361882" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361887" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n (-1) ((2 n + 1) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-4 n - 4) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------------------------} 2 (n + 1) "A361895" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361896" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A361932" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A361938" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { (n/2) 2 { (-1) (n/2)! ((1/4 n + 1/2 n - 1) BesselJ(n/2, -2) + (n/2 + 1) BesselJ(n/2 - 1, -2)) n::even { {{ (n/2 + 1/2) , { (-1) (n/2 + 1/2)! ((n + 3) (n - 1) BesselJ(n/2 + 1/2, -2) + (2 n + 2) BesselJ(n/2 - 1/2, -2)) { 1/2 -------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) 2 { (-1) (n/2)! ((1/4 n + 1/2 n - 1) BesselY(n/2, -2) + (n/2 + 1) BesselY(n/2 - 1, -2)) n::even { { (n/2 + 1/2) , { { (-1) (n/2 + 1/2)! ((n + 3) (n - 1) BesselY(n/2 + 1/2, -2) + (2 n + 2) BesselY(n/2 - 1/2, -2)) { { 1/2 -------------------------------------------------------------------------------------------------------- n::odd { n + 1 (n/2) (-1/4) binomial(n, n/2) (n/2)! (1/2 (n + 3) (n - 1) BesselJ(n/2 + 1/2, -2) + BesselJ(n/2 - 1/2, -2) (n + 1)) , n::even (n/2 - 1/2) 2 , { 1/4 (-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n ((n + 2 n - 4) BesselJ(n/2, -2) + (2 n + 4) BesselJ(n/2 - 1, -2)) , n::odd { (n/2) (-1/4) binomial(n, n/2) (n/2)! (1/2 (n + 3) (n - 1) BesselY(n/2 + 1/2, -2) + BesselY(n/2 - 1/2, -2) (n + 1)) , n::even (n/2 - 1/2) 2 } 1/4 (-1/4) binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n ((n + 2 n - 4) BesselY(n/2, -2) + (2 n + 4) BesselY(n/2 - 1, -2)) , n::odd "A361962" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A362154" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z - _Z + 4, index = 1) , RootOf(_Z - _Z + 4, index = 2) , RootOf(_Z - _Z + 4, index = 3) } "A362156" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z - _Z + 4, index = 1) , RootOf(_Z - _Z + 4, index = 2) , RootOf(_Z - _Z + 4, index = 3) } "A362176" LREtools/SearchTable: "SearchTable successful" 1/2 (n/2) 2 {2 HermiteH(n, ----)} 4 "A362177" LREtools/SearchTable: "SearchTable successful" 1/2 (n/2) 3 {3 HermiteH(n, ----)} 6 "A362269" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (n/3) { (n/3) { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 0 { 3 GAMMA(5/3 + n/3) { { ----------------------- irem(n, 3) = 0 { (n/3 + 2/3) { n + 2 { 3 GAMMA(n/3 + 2) { {{ --------------------------- irem(n, 3) = 1, { (n/3 - 1/3) , { n + 3 { 3 GAMMA(n/3 + 4/3) irem(n, 3) = 1 { { { (n/3 + 1/3) { (n/3 + 1/3) { 3 GAMMA(5/3 + n/3) { 3 GAMMA(n/3 + 2) { ----------------------------- irem(n, 3) = 2 { --------------------------- irem(n, 3) = 2 { n + 2 { n + 3 { (n/3) { 1/3 3 (n/3)! irem(n, 3) = 0 { { (n/3 - 1/3) } { 1/3 3 (n/3 - 1/3)! irem(n, 3) = 1 { { (n/3 - 2/3) { (n/3 + 1/3) 3 (n/3 - 2/3)! irem(n, 3) = 2 "A362278" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |6 | 6 {|----| HermiteH(n, ----)} \ 2 / 6 "A362279" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 |10 | 10 {|-----| HermiteH(n, -----)} \ 2 / 10 "A362309" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A362413" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A362595" 2 n binomial(2 n, n) (n + n + 4) {4 , -----------------------------} n + 1 "A362596" 2 n binomial(2 n, n) (n - 3 n + 4) {4 , -------------------------------} n + 1 "A362597" LREtools/SearchTable: "SearchTable successful" (4 n + 3) LegendreP(n + 1, 3) + (-24 n - 9) LegendreP(n, 3) (4 n + 3) LegendreQ(n + 1, 3) + (-24 n - 9) LegendreQ(n, 3) {-----------------------------------------------------------, -----------------------------------------------------------} n n "A362676" LREtools/SearchTable: "SearchTable successful" n 2 2 4 (2 (n + 1) hypergeom([1/2, -n - 1, -n - 1], [1, 1], 1) + (-14 n - 11 n - 3) hypergeom([1/2, -n, -n], [1, 1], 1)) {---------------------------------------------------------------------------------------------------------------------} n (4 n + 1) "A362718" LREtools/SearchTable: "SearchTable successful" (-n) {2 binomial(2 n, n) n! hypergeom([-n], [1/2], 1/2)} "A362741" LREtools/SearchTable: "SearchTable successful" n hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) + (-8 n - 24) hypergeom([-n, -n, -n], [1, 1], -1) {---------------------------------------------------------------------------------------------------} 2 (n + 2) "A362744" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) 2 | n n | \ 2 (3 n1 + 1) binomial(3 n1, n1) (23 n1 + 40 n1 + 12)| {(1/2) , (1/2) | ) -------------------------------------------------------------|} | / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A362819" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A362825" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A363006" LREtools/SearchTable: "SearchTable successful" /46656\n {|-----| (72 (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1) hypergeom([-n - 1, -5 n - 5], [-6 n - 6], -1) \3125 / 4 3 2 - (5 n + 1) (15211 n + 35210 n + 29417 n + 10450 n + 1320) hypergeom([-n, -5 n], [-6 n], -1)) GAMMA(n) GAMMA(1/6 + n) GAMMA(n + 1/2) / n - 1 4 31 3 8675 2 1094 457 \ | --------' k + -- k + ---- k + ---- k + ---- | |' | | 13 4173 1391 4173 | / 2 GAMMA(n + 1/3) GAMMA(n + 2/3) GAMMA(n + 5/6) | | | --------------------------------------| / (GAMMA(n + 1) GAMMA(n + 2/5) GAMMA(n + 3/5) | | | 4 83 3 63566 2 67177 8846| / | | | k + -- k + ----- k + ----- k + ----| \ k = 0 13 4173 4173 1391/ GAMMA(n + 4/5) GAMMA(n + 6/5))} "A363180" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 4 { 4 4 { 16 ((n/2)!) { 4 ((n/2)!) binomial(n, n/2) n::even { ------------- n::even { {{ 2 , { 4 4 } { n { ((n/2 + 1/2)!) binomial(n + 1, n/2 + 1/2) { { ------------------------------------------- n::odd { (4 n - 4) 4 { 2 { 4 2 ((n/2 - 1/2)!) n::odd { n "A363181" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 (-1/2 I 2 ) ((2 n + 2) HermiteH(n + 1, 1/4 I 2 ) - 2 (2 n + 1) HermiteH(n, 1/4 I 2 ) I) {-------------------------------------------------------------------------------------------------} n "A363304" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A363308" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" binomial(2 n, n) (n - 1) {------------------------} (n + 1) (2 n - 1) "A363310" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A363311" memory used=275111.6MB, alloc=3575.5MB, time=2103.47 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A363409" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 2 ) GAMMA(n + 1 + 1/2 I 2 ), (2 I) GAMMA(n + 1 - 1/2 I 2 )} "A363410" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 2 ) GAMMA(n + 1 + 1/2 I 2 ), (2 I) GAMMA(n + 1 - 1/2 I 2 )} "A363411" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 3 ) GAMMA(n + 1 + 1/3 I 3 ), (3 I) GAMMA(n + 1 - 1/3 I 3 )} "A363412" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 3 ) GAMMA(n + 1 + 1/3 I 3 ), (3 I) GAMMA(n + 1 - 1/3 I 3 )} "A363413" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-2 I) GAMMA(n + 1 + 1/2 I), (2 I) GAMMA(n + 1 - 1/2 I)} "A363414" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" n n {(-2 I) GAMMA(n + 1 + 1/2 I), (2 I) GAMMA(n + 1 - 1/2 I)} "A363415" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 5 ) GAMMA(n + 1 + 1/5 I 5 ), (5 I) GAMMA(n + 1 - 1/5 I 5 )} "A363416" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 1/2 n 1/2 {(-I 5 ) GAMMA(n + 1 + 1/5 I 5 ), (5 I) GAMMA(n + 1 - 1/5 I 5 )} "A363448" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 1)} "A363449" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n binomial(2 n, n) {(-1) (n + 1), ----------------} n + 1 "A363555" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A363570" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {1, 2 LegendreP(n, 2), 2 LegendreQ(n, 2)} "A363571" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" {1, (2 n + 2) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8)} "A363582" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" / /{ 0 irem(n1, 3) = 0\\ | |{ || | |{ 0 irem(n1, 3) = 1|| |n - 1 |{ || |----- |{ / n1 \ || n n | \ (-n1 - 1) |{ |---- - 2/3| || {2 , 2 | ) 2 |{ \ 3 / n1 n1 ||, | / |{ (27/4) GAMMA(---- + 1/3) GAMMA(---- + 1) || |----- |{ 3 3 || |n1 = 0 |{ ---------------------------------------------------- irem(n1, 3) = 2|| | |{ n1 n1 || | |{ GAMMA(---- + 7/6) GAMMA(5/3 + ----) || \ \{ 3 3 // / /{ 0 irem(n1, 3) = 0\\ |n - 1 |{ || |----- |{ n1 || n | \ (-n1 - 1) |{ 9 binomial(n1 - 1, ---- - 1/3) n1 || 2 | ) 2 |{ 3 ||, | / |{ --------------------------------- irem(n1, 3) = 1|| |----- |{ (n1 + 2) (2 n1 + 1) || |n1 = 0 |{ || \ \{ 0 irem(n1, 3) = 2// / /{ / n1 \ \\ | |{ |----| || | |{ \ 3 / n1 n1 || |n - 1 |{ (27/4) GAMMA(---- + 1/3) GAMMA(---- + 1) || |----- |{ 3 3 || n | \ (-n1 - 1) |{ ---------------------------------------------- irem(n1, 3) = 0|| 2 | ) 2 |{ n1 n1 ||} | / |{ GAMMA(5/3 + ----) GAMMA(---- + 7/6) || |----- |{ 3 3 || |n1 = 0 |{ || | |{ 0 irem(n1, 3) = 1|| | |{ || \ \{ 0 irem(n1, 3) = 2// "A363812" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(1/2) } "A363816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {n + 1} "A363817" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {(n + 1) (n + 2)} "A363867" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A363868" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A363869" LREtools/SearchTable: "SearchTable not successful" {} "A363870" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A363871" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A363982" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A363985" LREtools/SearchTable: "SearchTable not successful" {} "A364167" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364324" memory used=276553.8MB, alloc=3575.5MB, time=2113.74 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 2 n 2 n {(RootOf(%1, index = 1) + RootOf(%1, index = 1) + 1) n!, (RootOf(%1, index = 2) + RootOf(%1, index = 2) + 1) n!, 2 n (RootOf(%1, index = 3) + RootOf(%1, index = 3) + 1) n!} 3 2 %1 := _Z + _Z + _Z - 1 "A364336" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364337" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A364338" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A364371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A364374" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364394" LREtools/SearchTable: "SearchTable successful" n 2 (-1) (4 (2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) - (3 n + 2) (14 n + 5) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {--------------------------------------------------------------------------------------------------------------------------------------------} n (2 n - 1) (10 n + 7) "A364403" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364422" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | 1/2 |5 | 1/2 {|1/2 - ----| n! (5 + 5 n), |---- + 1/2| n! (-5 + 5 n)} \ 2 / \ 2 / "A364437" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364472" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A364474" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A364475" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364520" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /52706752\(n/2 - 1/2) 13 11 {{ |--------| GAMMA(n/2 + 9/14) GAMMA(n/2 + --) GAMMA(n/2 + 5/14) GAMMA(n/2 + 1/14) GAMMA(n/2 + --) GAMMA(n/2 + 3/14) , { \ 729 / 14 14 { ----------------------------------------------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) GAMMA(1/6 + n/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/3) { /52706752\(n/2) 13 11 { |--------| GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + 1/14) GAMMA(n/2 + 9/14) GAMMA(n/2 + 5/14) GAMMA(n/2 + 3/14) { \ 729 / 14 14 { ----------------------------------------------------------------------------------------------------------------------- n::even} { GAMMA(n/2 + 1/3) GAMMA(n/2 + 1) GAMMA(1/6 + n/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/2) GAMMA(n/2 + 5/6) { { 0 n::odd "A364522" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364523" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A364552" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364589" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364590" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A364593" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364594" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364595" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364623" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {1} "A364625" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364634" LREtools/SearchTable: "SearchTable successful" {LegendreP(n, 3) n, LegendreQ(n, 3) n} "A364641" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364643" LREtools/SearchTable: "SearchTable successful" 2 2 (243 n - 51 n - 114) LegendreP(n, 3) + (-41 n - 15 n + 38) LegendreP(n + 1, 3) {- --------------------------------------------------------------------------------, (n - 1) n (n - 2) 2 2 (243 n - 51 n - 114) LegendreQ(n, 3) + (-41 n - 15 n + 38) LegendreQ(n + 1, 3) - --------------------------------------------------------------------------------} (n - 1) n (n - 2) "A364645" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364646" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364647" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364735" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364739" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364740" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364742" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364743" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364744" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A364747" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364748" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A364758" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364760" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364765" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A364825" LREtools/SearchTable: "SearchTable not successful" {} "A364826" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A364827" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A364849" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { ((n/2)!) n::even { (-n) 2 2 { { 4 binomial(n, n/2) ((n/2)!) (2 n + 2) n::even {{ 2 , { } { 2 ((n/2 + 1/2)!) { (-2 n + 2) 2 2 2 { ----------------- n::odd { 2 n binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd { n + 1 "A364864" LREtools/SearchTable: "SearchTable not successful" {} "A364866" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" memory used=278069.4MB, alloc=3575.5MB, time=2124.37 RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A365026" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" memory used=279197.8MB, alloc=3575.5MB, time=2132.76 { {{ 0 , n::even { (9 n - 9) (-4 n + 4) 3 2 GAMMA(n/2 + 1/3) GAMMA(n/2 + 1/9) GAMMA(n/2 + 1/10) GAMMA(n/2 + 2/3) GAMMA(n/2 + 2/9) GAMMA(n/2 + 3/10) GAMMA(n/2 + 4/9) / 2 2 2 GAMMA(n/2 + 5/9) GAMMA(n/2 + 7/9) GAMMA(n/2 + 7/10) GAMMA(n/2 + 8/9) GAMMA(n/2 + 9/10) / (GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 1/4) / 2 { n (-n) GAMMA(n/2 + 1/5) GAMMA(n/2 + 2/5) GAMMA(n/2 + 3/4) GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5)) , n::odd, { 19683 16 GAMMA(n/2 + 1/3) { GAMMA(n/2 + 1/9) GAMMA(n/2 + 1/10) GAMMA(n/2 + 2/3) GAMMA(n/2 + 2/9) GAMMA(n/2 + 3/10) GAMMA(n/2 + 4/9) GAMMA(n/2 + 5/9) GAMMA(n/2 + 7/9) / 2 2 2 GAMMA(n/2 + 7/10) GAMMA(n/2 + 8/9) GAMMA(n/2 + 9/10) / (GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 1/4) GAMMA(n/2 + 1/5) GAMMA(n/2 + 2/5) / 2 GAMMA(n/2 + 3/4) GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5)) , n::even 0 , n::odd} "A365028" LREtools/SearchTable: "SearchTable not successful" {} "A365085" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365109" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365118" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365178" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365183" LREtools/SearchTable: "SearchTable not successful" {} "A365184" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A365189" LREtools/SearchTable: "SearchTable not successful" {} "A365192" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365193" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365194" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365218" LREtools/SearchTable: "SearchTable not successful" {} "A365252" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A365516" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365574" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-2) } "A365694" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A365696" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A365698" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A365699" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" 4 n 4 n 4 n 4 n {RootOf(_Z + 1, index = 1) , RootOf(_Z + 1, index = 2) , RootOf(_Z + 1, index = 3) , RootOf(_Z + 1, index = 4) } "A365723" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A365724" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365727" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A365732" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A365733" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A365744" LREtools/SearchTable: "SearchTable not successful" {} "A365754" LREtools/SearchTable: "SearchTable successful" n {(-8) (9 (3 n + 4) (3 n + 2) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) + 4 (4 n + 3) (2 n + 3) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1)/((n + 1) (2 n + 1) (2 n + 3) (17 n + 24))} "A365764" LREtools/SearchTable: "SearchTable successful" (42 n + 18) hypergeom([-n - 1, 3 n + 4], [1], -1) + (-21 n - 7) hypergeom([-n, 3 n + 1], [1], -1) {-------------------------------------------------------------------------------------------------} 2 238 n + 289 n + 83 "A365765" LREtools/SearchTable: "SearchTable successful" 2 (340 n + 312 n + 68) hypergeom([-n - 1, 4 n + 5], [1], -1) - 9 (4 n + 1) (2 n + 1) hypergeom([-n, 4 n + 1], [1], -1) {---------------------------------------------------------------------------------------------------------------------} 3 2 4633 n + 8136 n + 4541 n + 798 "A365766" memory used=280595.7MB, alloc=3575.5MB, time=2142.44 LREtools/SearchTable: "SearchTable successful" 3 2 {((14660 n + 20800 n + 9400 n + 1340) hypergeom([-n - 1, 5 n + 6], [1], -1) - 11 (5 n + 1) (5 n + 2) (5 n + 3) hypergeom([-n, 5 n + 1], [1], -1)) / 4 3 2 / (57708 n + 131904 n + 108458 n + 37767 n + 4657)} / "A365772" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365773" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365774" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365775" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A365816" LREtools/SearchTable: "SearchTable successful" n (-1/9) ((2 n + 1) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -3) + (74 n + 26) hypergeom([-3 n, -n], [-2 n], -3)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------} (26 n + 9) (n + 1) "A365817" LREtools/SearchTable: "SearchTable successful" / 1/2\n / | 21 21 | | {|81/8 - --------| | \ 8 / \ 1/2 1/2 \ 343 63 21 1/2 343 63 21 | (600 n + 800) hypergeom([5/6, - 2/3 - n], [5/3], --- + --------) + 3 (27 + 7 21 ) (39 n + 19) hypergeom([5/6, 1/3 - n], [5/3], --- + --------)| 50 50 50 50 / / 1/2\ | 293 63 21 | GAMMA(n + 1/3) |- --- + --------|/GAMMA(n + 2)} \ 50 50 / "A365818" LREtools/SearchTable: "SearchTable successful" / 1/2\n / 1/2 |308 62 31 | | 59582 9548 31 {|--- - --------| |(12150 n + 16200) hypergeom([5/6, - 2/3 - n], [5/3], ----- + ----------) \25 25 / \ 6075 6075 1/2 \ / 1/2\ 1/2 59582 9548 31 | | 53507 9548 31 | + 11 (154 + 31 31 ) (59 n + 29) hypergeom([5/6, 1/3 - n], [5/3], ----- + ----------)| GAMMA(n + 1/3) |- ----- + ----------|/GAMMA(n + 2)} 6075 6075 / \ 2025 2025 / "A365839" LREtools/SearchTable: "SearchTable successful" (2 n + 1) binomial(2 n, n) hypergeom([-n, 3 n + 3], [n + 3], -1) {----------------------------------------------------------------} (n + 1) (n + 2) "A365842" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (3 n1 + 1) (3 n1 + 2) 2 binomial(3 n1, n1) {1, ) --------------------------------------------} / (n1 + 2) (2 n1 + 3) (2 n1 + 1) ----- n1 = 0 "A365971" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366051" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366052" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366081" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366082" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366083" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366084" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366085" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366086" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366089" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A366107" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 4 n 4 n 4 n 4 n {RootOf(3 _Z + 1, index = 1) , RootOf(3 _Z + 1, index = 2) , RootOf(3 _Z + 1, index = 3) , RootOf(3 _Z + 1, index = 4) } "A366112" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A366114" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) 3 GAMMA(n1 - 1/3) / ----- n1 = 0 ((86 n1 + 25) hypergeom([n1 - 1/3, - 4/3 - n1], [-1/3], 1/9) + (-90 n1 - 39) hypergeom([n1 - 4/3, - 1/3 - n1], [-1/3], 1/9))/GAMMA(n1 + 3), n - 1 ----- \ n1 ) 3 GAMMA(n1 - 1/3) / ----- n1 = 0 (n1 + 1) n1 ((-1) (86 n1 + 25) hypergeom([n1 - 1/3, - 4/3 - n1], [-1/3], 8/9) - 3 (-1) (30 n1 + 13) hypergeom([n1 - 4/3, - 1/3 - n1], [-1/3], 8/9)) /GAMMA(n1 + 3)} "A366115" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366118" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A366119" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366236" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) (n + 2)} "A366237" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(n + 2) (n + 3) (-1) } "A366266" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366325" LREtools/SearchTable: "SearchTable successful" n 2 (-1) ((n + 1) (4 n - 3) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3 n - 3) hypergeom([-1/2, -n], [1], -4)) {---------------------------------------------------------------------------------------------------------------} n (n - 1) "A366326" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366356" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A366363" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 n - 1 ----- n1 ----- n1 \ -I (2 I) (LegendreP(n1, I) + LegendreP(n1 + 1, I) I) \ -I (2 I) (LegendreQ(n1, I) + LegendreQ(n1 + 1, I) I) {1, ) ------------------------------------------------------, ) ------------------------------------------------------} / n1 / n1 ----- ----- n1 = 0 n1 = 0 "A366403" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 1) hypergeom([1/2, -n], [1], 4) {1, ---------------------------------------------------------------------------------} n "A366554" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A366555" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A366556" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A366588" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A366589" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(-1) } "A366590" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A366591" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(-1) } "A366592" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(-1) } "A366646" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366694" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366695" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366706" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n {4 } "A366774" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A366950" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A366957" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 3 n 3 n 3 n {(-RootOf(_Z - _Z + 3, index = 1)) n!, (-RootOf(_Z - _Z + 3, index = 2)) n!, (-RootOf(_Z - _Z + 3, index = 3)) n!} "A367027" LREtools/SearchTable: "SearchTable not successful" {} "A367028" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367031" LREtools/SearchTable: "SearchTable not successful" {} "A367042" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367056" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A367071" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367072" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A367073" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A367074" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A367111" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367112" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367113" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367114" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367115" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367233" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A367235" memory used=282119.2MB, alloc=3607.5MB, time=2153.15 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A367330" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2\| n n | \ | (-1) GAMMA(n1 + 4/3) || {27 , 27 | ) |1/27 -----------------------||} | / | 2 || |----- \ GAMMA(n1 + 2) /| \n1 = 0 / "A367331" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2\| n n | \ | (-1) GAMMA(n1 + 2/3) || {27 , 27 | ) |1/27 -----------------------||} | / | 2 || |----- \ GAMMA(n1 + 2) /| \n1 = 0 / "A367332" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2\| n n | \ | GAMMA(n1 + 2/3) || {27 , 27 | ) |1/27 ----------------||} | / | 2 || |----- \ GAMMA(n1 + 2) /| \n1 = 0 / "A367333" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 2\| n n | \ | GAMMA(n1 + 4/3) || {27 , 27 | ) |1/27 ----------------||} | / | 2 || |----- \ GAMMA(n1 + 2) /| \n1 = 0 / "A367389" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ / n1 \ \\ | |{ |----| || | |{ \ 2 / 3 n1 n1 || | |{ 2 2 binomial(----, ----) (3 n1 + 2) || |n - 1 |{ 2 2 || |----- |{ ----------------------------------------- n1::even|| n n | \ (-2 n1 - 2) |{ (n1 + 1) (n1 + 2) || n {4 , 4 | ) 2 |{ ||, 4 | / |{ / n1 \ || |----- |{ |---- + 1/2| || |n1 = 0 |{ \ 2 / 3 n1 n1 || | |{ 2 binomial(---- + 3/2, ---- + 1/2) || | |{ 2 2 || | |{ ---------------------------------------------- n1::odd || \ \{ n1 + 2 // / /{ / 3 n1\ \\ | |{ |- ----| || | |{ \ 2 / 3 n1 3 n1 n1 || | |{ 2 2 (3 n1 + 1) binomial(3 n1, ----) binomial(----, ----) || | |{ 2 2 2 || | |{ ---------------------------------------------------------------- n1::even|| |n - 1 |{ n1 || |----- |{ (n1 + 1) (n1 + 2) binomial(n1, ----) || | \ (-2 n1 - 2) |{ 2 || | ) 2 |{ || | / |{ / 3 n1 \ || |----- |{ |- ---- + 3/2| || |n1 = 0 |{ \ 2 / 3 n1 3 n1 n1 || | |{ 4 2 (3 n1 - 2) (3 n1 + 2) binomial(3 n1 - 3, ---- - 3/2) binomial(---- - 3/2, ---- - 1/2) || | |{ 2 2 2 || | |{ ------------------------------------------------------------------------------------------------------- n1::odd || | |{ n1 || | |{ n1 (n1 + 1) (n1 + 2) binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // } "A367393" LREtools/SearchTable: "SearchTable successful" n 2 3 2 {(-1) (2 (2 n + 3) (2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-42 n - 111 n - 78 n - 15) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n)/(n (n - 1) (10 n + 7))} "A367415" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367462" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A367548" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (-1) binomial(2 n1, n1) (4 n1 + 1)| {(4/3) , (4/3) | ) ------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (4/3) | \n1 = 0 / "A367639" LREtools/SearchTable: "SearchTable successful" n n -I (-2 I) ((-n - 3) LegendreP(n, I) + (n - 3) LegendreP(n + 1, I) I) -I (-2 I) ((-n - 3) LegendreQ(n, I) + (n - 3) LegendreQ(n + 1, I) I) {---------------------------------------------------------------------, ---------------------------------------------------------------------} (n - 1) n (n - 1) n "A367780" 2 n (2 n + 1) binomial(2 n, n) (5 n + 14 n + 6) {4 (2 n + 1), --------------------------------------------} n + 1 "A367857" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 1/2 ------------------------ n::even { 4 binomial(n, n/2) (n + 1) { (n + 1) binomial(n, n/2) { -------------------------- n::even {1, { , { n + 2 } { (2 n - 2) { { 2 (n + 1) { 2 binomial(n + 1, n/2 + 1/2) n::odd { ------------------------------------ n::odd { n (n + 2) binomial(n - 1, n/2 - 1/2) "A367963" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (2 n1 + 1) binomial(2 n1, n1) | {n! binomial(2 n, n), n! binomial(2 n, n) | ) ---------------------------------------------|} | / (n1 + 1) (n1 + 1)! binomial(2 n1 + 2, n1 + 1)| |----- | \n1 = 0 / "A368079" memory used=283173.3MB, alloc=3607.5MB, time=2160.92 LREtools/SearchTable: "SearchTable not successful" {} "A368164" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 - 1) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A368178" LREtools/SearchTable: "SearchTable successful" ((4 n + 2) hypergeom([-n - 1, -n - 1, -n - 1], [1, -2 n - 2], 1) + (-3 n - 3) hypergeom([-n, -n, -n], [1, -2 n], 1)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------------------} (n + 1) n "A368234" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 + 5) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / (n1 + 1) (n1 + 2) | |----- | \n1 = 0 / "A368378" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 8 binomial(n, n/2) (n + 1) n { -------------------------------- n::even { ---------------------------- n::even { (n + 1) (n + 3) binomial(n, n/2) { (n + 2) (n + 6) (n + 4) {{ , { } { (2 n + 2) { 4 binomial(n - 1, n/2 - 1/2) n { 2 2 n { ------------------------------ n::odd { -------------------------------------------------- n::odd { (n + 1) (n + 3) { (n + 2) (n + 4) (n + 6) binomial(n + 1, n/2 + 1/2) "A368379" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { 16 binomial(n, n/2) (n + 1) (n - 2) n { 1/2 -------------------------------- n::even { ------------------------------------- n::even { (n + 3) (n + 5) binomial(n, n/2) { (n + 2) (n + 4) (n + 6) (n + 8) {{ , { } { (2 n - 2) { 4 binomial(n + 1, n/2 + 1/2) (n + 1) { 2 2 (n - 2) (n + 1) { ------------------------------------ n::odd { ---------------------------------------------------------- n::odd { (n + 5) (n + 3) { (n + 2) (n + 4) (n + 6) (n + 8) binomial(n - 1, n/2 - 1/2) "A368380" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 (n - 1) { 64 binomial(n, n/2) n (n + 1) (n - 4) (n - 2) { 5/2 ---------------------------------------- n::even { --------------------------------------------- n::even { (n + 3) (n + 5) (n + 7) binomial(n, n/2) { (n + 2) (n + 4) (n + 6) (n + 8) (n + 10) {{ , { { (2 n - 2) { 20 binomial(n + 1, n/2 + 1/2) (n - 1) (n + 1) { 8 2 (n - 4) (n - 2) (n + 1) { --------------------------------------------- n::odd { ------------------------------------------------------------------- n::odd { (n + 3) (n + 5) (n + 7) { (n + 2) (n + 4) (n + 6) (n + 8) (n + 10) binomial(n - 1, n/2 - 1/2) } "A368555" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (-n1 - 1) | {3 , 3 | ) 3 (n1 + 1) n1!|} | / | |----- | \n1 = 0 / "A368567" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 3 n { binomial(---, n/2) (n + 2) { 2 { -------------------------- n::even { n + 1 {{ , { 3 n { binomial(--- - 3/2, n/2 - 1/2) (n + 3) (3 n - 1) { 2 { 1/2 ------------------------------------------------ n::odd { (n + 1) n { (-n) 3 n 3 n { 4 (n + 3) binomial(3 n, ---) binomial(---, n/2) { 2 2 { --------------------------------------------------- n::even { (n + 1) binomial(n, n/2) { } { (-2 n - 2) 3 n 3 n { 2 2 (n + 2) binomial(3 n + 3, --- + 3/2) binomial(--- + 3/2, n/2 + 1/2) { 2 2 { --------------------------------------------------------------------------------- n::odd { (3 n + 2) binomial(n + 1, n/2 + 1/2) "A368574" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 3) (n1 + 2) (n1 + 1)| {n! | ) --------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368575" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)| {n! | ) -----------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368576" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)| {n! | ) --------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368585" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 3) (n1 + 2) (n1 + 1)|| {(-1) n!, (-1) n! | ) |- ---------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368586" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)|| {(-1) n!, (-1) n! | ) |- ------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368587" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ | (-1) (n1 + 5) (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)|| {(-1) n!, (-1) n! | ) |- ---------------------------------------------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368626" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A368633" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A368634" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A368635" memory used=284072.5MB, alloc=3607.5MB, time=2167.89 LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A368708" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368709" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368716" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 3\| n n | \ | (-1) (n1 + 1) || {(-1) n!, (-1) n! | ) |- ----------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368717" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 4\| n n | \ | (-1) (n1 + 1) || {(-1) n!, (-1) n! | ) |- ----------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368718" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 5\| n n | \ | (-1) (n1 + 1) || {(-1) n!, (-1) n! | ) |- ----------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A368719" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 5| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A368760" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3| | \ (n1 + 1) | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A368762" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 1) (n1 + 2)| {n! | ) -----------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368763" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 3) (n1 + 2) (n1 + 1)| {n! | ) --------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368764" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1)| {n! | ) -----------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368765" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (-1) (n1 + 1)| {n! | ) ---------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 1) (n1 + 2) (-1) | {n! | ) ------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368767" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 3) (n1 + 2) (n1 + 1) (-1) | {n! | ) ---------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368768" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1| | \ (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (-1) | {n! | ) ------------------------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A368773" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" { 2 { 3 ((n + 2) (8 n - 1) hypergeom([-1/2, - n/2 - 1], [1], -8) + (-8 n - 23 n - 6) hypergeom([-1/2, - n/2], [1], -8)) { ------------------------------------------------------------------------------------------------------------------ n::even { n {{ , { 2 { (n + 3) (8 n + 7) hypergeom([-1/2, - n/2 - 3/2], [1], -8) + (-8 n - 39 n - 37) hypergeom([-1/2, - n/2 - 1/2], [1], -8) { ----------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { 2 { (n + 3) (8 n + 7) hypergeom([-1/2, - n/2 - 3/2], [1], -8) + (-8 n - 39 n - 37) hypergeom([-1/2, - n/2 - 1/2], [1], -8) { 1/2 ----------------------------------------------------------------------------------------------------------------------- n::even { n + 1 { { , { { 2 { { (n + 2) (8 n - 1) hypergeom([-1/2, - n/2 - 1], [1], -8) + (-8 n - 23 n - 6) hypergeom([-1/2, - n/2], [1], -8) { { 3/2 -------------------------------------------------------------------------------------------------------------- n::odd { n (n/2 + 1) (n/2) 2 3 (9 (n + 2) (8 n - 1) hypergeom([3/2, - n/2 - 1], [1], 8/9) - 9 (8 n + 23 n + 6) hypergeom([3/2, - n/2], [1], 8/9)) ----------------------------------------------------------------------------------------------------------------------------------- , n::even n (n/2 + 3/2) (n/2 + 1/2) 2 (9 (n + 3) (8 n + 7) hypergeom([3/2, - n/2 - 3/2], [1], 8/9) - 9 (8 n + 39 n + 37) hypergeom([3/2, - n/2 - 1/2], [1], 8/9) { , { )/(n + 1) , n::odd { 1/2 ( { (n/2 + 3/2) (n/2 + 1/2) 2 9 (n + 3) (8 n + 7) hypergeom([3/2, - n/2 - 3/2], [1], 8/9) - 9 (8 n + 39 n + 37) hypergeom([3/2, - n/2 - 1/2], [1], 8/9)) /(n + 1) , n::even (n/2 + 1) (n/2) 2 9 (n + 2) (8 n - 1) hypergeom([3/2, - n/2 - 1], [1], 8/9) - 9 (8 n + 23 n + 6) hypergeom([3/2, - n/2], [1], 8/9) } 3/2 ------------------------------------------------------------------------------------------------------------------------------- , n::odd n "A368787" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A368790" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 | {3 GAMMA(n + 1/3), 3 GAMMA(n + 1/3) | ) ---------------|} | / GAMMA(n1 + 4/3)| |----- | \n1 = 0 / "A368791" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 | {3 GAMMA(n + 2/3), 3 GAMMA(n + 2/3) | ) ---------------|} | / GAMMA(n1 + 5/3)| |----- | \n1 = 0 / "A368792" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | n n | \ 2 (n1 + 1) | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A368793" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 (n1 + 1)| {3 GAMMA(n + 1/3), 3 GAMMA(n + 1/3) | ) -------------------|} | / GAMMA(n1 + 4/3) | |----- | \n1 = 0 / "A368794" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (n1 + 1) | n n | \ (-1) 2 | {(1/2) n! binomial(2 n, n), (1/2) n! binomial(2 n, n) | ) ------------------------------------|} | / binomial(2 n1 + 2, n1 + 1) (n1 + 1)!| |----- | \n1 = 0 / "A368837" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) (n + 2) (n!) , (n + 1) (n + 2) (n!) | ) ------------------------------|} | / 2| |----- (n1 + 2) (n1 + 3) ((n1 + 1)!) | \n1 = 0 / "A368838" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 3) (n + 2) (n + 1) (n!) , (n + 3) (n + 2) (n + 1) (n!) | ) ---------------------------------------|} | / 2| |----- (n1 + 4) (n1 + 3) (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A368840" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (n1 + 1) | 2 n 3 2 n 3 | \ 6 | {(n + 2) (n + 1) (1/6) (n!) , (n + 2) (n + 1) (1/6) (n!) | ) -------------------------------|} | / 2 3| |----- (n1 + 2) (n1 + 3) ((n1 + 1)!) | \n1 = 0 / "A368853" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-1) | {(n + 1) (n!) , (n + 1) (n!) | ) ---------------------|} | / 2| |----- (n1 + 2) ((n1 + 1)!) | \n1 = 0 / "A368957" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368961" LREtools/SearchTable: "SearchTable not successful" {} "A368962" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368963" LREtools/SearchTable: "SearchTable not successful" {} "A368964" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368965" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368966" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368967" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368968" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368969" LREtools/SearchTable: "SearchTable not successful" {} "A368970" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368971" LREtools/SearchTable: "SearchTable not successful" {} "A368972" memory used=285446.6MB, alloc=3607.5MB, time=2177.43 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368973" LREtools/SearchTable: "SearchTable not successful" {} "A368974" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368975" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A368976" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369011" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369012" LREtools/SearchTable: "SearchTable successful" 2 (22 n + 16 n + 2) hypergeom([-n - 1, 3 n + 4], [1], -1) - 5 (3 n + 1) (3 n + 2) hypergeom([-n, 3 n + 1], [1], -1) {------------------------------------------------------------------------------------------------------------------} 2 (238 n + 289 n + 83) n "A369013" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369023" LREtools/SearchTable: "SearchTable successful" 3 (3 n + 4) (2 n + 1) hypergeom([-n, -3 n - 3], [2], 2) + (3 n + 1) n hypergeom([-3 n, -n + 1], [2], 2) {-------------------------------------------------------------------------------------------------------} 2 35 n + 42 n + 12 "A369024" LREtools/SearchTable: "SearchTable successful" (4 n + 1) (2 n + 1) binomial(4 n, n) hypergeom([-n, 4 n + 4], [3 n + 3], -1) {----------------------------------------------------------------------------} (n + 1) (3 n + 1) (3 n + 2) "A369080" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { (- n/2) 3 4 { 2 binomial(n/2, n/4) ((n/4)!) binomial(n, n/2) irem(n, 4) = 0 { { (- n/2 - 3/2) 3 4 { 2 binomial(n/2 + 3/2, n/4 + 3/4) ((n/4 + 3/4)!) binomial(n + 3, n/2 + 3/2) { ----------------------------------------------------------------------------------------- irem(n, 4) = 1 { n (n + 1) (n + 2) { {{ (- n/2 - 1) 3 4 , { 2 binomial(n/2 + 1, n/4 + 1/2) ((n/4 + 1/2)!) binomial(n + 2, n/2 + 1) { ----------------------------------------------------------------------------------- irem(n, 4) = 2 { n (n + 1) { { (- n/2 - 1/2) 3 4 { 2 binomial(n/2 + 1/2, n/4 + 1/4) ((n/4 + 1/4)!) binomial(n + 1, n/2 + 1/2) { ----------------------------------------------------------------------------------------- irem(n, 4) = 3 { n { (n/2) 4 3 n { 2 ((n/4)!) binomial(n, n/4) binomial(---, n/4) { 4 { ---------------------------------------------------- irem(n, 4) = 0 { n { { (n/2 - 1/2) 4 3 n { 2 ((n/4 - 1/4)!) binomial(n - 1, n/4 - 1/4) binomial(--- - 3/4, n/4 - 1/4) irem(n, 4) = 1 { 4 { , { (n/2 - 1) 4 3 n { 2 ((n/4 - 1/2)!) (n - 1) binomial(n - 2, n/4 - 1/2) binomial(--- - 3/2, n/4 - 1/2) irem(n, 4) = 2 { 4 { { (n/2 + 1/2) 4 3 n { 2 ((n/4 + 1/4)!) binomial(n + 1, n/4 + 1/4) binomial(--- + 3/4, n/4 + 1/4) { 4 { -------------------------------------------------------------------------------------- irem(n, 4) = 3 { n (n + 1) { n 2 { 4 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { ---------------------------------------------------- irem(n, 4) = 0 { n { { (2 n - 2) 2 { 2 GAMMA(n/4 + 1/2) GAMMA(n/4 + 1/4) GAMMA(n/4 + 3/4) irem(n, 4) = 1 { { n 2 , { 16 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) { ------------------------------------------------------- irem(n, 4) = 2 { n (n + 1) (n + 2) { { (2 n + 2) 2 { 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { ------------------------------------------------------------ irem(n, 4) = 3 { n (n + 1) { n 2 { 4 GAMMA(n/4 + 5/4) GAMMA(n/4 + 1) GAMMA(n/4 + 3/2) { ---------------------------------------------------- irem(n, 4) = 0 { n (n + 1) (n + 2) { { (2 n - 2) 2 { 2 GAMMA(n/4 + 1) GAMMA(n/4 + 3/4) GAMMA(n/4 + 5/4) { ------------------------------------------------------------ irem(n, 4) = 1 { n (n + 1) } { { n 2 { 4 GAMMA(n/4 + 3/4) GAMMA(n/4 + 1) GAMMA(n/4 + 1/2) { 1/16 ---------------------------------------------------- irem(n, 4) = 2 { n { { n 2 { 1/64 4 GAMMA(n/4 + 1/2) GAMMA(n/4 + 1/4) GAMMA(n/4 + 3/4) irem(n, 4) = 3 "A369082" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 3 16 binomial(2 n, n/2) { ----------------------------------------------------- n::even { 16 binomial(2 n, n/2) { 3 n { --------------------- n::even { (n + 1) (3 n + 1) binomial(n, n/2) binomial(3 n, ---) { 3 n + 2 { 2 {{ , { } { 6 binomial(2 n + 2, n/2 + 1/2) { (4 n - 4) { ------------------------------ n::odd { 8 2 (2 n - 1) binomial(2 n - 2, n/2 - 1/2) { 2 n + 1 { ----------------------------------------------------------------------------- n::odd { 3 n { n (3 n - 2) (3 n + 2) binomial(n - 1, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 "A369083" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A369123" LREtools/SearchTable: "SearchTable successful" 1/2 n 10 1/2 (5 - 2 I) GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], 4/27 - -- I 2 ) 27 {-------------------------------------------------------------------------------} GAMMA(n + 2) "A369126" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369128" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A369156" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369208" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369212" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369215" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369316" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" 3 n 3 n 3 n {RootOf(_Z - 4 _Z - 4, index = 1) , RootOf(_Z - 4 _Z - 4, index = 2) , RootOf(_Z - 4 _Z - 4, index = 3) } "A369325" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (3 n1 + 1) binomial(2 n1, n1)|| {1, (-1) , (-1) | ) |- ----------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A369335" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 / |----- |----- | n n 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ | n2 {2 , 8 , (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | ) |2 16 | / | / | |----- |----- | \n1 = 0 \n2 = 0 \ /n2 - 1 \\\\ |----- (-3 n3 - 3) 5 4 3 2 |||| 1/2 (-n2 - 1) | \ 2 (3 n3 + 1) binomial(3 n3, n3) (2275 n3 + 18250 n3 + 55097 n3 + 77834 n3 + 51024 n3 + 12240)|||| (1 + 3 I) | ) ------------------------------------------------------------------------------------------------------------||||} | / (2 n3 + 5) (2 n3 + 3) (2 n3 + 1) (n3 + 3) (n3 + 2) (n3 + 1) |||| |----- |||| \n3 = 0 //// "A369359" LREtools/SearchTable: "SearchTable successful" n (-1) ((4 n + 1) hypergeom([1/2, -n - 1], [1], 4) + (2 n + 1) hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------------------------} n "A369360" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 2 n (-1) ((29 n + 210 n + 487 n + 360) hypergeom([1/2, -n - 1], [1], 4) - 3 (2 n + 5) (5 n + 16) (n + 1) hypergeom([1/2, -n], [1], 4)) {3 , -------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A369432" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { n1 \ | { 4 | | { ------------------------------------ n1::even| | { n1 | | { (n1 + 1) (n1 + 3) binomial(n1, ----) | | { 2 | | { | | { (2 n1 - 2) | | { 3 2 | | { - ---------------------------------------- n1::odd | |n - 1 { n1 | |----- { n1 (n1 + 2) binomial(n1 - 1, ---- - 1/2) | n n | \ { 2 | {(5/2) , (5/2) | ) ------------------------------------------------------------|, | / (n1 + 1) | |----- (5/2) | \n1 = 0 / / { n1 \ | { 6 binomial(n1, ----) | | { 2 | | { - -------------------- n1::even| | { n1 + 2 | | { | | { n1 | | { 2 binomial(n1 + 1, ---- + 1/2) | |n - 1 { 2 | |----- { ------------------------------ n1::odd | n | \ { n1 + 3 | (5/2) | ) ------------------------------------------------|} | / (n1 + 1) | |----- (5/2) | \n1 = 0 / "A369436" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 7 _Z + 14 _Z - 9, index = 1) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 2) , RootOf(_Z - 7 _Z + 14 _Z - 9, index = 3) } "A369472" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 (2 n + 1) binomial(2 n, n/2) { --------------------------------------------------------------- n::even { 3 n { (n + 1) (3 n + 1) (3 n + 5) binomial(n, n/2) binomial(3 n, ---) { 2 {{ , { (4 n - 4) { 4 2 (2 n - 1) (2 n + 1) binomial(2 n - 2, n/2 - 1/2) { --------------------------------------------------------------------------------------- n::odd { 3 n { n (3 n - 2) (3 n + 2) (3 n + 4) binomial(n - 1, n/2 - 1/2) binomial(3 n - 3, --- - 3/2) { 2 { 16 binomial(2 n, n/2) (2 n + 1) { ------------------------------- n::even { (3 n + 4) (3 n + 2) { } { 4 binomial(2 n + 2, n/2 + 1/2) { ------------------------------ n::odd { 3 n + 5 "A369502" LREtools/SearchTable: "SearchTable not successful" {} "A369503" LREtools/SearchTable: "SearchTable successful" (2 n + 1) (4 n + 3) (4 n + 1) binomial(4 n, n) hypergeom([- n/2, - n/2 + 1/2], [-2 n - 3/2], -1) {------------------------------------------------------------------------------------------------} (n + 1) (3 n + 4) (3 n + 1) (3 n + 2) "A369528" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369580" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {9 , (2 n + 2) hypergeom([-1/2, -n - 1], [1], -8) + (-2 n - 3) hypergeom([-1/2, -n], [1], -8)} "A369616" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369617" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369722" LREtools/SearchTable: "SearchTable successful" n n {(-3) BesselI(n - 1/2, 3/2), (-3) BesselK(n - 1/2, -3/2)} "A369723" LREtools/SearchTable: "SearchTable successful" n n {(-4) BesselI(n - 1/2, 2), (-4) BesselK(n - 1/2, -2)} "A369724" LREtools/SearchTable: "SearchTable successful" n n {(-5) BesselI(n - 1/2, 5/2), (-5) BesselK(n - 1/2, -5/2)} "A369737" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselJ(n, -1/2), (-1) BesselY(n, -1/2)} "A369746" LREtools/SearchTable: "SearchTable successful" n n {(-3) BesselI(n - 1/2, 3), (-3) BesselK(n - 1/2, -3)} "A369751" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A369822" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" n 2 2 (n + 1) 4 (n!) (2 n + 1) (hypergeom([1/2, -2 n - 2], [1], 4) + 3 hypergeom([1/2, -2 n], [1], 4)) {(2 n + 1) (n!) binomial(2 n, n), --------------------------------------------------------------------------------------------------} 4 n + 3 "A369906" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n n \ binomial(2 n1, n1) n1 {1, 2 , 4 , ) ---------------------} / n1 + 1 ----- n1 = 0 "A369930" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ n1 \\ | |{ 8 binomial(2 n1, ----) || |n - 1 |{ 2 || |----- |{ ---------------------- n1::even|| n n | \ (-2 n1 - 2) |{ 3 n1 + 2 || {4 , 4 | ) 2 |{ ||, | / |{ n1 || |----- |{ 64 binomial(2 n1 - 2, ---- - 1/2) (2 n1 - 1) (2 n1 + 1) || |n1 = 0 |{ 2 || | |{ ------------------------------------------------------- n1::odd || \ \{ (n1 + 1) (3 n1 - 1) (3 n1 + 1) // / /{ n1 n1 \\ | |{ 16 16 (2 n1 + 1) binomial(2 n1, ----) || | |{ 2 || | |{ -------------------------------------------------------------- n1::even|| |n - 1 |{ n1 3 n1 || |----- |{ n1 (n1 + 1) (3 n1 + 1) binomial(n1, ----) binomial(3 n1, ----) || n | \ (-2 n1 - 2) |{ 2 2 || 4 | ) 2 |{ ||} | / |{ (4 n1 + 4) n1 || |----- |{ 2 2 binomial(2 n1 + 2, ---- + 1/2) || |n1 = 0 |{ 2 || | |{ ------------------------------------------------------------------------------- n1::odd || | |{ n1 3 n1 || | |{ (n1 + 1) (2 n1 + 1) binomial(n1 + 1, ---- + 1/2) binomial(3 n1 + 3, ---- + 3/2) || \ \{ 2 2 // "A369982" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / /{ n1 \\ | |{ 4 || | |{ --------------------------- n1::even|| |n - 1 |{ n1 || |----- |{ (n1 + 1) binomial(n1, ----) || n n | \ (-n1 - 1) |{ 2 || {3 , 3 | ) 3 |{ ||, | / |{ (2 n1 - 2) || |----- |{ 2 2 || |n1 = 0 |{ - ------------------------------- n1::odd || | |{ n1 || | |{ n1 binomial(n1 - 1, ---- - 1/2) || \ \{ 2 // /n - 1 /{ n1 \\ |----- |{ -2 binomial(n1, ----) n1::even|| n | \ (-n1 - 1) |{ 2 || 3 | ) 3 |{ ||} | / |{ n1 || |----- |{ binomial(n1 + 1, ---- + 1/2) n1::odd || \n1 = 0 \{ 2 // "A370048" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370061" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /3125\(n/2) { |----| GAMMA(n/2 + 4/5) GAMMA(n/2 + 8/5) GAMMA(n/2 + 7/5) GAMMA(n/2 + 6/5) { \256 / { ------------------------------------------------------------------------------- n::even { GAMMA(n/2 + 3/2) GAMMA(n/2 + 2) GAMMA(n/2 + 7/4) GAMMA(n/2 + 5/4) {{ , { /3125\(n/2 + 1/2) 13 17 19 21 { 12 |----| GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) (2 n + 5) (n + 3) (n + 1) { \256 / 10 10 10 10 { -------------------------------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 2) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/4) GAMMA(n/2 + 9/4) (5 n + 11) (5 n + 9) (5 n + 7) { /3125\(n/2) 13 17 19 21 { 12 |----| GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) GAMMA(n/2 + --) (2 n + 5) (n + 3) (n + 1) { \256 / 10 10 10 10 { -------------------------------------------------------------------------------------------------------- n::even { GAMMA(n/2 + 2) GAMMA(n/2 + 5/2) GAMMA(n/2 + 7/4) GAMMA(n/2 + 9/4) (5 n + 11) (5 n + 9) (5 n + 7) { } { /3125\(n/2 - 1/2) { |----| GAMMA(n/2 + 6/5) GAMMA(n/2 + 8/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 7/5) { \256 / { ------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 3/2) GAMMA(n/2 + 5/4) GAMMA(n/2 + 7/4) GAMMA(n/2 + 2) "A370097" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 2 \ 2 binomial(3 n1, n1) (5 n1 + 5 n1 + 1) {1, ) -----------------------------------------} / (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A370099" LREtools/SearchTable: "SearchTable successful" 2 2 (2 (2 n + 1) hypergeom([2 n + 3, -n - 1], [n + 2], -1) + (-26 n - 25 n - 5) hypergeom([-n, 2 n + 1], [n + 1], -1)) binomial(2 n, n) {-------------------------------------------------------------------------------------------------------------------------------------} n (10 n + 7) "A370107" memory used=287024.5MB, alloc=3607.5MB, time=2188.25 LREtools/SearchTable: "SearchTable not successful" {} "A370145" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370147" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370149" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370159" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370160" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370170" LREtools/SearchTable: "SearchTable successful" {(5 (5 n + 6) (5 n + 7) (5 n + 3) (5 n + 4) hypergeom([- n/2, - n/2 - 1/2], [2 n + 4], 4) 3 2 / - 6 (843 n + 2060 n + 1561 n + 360) (2 n + 3) hypergeom([- n/2, - n/2 + 1/2], [2 n + 2], 4)) binomial(3 n, n) / ((2 n + 1) (2 n + 3) / 2 (57 n + 114 n + 55))} "A370171" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370197" (2 n + 1) binomial(2 n, n) 6 5 4 3 2 {--------------------------, n - 3 n + 10 n + 3 n + 25 n + 36 n + 36} n + 1 "A370246" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A370253" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n {(-2) n! LaguerreL(n, -n - 1/2, -1/2), (1/2) n! binomial(2 n, n)} "A370280" LREtools/SearchTable: "SearchTable successful" / 1/2\n | 12 6 | {|27/5 - -------| \ 5 / / 1/2 1/2 \ | 24 6 1/2 24 6 | |(45 n + 60) hypergeom([5/6, - 2/3 - n], [5/3], 64/5 + -------) + (9 + 4 6 ) (39 n + 22) hypergeom([5/6, 1/3 - n], [5/3], 64/5 + -------)| \ 5 5 / / 1/2\ | 24 6 | GAMMA(n + 1/3) |- 59/5 + -------|/GAMMA(n + 1)} \ 5 / "A370285" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370286" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370287" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370357" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370358" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" n 2 {(1/6) (n!) binomial(2 n, n) binomial(3 n, n)} "A370369" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ | {1, (n + 1) | ) (n1 + 1) n1!|, n + 1} | / | |----- | \n1 = 0 / "A370375" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /216\n /216\n | \ binomial(3 n1, n1) (21 n1 + 11) | {|---| , |---| | ) ---------------------------------|} \25 / \25 / | / /216\(n1 + 1)| |----- (n1 + 1) (2 n1 + 1) |---| | \n1 = 0 \25 / / "A370376" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(3 n1, n1) (13 n1 + 7) | {(64/9) , (64/9) | ) ----------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (2 n1 + 1) (64/9) | \n1 = 0 / "A370390" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 5 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" /n - 1 \ |----- n1 | | \ (-1) | n! (n + 1) | ) ------------------| | / (n1 + 1)! (n1 + 2)| |----- | n! (n + 1) \n1 = 0 / {----------, --------------------------------------} n n "A370426" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n - 1, 2), (-1) BesselK(n - 1, -2)} "A370479" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (2 n + 10 n + 7 n + 8) LegendreP(n + 1, 3) + (-14 n - 42 n - 37 n - 24) LegendreP(n, 3) {-------------------------------------------------------------------------------------------, (n + 2) n (n - 1) (n - 2) 3 2 3 2 (2 n + 10 n + 7 n + 8) LegendreQ(n + 1, 3) + (-14 n - 42 n - 37 n - 24) LegendreQ(n, 3) -------------------------------------------------------------------------------------------} (n + 2) n (n - 1) (n - 2) "A370480" LREtools/SearchTable: "SearchTable successful" 5 4 3 2 5 4 3 2 (2 n + 18 n + 29 n + 78 n + 77 n + 120) LegendreP(n + 1, 3) + (-14 n - 82 n - 175 n - 302 n - 471 n - 360) LegendreP(n, 3) {----------------------------------------------------------------------------------------------------------------------------------, (n + 3) (n + 2) n (n - 1) (n - 2) (n - 3) 5 4 3 2 5 4 3 2 (2 n + 18 n + 29 n + 78 n + 77 n + 120) LegendreQ(n + 1, 3) + (-14 n - 82 n - 175 n - 302 n - 471 n - 360) LegendreQ(n, 3) ----------------------------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n (n - 1) (n - 2) (n - 3) "A370487" {1, binomial(2 n, n)} "A370498" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" (n + 1) LegendreP(2 n + 2, 3) + (-17 n - 13) LegendreP(2 n, 3) (n + 1) LegendreQ(2 n + 2, 3) + (-17 n - 13) LegendreQ(2 n, 3) {--------------------------------------------------------------, --------------------------------------------------------------} (4 n + 3) n (4 n + 3) n "A370509" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A370524" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) BesselI(n, 2), (-1) BesselK(n, -2)} "A370528" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 5 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370570" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370616" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370669" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n n {(-1) (n BesselI(n, 2) - 2 BesselI(n - 1, 2)), (-1) (n BesselK(n, -2) - 2 BesselK(n - 1, -2))} "A370695" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { / n1 \ \ | { | | { |----| | | { / n1 \ | | { \ 2 / / n1 \ | | { |---- - 1/2| | | { (-4) |----|! n1::even| |n - 1 { \ 2 / / n1 \ n1 | |n - 1 { \ 2 / | |----- { n1 (-1) |---- - 1/2|! binomial(n1 - 1, ---- - 1/2) n1::odd | |----- { | | \ { \ 2 / 2 | | \ { 0 n1::odd | {n! | ) --------------------------------------------------------------------------------|, n! | ) ------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A370704" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful after finding recurrence for u(n/2)" {BesselJ(2 n + 1, -2) + BesselJ(2 n - 1, -2), BesselY(2 n + 1, -2) + BesselY(2 n - 1, -2)} "A370720" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370767" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370768" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A370769" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 8" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 6 to 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" n n {(-I) , I } "A370779" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370780" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370781" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370782" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370799" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370836" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A370888" {(n + 1) n!, n + 1} "A370943" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 4 { ---------------- n::even n { (2 n + 1) binomial(n, n/2) n::even { binomial(n, n/2) {2 , { , { } { 4 n binomial(n - 1, n/2 - 1/2) n::odd { (2 n + 2) { 2 (2 n + 1) { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A370973" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A371003" (2 n + 1) binomial(2 n, n) 4 2 {--------------------------, n + 3 n + 4 n + 4} n + 1 "A371036" (2 n + 1) binomial(2 n, n) {--------------------------, n + 2} n + 1 "A371208" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 2 { (-n) 2 { 1/8 (n/2)! (n + 4) (n + 3) (n + 2) n::even { 2 2 (n + 3) (n + 1) binomial(n, n/2) (n/2)! n::even {{ , { } { 2 { (-n - 1) 2 { 1/4 (n/2 - 1/2)! (n + 1) (n + 3) n::odd { 2 (n + 4) (n + 2) (n + 3) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd "A371210" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { 2 (2 n + 2) binomial(n, n/2) (n/2)! n::even { 2 {{ , { 1/2 (n/2)! (n + 2) n::even} { (-n - 1) 2 { { 2 (n + 2) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd { (n/2 - 1/2)! (n + 1) n::odd "A371217" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ /n - 1 \ |----- |----- 3 2 || |----- | | \ | \ 2 n2 + 7 n2 + 6 n2 + 2 || | \ | {n, n | ) n1! | ) ------------------------------||, n | ) n1!|} | / | / (n2 + 2) (n2 + 1)! (n2 + 1) n2|| | / | |----- |----- || |----- | \n1 = 0 \n2 = 0 // \n1 = 0 / "A371250" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 5) (3 n1 + 2) (3 n1 + 4) (3 n1 + 1) binomial(3 n1, n1) (3 n1 + 11)| {(27/4) (n + 2), (27/4) (n + 2) | ) --------------------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 3) (2 n1 + 1) (27/4) (n1 + 3) (4 n1 + 8) | \n1 = 0 / "A371252" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371358" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A371362" LREtools/SearchTable: "SearchTable successful" 2 2 (2 n + 5) (n + 2) hypergeom([-n - 1, -2 n - 4], [2], 3) + (-123 n - 369 n - 276) hypergeom([-n, -2 n - 2], [2], 3) {---------------------------------------------------------------------------------------------------------------------} (11 n + 17) (2 n + 1) "A371364" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371366" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371367" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371368" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371369" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371380" LREtools/SearchTable: "SearchTable successful" n (-1) ((8 n + 12) hypergeom([-n, 2 n + 4], [2], 3) + 5 n hypergeom([2 n + 2, -n + 1], [2], 3)) {----------------------------------------------------------------------------------------------} 19 n + 12 "A371386" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A371391" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 3 binomial(2 n1, n1)| {(-4) , (-4) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-4) | \n1 = 0 / "A371392" LREtools/SearchTable: "SearchTable successful" {((2 n + 1) (55 n + 26) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -2) - 54 (3 n + 1) (3 n + 2) hypergeom([-3 n, -n], [-2 n], -2)) binomial(2 n, n)/( (n + 1) (n + 2) (7 n + 2))} "A371393" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 4 binomial(2 n1, n1)| {(-9) , (-9) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-9) | \n1 = 0 / "A371394" LREtools/SearchTable: "SearchTable successful" {((2 n + 1) (44 n + 21) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -3) - 60 (3 n + 1) (3 n + 2) hypergeom([-3 n, -n], [-2 n], -3)) binomial(2 n, n)/( (n + 1) (n + 2) (26 n + 9))} "A371399" LREtools/SearchTable: "SearchTable successful" n (-1/9) ((2 n + 1) hypergeom([-n - 1, -3 n - 3], [-2 n - 2], -2) + (-18 n - 6) hypergeom([-3 n, -n], [-2 n], -2)) binomial(2 n, n) {----------------------------------------------------------------------------------------------------------------------------------} 7 n + 2 "A371404" memory used=288618.6MB, alloc=3607.5MB, time=2199.26 LREtools/SearchTable: "SearchTable successful" {hypergeom([-n, -2 n - 2], [2], 3)} "A371406" LREtools/SearchTable: "SearchTable successful" {(9 (3 n + 5) (2 n + 3) (n + 2) hypergeom([-n - 1, 3 n + 6], [n + 4], -1) + (5 n + 7) (n + 3) (2 n + 1) hypergeom([-n, 3 n + 3], [n + 3], -1)) binomial(2 n, n) (2 n + 1)/((n + 1) (n + 2) (n + 3) (3 n + 4) (22 n + 29))} "A371407" LREtools/SearchTable: "SearchTable not successful" {} "A371408" LREtools/SearchTable: "SearchTable successful" n 2 3 2 (-1) ((n + 1) (13 n - 29 n + 12) hypergeom([1/2, -n - 1], [1], 4) + (41 n - 72 n - 5 n + 12) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------------------------------------} n "A371411" LREtools/SearchTable: "SearchTable successful" n (-1) binomial(2 n, n) (hypergeom([1/2, -n - 1], [1], 4) + hypergeom([1/2, -n], [1], 4)) {----------------------------------------------------------------------------------------} n "A371426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371428" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371431" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371433" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371458" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 6, 6 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" 3 n 3 n 3 n {RootOf(_Z - 10, index = 1) , RootOf(_Z - 10, index = 2) , RootOf(_Z - 10, index = 3) } "A371494" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371516" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371542" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371564" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 2 } "A371570" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, 2 } "A371583" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A371585" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A371605" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 2 n 2 n {(1/3 RootOf(%1, index = 1) + 1/6 RootOf(%1, index = 1) + 5/6) n!, (1/3 RootOf(%1, index = 2) + 1/6 RootOf(%1, index = 2) + 5/6) n!, 2 n (1/3 RootOf(%1, index = 3) + 1/6 RootOf(%1, index = 3) + 5/6) n!} 3 %1 := 4 _Z + 9 _Z + 7 "A371655" LREtools/SearchTable: "SearchTable successful" n 2 ((2 n + 2) hypergeom([2 n + 3, -n - 1], [1], -1) + (-11 n - 8) hypergeom([-n, 2 n + 1], [1], -1)) {----------------------------------------------------------------------------------------------------} (17 n + 11) n "A371657" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371658" LREtools/SearchTable: "SearchTable successful" n {2 ((5 n + 2) (3 n + 5) (2 n + 3) (n + 2) hypergeom([-n - 1, 3 n + 6], [n + 4], -1) 2 - (51 n + 105 n + 50) (2 n + 1) (n + 3) hypergeom([-n, 3 n + 3], [n + 3], -1)) binomial(2 n, n)/(n (n + 1) (n + 2) (n + 3) (22 n + 29))} "A371660" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371661" LREtools/SearchTable: "SearchTable not successful" {} "A371668" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" n {2 } "A371682" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2, 6 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" / 1/2\n / 1/2 \n n | 5 | |5 | {2 , |1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A371724" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371758" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ (2 n2 + 1) binomial(2 n2, n2) (7 n2 + 10)| | {(-1/3 I 3 ) , (1/3 I 3 ) , (-1/3 I 3 ) | ) (-1) 3 | ) -----------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (n2 + 1) (n2 + 2) (1/3 I 3 ) | | \n1 = 0 \n2 = 0 / / "A371770" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/6 - 1/6 I 3 ) , (1/6 + 1/6 I 3 ) , (1/6 - 1/6 I 3 ) | ) (1/6 + 1/6 I 3 ) (1/6 - 1/6 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) || | \ (1/6 + 1/6 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (37 n2 + 60) (13 n2 + 17)|| | ) ----------------------------------------------------------------------------------------------||} | / (n2 + 2) (2 n2 + 3) (n2 + 1) (2 n2 + 1) || |----- || \n2 = 0 // "A371771" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/6 - 1/18 I 3 ) , (1/6 + 1/18 I 3 ) , (1/6 - 1/18 I 3 ) | ) (1/6 + 1/18 I 3 ) (1/6 - 1/18 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 4 3 2 || | \ (1/6 + 1/18 I 3 ) (4 n2 + 1) (2 n2 + 1) binomial(4 n2, n2) (9037 n2 + 47053 n2 + 89514 n2 + 73332 n2 + 21640)|| | ) ---------------------------------------------------------------------------------------------------------------------------||} | / (n2 + 1) (3 n2 + 1) (3 n2 + 2) (n2 + 2) (3 n2 + 4) (3 n2 + 5) || |----- || \n2 = 0 // "A371773" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A371777" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n \ (2 n1 + 1) binomial(2 n1, n1) {1, 4 , ) -----------------------------} / n1 + 1 ----- n1 = 0 "A371780" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 / /n1 - 1 |----- | |----- n n n | \ | n1 n1 | \ n2 (-13 n2 - 5) {(-1) , 32 , (-1) | ) |-(-1) 32 | ) 3125 2 | / | | / |----- | |----- \n1 = 0 \ \n2 = 0 6 5 4 3 2 (271929 n2 + 2174930 n2 + 7034281 n2 + 11746114 n2 + 10649582 n2 + 4955484 n2 + 922320) GAMMA(n2 + 3/5) GAMMA(n2 + 4/5) GAMMA(n2 + 6/5) \\\ ||| ||| GAMMA(n2 + 7/5)/(GAMMA(n2 + 3) GAMMA(n2 + 5/2) GAMMA(n2 + 9/4) GAMMA(n2 + 11/4))|||} ||| ||| /// "A371787" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /3125\n1 3 2 \ |----- |----| GAMMA(n1 + 3/5) GAMMA(n1 + 4/5) GAMMA(n1 + 2/5) GAMMA(n1 + 6/5) (161 n1 + 304 n1 + 185 n1 + 36)| n n | \ \256 / | {(-1) , (-1) | ) ----------------------------------------------------------------------------------------------------------|} | / (n1 + 1) | |----- GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 5/4) (-1) | \n1 = 0 / "A371798" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" n n {(1/2 - 1/2 I) , (1/2 + 1/2 I) , /n - 1 /n1 - 1 \\ |----- |----- (-n2 - 1) || n | \ | \ (1/2 + 1/2 I) (2 n2 + 1) binomial(2 n2, n2) (5 n2 + 6)|| (1/2 - 1/2 I) | ) (1 + I) exp(1/2 I n1 Pi) | ) ---------------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A371813" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(3 n1, n1) (28 n1 + 25 n1 + 5)| {(-1/4) , (-1/4) | ) ---------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (-1/4) | \n1 = 0 / "A371814" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 3 2 | n n | \ binomial(4 n1, n1) (415 n1 + 592 n1 + 259 n1 + 34)| {(-1/8) , (-1/8) | ) ----------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) (-1/8) | \n1 = 0 / "A371815" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- n n n | \ {(1/4 - 1/4 I) , (1/4 + 1/4 I) , (1/4 - 1/4 I) | ) (2 + 2 I) exp(1/2 I n1 Pi) | / |----- \n1 = 0 /n1 - 1 \\ |----- (-n2 - 1) 5 4 3 2 || | \ (1/4 + 1/4 I) (4 n2 + 1) binomial(4 n2, n2) (29257 n2 + 167584 n2 + 368855 n2 + 387004 n2 + 191908 n2 + 35760)|| | ) ---------------------------------------------------------------------------------------------------------------------------||} | / (n2 + 1) (3 n2 + 1) (3 n2 + 2) (n2 + 2) (3 n2 + 4) (3 n2 + 5) || |----- || \n2 = 0 // "A371818" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 3 _Z + 2 _Z + 1, index = 1) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 2) , RootOf(_Z - 3 _Z + 2 _Z + 1, index = 3) } "A371819" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 6 _Z + 1, index = 1) , RootOf(_Z - 5 _Z + 6 _Z + 1, index = 2) , RootOf(_Z - 5 _Z + 6 _Z + 1, index = 3) } "A371820" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 3 (2 n1 + 1) binomial(2 n1, n1)| {3 , 3 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A371842" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A371854" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 4 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 4 _Z - 1, index = 3) } "A371870" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) 2 || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (n2 + 6 n2 + 2)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A371871" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \ \ |----- |----- | | 1/2 n 1/2 n 1/2 n | \ n1 1/2 | \ (2 n2 + 1) binomial(2 n2, n2) (7 n2 + 8)| | {(-1/3 I 3 ) , (1/3 I 3 ) , (-1/3 I 3 ) | ) (-1) 3 | ) ----------------------------------------| I|} | / | / 1/2 (n2 + 1) | | |----- |----- (n2 + 1) (n2 + 2) (1/3 I 3 ) | | \n1 = 0 \n2 = 0 / / "A371872" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 5 _Z + 2 _Z - 1, index = 1) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 2) , RootOf(_Z - 5 _Z + 2 _Z - 1, index = 3) } "A371873" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n \ (2 n1 + 1) binomial(2 n1, n1) {1, 4 , ) -----------------------------} / n1 + 1 ----- n1 = 0 "A371882" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371888" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 LegendreP(n1 + 1, ----) - LegendreP(n1, ----)| \ \ 5 5 / {1, ) ------------------------------------------------------------------, / n1 ----- n1 = 0 / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 LegendreQ(n1 + 1, ----) - LegendreQ(n1, ----)| \ \ 5 5 / ) ------------------------------------------------------------------} / n1 ----- n1 = 0 "A371889" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A371943" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SearchTable: "SearchTable successful" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), (-1/2 I 2 ) HermiteH(n + 1, 1/2 I 2 )} "A371963" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (n1 - 2) (n1 - 1) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (n1 + 2) ----- n1 = 0 "A371964" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (n1 - 1) (n1 - 2) binomial(2 n1, n1) {1, ) ------------------------------------} / (n1 + 1) (2 n1 - 1) ----- n1 = 0 "A371965" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ binomial(2 n1, n1) (n1 - 1) {1, ) ---------------------------} / n1 + 1 ----- n1 = 0 "A371978" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A371986" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2 \n | 5 | |5 | |1/2 - ----| binomial(2 n, n) |---- + 1/2| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A371987" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A371988" LREtools/SearchTable: "SearchTable successful" 1/2 n {(4 I 2 - 20) / 10 1/2 1/2 10 1/2 \ |(27 n + 36) hypergeom([5/6, - 2/3 - n], [5/3], 4/27 - -- I 2 ) - 5 (5 + 2 I) (2 n + 1) hypergeom([5/6, 1/3 - n], [5/3], 4/27 - -- I 2 )| \ 27 27 / 1/2 GAMMA(n - 1/3) GAMMA(n + 1/3) (1/9 I 2 - 5/9)/(GAMMA(n + 1) GAMMA(n + 2/3))} "A371998" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Applying recursion to the left-factor" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { (n/2)! n::even {{ , { } { (-n + 1) { (n/2 + 1/2)! n::odd { 2 n binomial(n - 1, n/2 - 1/2) (n/2 - 1/2)! n::odd "A372002" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372012" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372018" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" RightFactors: "Input is not over Q" RightFactors: "Input is not over Q" LREtools/SearchTable: "SearchTable not successful" {} "A372023" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 1) hypergeom([1/2, -n], [1], 4) {---------------------------------------------------------------------------------} n "A372024" LREtools/SearchTable: "SearchTable successful" (n + 1) hypergeom([-1/3, -n - 1], [1], 9) + (-n - 4) hypergeom([-1/3, -n], [1], 9) {----------------------------------------------------------------------------------} n "A372035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 LegendreP(n1 + 1, ----) - LegendreP(n1, ----)| \ \ 5 5 / {1, ) ------------------------------------------------------------------, / n1 ----- n1 = 0 / n1 \ |---- + 1/2| / 1/2 1/2 \ n - 1 \ 2 / | 1/2 5 5 | ----- 5 |5 LegendreQ(n1 + 1, ----) - LegendreQ(n1, ----)| \ \ 5 5 / ) ------------------------------------------------------------------} / n1 ----- n1 = 0 "A372036" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372037" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372039" LREtools/SearchTable: "SearchTable successful" n (-3) GAMMA(n - 2/3) hypergeom([-n, n - 2/3], [1/6], -1/4) {----------------------------------------------------------} GAMMA(n + 1) "A372102" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n {(-1/2 I) (HermiteH(n + 1, 1/2 I) + 2 I (n + 1) HermiteH(n, 1/2 I)) I} "A372104" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372216" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" 1/2 n 1/2 n (2 n + 1) (1 - 2 ) binomial(2 n, n) (2 n + 1) (1 + 2 ) binomial(2 n, n) {--------------------------------------, --------------------------------------} (n + 1) (n + 2) (n + 1) (n + 2) "A372239" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) (3 n1 + 2)| {2 , 2 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A372255" n! (n + 1) {n, ----------} n "A372264" 2 {n , (n + 1) n!} "A372310" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / | 1/2 n 1/2 n 1/2 n | {(9/2 - 3/2 I 3 ) , (9/2 + 3/2 I 3 ) , (9/2 - 3/2 I 3 ) | | | \ n - 1 /n1 - 1 \ ----- |----- 1/2 (-n2 - 1) | \ 1/2 n1 1/2 (-n1 - 1) | \ (9/2 + 3/2 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (7 n2 + 11)| ) (9/2 + 3/2 I 3 ) (9/2 - 3/2 I 3 ) | ) --------------------------------------------------------------------------------| / | / (n2 + 3) (n2 + 2) (n2 + 1) (2 n2 + 3) (2 n2 + 1) | ----- |----- | n1 = 0 \n2 = 0 / \ | | |} | | / "A372324" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { (-n1) n1 2 // n1 \ \2 \ | { | | { 2 binomial(n1, ----) ||----|!| n1::even| |n - 1 { (n1 - 1) // n1 \ \2 | |n - 1 { 2 \\ 2 / / | |----- { 2 ||---- - 1/2|!| n1::odd | |----- { | | \ { \\ 2 / / | | \ { 0 n1::odd | {n! | ) --------------------------------------------|, n! | ) -------------------------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A372342" LREtools/SearchTable: "SearchTable successful" n {(-1) ((n + 1) hypergeom([1/2, -n - 1], [1], 4) + (3 n + 1) hypergeom([1/2, -n], [1], 4))} "A372370" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n {(-1) binomial(2 n, n) hypergeom([-n, n/2, n/2 + 1/2], [n, n + 1], 4), binomial(2 n, n)} "A372413" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A372420" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) (5 n1 + 3)| {2 , 2 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A372506" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 1/2 4 binomial(---, n/2) n::even { 2 { {{ (2 n - 2) 3 n , { 3 2 (3 n - 1) binomial(--- - 3/2, n/2 - 1/2) { 2 { ----------------------------------------------------- n::odd { n { 3 n 3 n { 6 binomial(---, n/2) binomial(3 n, ---) { 2 2 { --------------------------------------- n::even { binomial(n, n/2) { } { 3 n 3 n { (n + 1) binomial(--- + 3/2, n/2 + 1/2) binomial(3 n + 3, --- + 3/2) { 2 2 { ------------------------------------------------------------------- n::odd { (3 n + 2) binomial(n + 1, n/2 + 1/2) "A372611" memory used=290135.0MB, alloc=3639.5MB, time=2210.20 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-n1 - 1) | n n | \ 2 binomial(2 n1, n1) (7 n1 + 5)| {2 , 2 | ) ----------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A372716" LREtools/SearchTable: "SearchTable successful" n n {(-1) binomial(2 n, n) n! BesselI(n + 1/2, 1/2), (-1) binomial(2 n, n) n! BesselK(n + 1/2, -1/2)} "A372829" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ / { (-n1) n1 / n1 \ \ | { | | { 2 (n1 + 1) binomial(n1, ----) |----|! n1::even| |n - 1 { / n1 \ / n1 \ | |n - 1 { 2 \ 2 / | |----- { |---- + 1/2| |---- - 1/2|! n1::odd | |----- { | | \ { \ 2 / \ 2 / | | \ { 0 n1::odd | {n! | ) --------------------------------------------|, n! | ) ------------------------------------------------------------|, n!} | / (n1 + 1)! | | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A372986" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { n {{ , { 4 GAMMA(n/2 + 1) GAMMA(n/2 + 3/4) n::even} { (2 n - 2) { { 2 GAMMA(n/2 + 1) GAMMA(n/2 + 3/4) n::odd { 0 n::odd "A372987" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { /3 n\ { { |---| {{ /3 n \ , { \ 2 / } { |--- - 3/2| { 2 GAMMA(n/2 + 3/4) n::even { \ 2 / { { 2 GAMMA(n/2 + 3/4) n::odd { 0 n::odd "A372988" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 { (2 n - 2) {{ , { , { 2 GAMMA(n/4 + 5/8) GAMMA(n/4 + 7/8) irem(n, 4) = 1, { 0 irem(n, 4) = 2 { 1/16 %1 irem(n, 4) = 2 { { { { 0 irem(n, 4) = 2 { 1/64 %1 irem(n, 4) = 3 { 0 irem(n, 4) = 3 { { 0 irem(n, 4) = 3 { %1 irem(n, 4) = 0 { { 0 irem(n, 4) = 1 { } { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 n %1 := 4 GAMMA(n/4 + 5/8) GAMMA(n/4 + 7/8) "A372991" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { 0 irem(n, 3) = 0 { { { 0 irem(n, 3) = 1 { /2 n \ {{ , { |--- - 2/3| , { /2 n \ { \ 3 / { |--- - 4/3| { 6 GAMMA(n/3 + 1) GAMMA(n/3 + 5/6) irem(n, 3) = 1 { \ 3 / { { 6 GAMMA(n/3 + 1) GAMMA(n/3 + 5/6) irem(n, 3) = 2 { 0 irem(n, 3) = 2 { /2 n\ { |---| { \ 3 / { 6 GAMMA(n/3 + 1) GAMMA(n/3 + 5/6) irem(n, 3) = 0} { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A372995" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { 0 irem(n, 4) = 0 { 0 irem(n, 4) = 0 { { { 0 irem(n, 4) = 1 { 0 irem(n, 4) = 1 {{ , { , { 0 irem(n, 4) = 2 { 1/2 n (n/4 - 1/2)! binomial(n/2 - 1, n/4 - 1/2) irem(n, 4) = 2 { { { (n/2 - 3/2) { 0 irem(n, 4) = 3 { 2 GAMMA(n/4 + 1) irem(n, 4) = 3 { 0 irem(n, 4) = 0 { (n/2) { { 2 (n/4)! irem(n, 4) = 0 { (n/2 - 1/2) { { 2 GAMMA(n/4 + 1) irem(n, 4) = 1, { 0 irem(n, 4) = 1} { { { 0 irem(n, 4) = 2 { 0 irem(n, 4) = 2 { { { 0 irem(n, 4) = 3 { 0 irem(n, 4) = 3 "A373006" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 4 { ------------------------ n::even n n n { (n + 1) binomial(n, n/2) { 2 binomial(n, n/2) n::even {1, 2 , (-I) , I , { , { } { (2 n - 2) { binomial(n + 1, n/2 + 1/2) n::odd { 2 2 { ---------------------------- n::odd { n binomial(n - 1, n/2 - 1/2) "A373175" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | n n | \ (16 n1 - 11) BesselI(n1 + 1/2, 1) - 12 BesselI(n1 - 1/2, 1)| {(2/3) n! (8 n - 1), (2/3) n! (8 n - 1) | ) -----------------------------------------------------------|, | / (n1 + 1) | |----- (2/3) (n1 + 1)! (8 n1 + 7) (24 n1 - 3) | \n1 = 0 / /n - 1 \ |----- | n | \ (16 n1 - 11) BesselK(n1 + 1/2, -1) - 12 BesselK(n1 - 1/2, -1)| (2/3) n! (8 n - 1) | ) -------------------------------------------------------------|} | / (n1 + 1) | |----- (2/3) (n1 + 1)! (8 n1 + 7) (24 n1 - 3) | \n1 = 0 / "A373176" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- 2 | n 2 n 2 | \ (10 n1 - 7 n1 + 10) BesselI(n1 + 1/2, 1) + (-5 n1 + 8) BesselI(n1 - 1/2, 1)| {(2/3) n! (8 n + 3 n + 4), (2/3) n! (8 n + 3 n + 4) | ) ----------------------------------------------------------------------------|, | / (n1 + 1) 2 2 | |----- (2/3) (n1 + 1)! (8 (n1 + 1) + 3 n1 + 7) (24 n1 + 9 n1 + 12) | \n1 = 0 / /n - 1 \ |----- 2 | n 2 | \ (10 n1 - 7 n1 + 10) BesselK(n1 + 1/2, -1) + (-5 n1 + 8) BesselK(n1 - 1/2, -1)| (2/3) n! (8 n + 3 n + 4) | ) ------------------------------------------------------------------------------|} | / (n1 + 1) 2 2 | |----- (2/3) (n1 + 1)! (8 (n1 + 1) + 3 n1 + 7) (24 n1 + 9 n1 + 12) | \n1 = 0 / "A373182" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A373339" n {2 (n - 2), n!} "A373340" n {2 (n - 2), n!} "A373578" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A373614" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | n |3/2 - ----| binomial(2 n, n) |3/2 + ----| binomial(2 n, n) (-1) binomial(2 n, n) \ 2 / \ 2 / {----------------------, ------------------------------, ------------------------------} n + 1 n + 1 n + 1 "A373622" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 5 | | 5 | |3/2 - ----| binomial(2 n, n) |3/2 + ----| binomial(2 n, n) \ 2 / \ 2 / {------------------------------, ------------------------------} n + 1 n + 1 "A373651" LREtools/SearchTable: "SearchTable successful" n {(n + 2) (n + 1) (-1) hypergeom([1/2, -n], [1], 4)} "A373713" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ (-9) GAMMA(n1 + 1/3) | n | \ (-9) GAMMA(n1 + 2/3) | {(9/7) , (9/7) | ) ---------------------------|, (9/7) | ) ---------------------------|} | / (n1 + 1)| | / (n1 + 1)| |----- GAMMA(n1 + 2) (9/7) | |----- GAMMA(n1 + 2) (9/7) | \n1 = 0 / \n1 = 0 / "A373714" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 3, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n / 1/2\n | 18 2 | | 18 2 | {|9/7 - -------| , |9/7 + -------| , \ 7 / \ 7 / /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- n2 ||| | 18 2 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ (-9) GAMMA(n2 + 2/3) ||| |9/7 - -------| | ) |-7/9 (-1) (-1 + 2 2 ) (1 + 2 2 ) | ) -------------------------------------|||, \ 7 / | / | | / / 1/2\(n2 + 1)||| |----- | |----- | 18 2 | ||| |n1 = 0 | |n2 = 0 GAMMA(n2 + 3) |9/7 + -------| ||| \ \ \ \ 7 / /// /n - 1 / /n1 - 1 \\\ / 1/2\n |----- | |----- n2 ||| | 18 2 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 | \ (-9) GAMMA(n2 + 4/3) ||| |9/7 - -------| | ) |-7/9 (-1) (-1 + 2 2 ) (1 + 2 2 ) | ) -------------------------------------|||} \ 7 / | / | | / / 1/2\(n2 + 1)||| |----- | |----- | 18 2 | ||| |n1 = 0 | |n2 = 0 GAMMA(n2 + 3) |9/7 + -------| ||| \ \ \ \ 7 / /// "A373715" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(3 _Z - 12 _Z - 48 _Z - 64, index = 1) , RootOf(3 _Z - 12 _Z - 48 _Z - 64, index = 2) , RootOf(3 _Z - 12 _Z - 48 _Z - 64, index = 3) } "A373740" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A373742" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A373760" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A373770" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A373783" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A373816" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n {RootOf(3 _Z - 48 _Z + 192 _Z - 256, index = 1) , RootOf(3 _Z - 48 _Z + 192 _Z - 256, index = 2) , 3 2 n RootOf(3 _Z - 48 _Z + 192 _Z - 256, index = 3) } "A373818" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ /n - 1 \ |----- n1 | |----- n1 | n n | \ 9 GAMMA(n1 + 1/3) | n | \ 9 GAMMA(n1 + 2/3) | {(72/7) , (72/7) | ) ----------------------------|, (72/7) | ) ----------------------------|} | / (n1 + 1)| | / (n1 + 1)| |----- GAMMA(n1 + 2) (72/7) | |----- GAMMA(n1 + 2) (72/7) | \n1 = 0 / \n1 = 0 / "A374162" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A374487" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-I 7 ) (7 LegendreP(n + 1, 1/7 I 7 ) I - 7 LegendreP(n, 1/7 I 7 )) (n + 1), 1/2 n 1/2 1/2 1/2 (-I 7 ) (7 LegendreQ(n + 1, 1/7 I 7 ) I - 7 LegendreQ(n, 1/7 I 7 )) (n + 1)} "A374488" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-I 11 ) (11 LegendreP(n + 1, 1/11 I 11 ) I - 11 LegendreP(n, 1/11 I 11 )) (n + 1), 1/2 n 1/2 1/2 1/2 (-I 11 ) (11 LegendreQ(n + 1, 1/11 I 11 ) I - 11 LegendreQ(n, 1/11 I 11 )) (n + 1)} "A374497" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) (LegendreP(n + 1, I) I - LegendreP(n, I)) (n + 1), (-2 I) (LegendreQ(n + 1, I) I - LegendreQ(n, I)) (n + 1)} "A374506" LREtools/SearchTable: "SearchTable successful" n {(-1) ((-15 n - 33) hypergeom([1/2, -n], [1], 4) + (13 n + 27) hypergeom([1/2, -n - 1], [1], 4)) (n + 1) (n + 4)} "A374508" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (n/2) | 5 1/2 5 | {5 (n + 1) |(-10 n - 35) LegendreP(n, ----) + 5 (4 n + 11) LegendreP(n + 1, ----)|, \ 5 5 / / 1/2 1/2 \ (n/2) | 5 1/2 5 | 5 (n + 1) |(-10 n - 35) LegendreQ(n, ----) + 5 (4 n + 11) LegendreQ(n + 1, ----)|} \ 5 5 / "A374509" LREtools/SearchTable: "SearchTable successful" / 1/2 1/2 \ (n/2) | 2 5 1/2 2 5 | {5 (n + 1) |(-15 n - 165 n - 400) LegendreP(n, ----) + 5 (11 n + 89 n + 160) LegendreP(n + 1, ----)|, \ 5 5 / / 1/2 1/2 \ (n/2) | 2 5 1/2 2 5 | 5 (n + 1) |(-15 n - 165 n - 400) LegendreQ(n, ----) + 5 (11 n + 89 n + 160) LegendreQ(n + 1, ----)|} \ 5 5 / "A374511" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) (2 I LegendreP(n + 1, I) - LegendreP(n, I)) (n + 1) (n + 2), (-2 I) (2 I LegendreQ(n + 1, I) - LegendreQ(n, I)) (n + 1) (n + 2)} "A374513" LREtools/SearchTable: "SearchTable successful" n 2 2 {(-2 I) ((-3 n - 15 n - 16) LegendreP(n, I) + (7 n + 37 n + 44) LegendreP(n + 1, I) I) (n + 1), n 2 2 (-2 I) ((-3 n - 15 n - 16) LegendreQ(n, I) + (7 n + 37 n + 44) LegendreQ(n + 1, I) I) (n + 1)} "A374541" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A374542" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 5 4 3 2 | | \ binomial(2 n1, n1) (3 n1 + 27 n1 + 10 n1 + 48 n1 + 278 n1 + 84)| {1, (3 n + 5) | ) -------------------------------------------------------------------|, 3 n + 5} | / (n1 + 4) (n1 + 3) (n1 + 2) (n1 + 1) (3 n1 + 8) (3 n1 + 5) | |----- | \n1 = 0 / "A374563" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A374574" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 ----- ----- \ 2 \ {1, ) (2 n1 + 3) (2 n1 + 1) (n1!) binomial(2 n1, n1), ) n1!} / / ----- ----- n1 = 0 n1 = 0 "A374585" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A374598" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A374599" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A374605" LREtools/SearchTable: "SearchTable not successful" {} "A374647" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 9 (n/3)! binomial(n, n/3) irem(n, 3) = 0 { { 2 binomial(n + 2, n/3 + 2/3) (n/3 + 2/3)! (2 n + 1) {{ --------------------------------------------------- irem(n, 3) = 1, { n + 1 { { 6 (n/3 + 1/3)! binomial(n + 1, n/3 + 1/3) irem(n, 3) = 2 { (n/3) { 6 (27/4) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { --------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 5/6) { { (n/3 - 1/3) { 9 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1/3) , { ----------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 1/2) { { (n/3 + 1/3) { 2 (27/4) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) (2 n + 1) { ------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 7/6) (n + 1) { (n/3) { 2 (27/4) GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) (2 n + 1) { ------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 7/6) (n + 1) { { (n/3 - 1/3) { 6 (27/4) GAMMA(n/3 + 2/3) GAMMA(n/3 + 1) } { --------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 5/6) { { (n/3 - 2/3) { 9 (27/4) GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) { ----------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1/2) "A374651" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n n | \ | (-1) (n1 + 1) (n1 + 2) (n1 + 3) n1!|| {(-1) , (-1) (n + 4), (-1) (n + 4) | ) |- -------------------------------------||} | / \ (n1 + 4) (n1 + 5) /| |----- | \n1 = 0 / "A374652" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n n | \ | (-1) (n1 + 1) (n1 + 2) (2 n1 + 9) n1!|| {(-1) , (-1) (n + 5), (-1) (n + 5) | ) |- ---------------------------------------||} | / \ (n1 + 5) (n1 + 6) /| |----- | \n1 = 0 / "A374653" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ (n1 + 2) (n1 + 1) n1! (2 n1 + 7)| {1, (n + 3) | ) --------------------------------|, n + 3} | / (n1 + 4) (n1 + 3) | |----- | \n1 = 0 / "A374835" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\ / 1/2 \n / 1/2 \ / 1/2\n |----- | | 5 | | 37 39 5 | |5 | | 39 5 37 | | 5 | 1/2 | \ | n1 1/2 (-n1 - 1) {|1/2 - ----| |n + --- - -------|, |---- + 1/2| |n + ------- + ---|, |1/2 - ----| (605 n + 185 - 39 5 ) | ) |- 2 (-1) (5 - 1) \ 2 / \ 121 605 / \ 2 / \ 605 121/ \ 2 / | / | |----- | \n1 = 0 \ 1/2 n1 2 (5 + 1) (121 n1 + 195 n1 + 38) / / 1/2 \(-n2 - 1) / 1/2\ \ |n1 - 1 |5 | | 158 39 5 | 2 | |----- |---- + 1/2| (3 n2 + 1) (3 n2 + 2) (3 n2 + 4) |n2 + --- - -------| (55 n2 + 35 n2 - 18) binomial(3 n2, n2)| / | \ \ 2 / \ 121 605 / | / | | ) --------------------------------------------------------------------------------------------------------------------| / | | / 2 2 | / \ |----- (n2 + 1) (n2 + 2) (2 n2 + 1) (2 n2 + 3) (121 (n2 + 1) + 195 n2 + 233) (121 n2 + 195 n2 + 38) | \n2 = 0 / \\ || / 1/2\ \|| | 37 39 5 | 1/2 ||| |n1 + --- - -------| (605 n1 + 790 - 39 5 )|||} \ 121 605 / /|| || // "A374836" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\n // 1/2\ 1/2\ / 1/2 \n // 1/2 \ 1/2 \ / 1/2\n | 5 | ||50 117 5 | 2 150 534 5 | |5 | ||117 5 50 | 2 534 5 150 | | 5 | {|1/2 - ----| ||--- - --------| n + n + ---- + --------|, |---- + 1/2| ||-------- + ---| n + n - -------- + ----|, |1/2 - ----| \ 2 / \\109 109 / 1199 5995 / \ 2 / \\ 109 109/ 5995 1199/ \ 2 / /n - 1 / |----- | 1/2 2 1/2 | \ | n1 1/2 (-n1 - 1) 1/2 n1 (-1287 5 n + 6655 n - 210 5 - 550 n - 642) | ) |- 2 (-1) (5 - 1) (5 + 1) | / | |----- | \n1 = 0 \ /n1 - 1 |----- / 1/2 \(-n2 - 1) 4 3 2 | \ |5 | (1331 n1 + 2442 n1 + 247 n1 - 1008 n1 - 216) | ) |---- + 1/2| (3 n2 + 1) (3 n2 + 2) (3 n2 + 4) | / \ 2 / |----- \n2 = 0 / 1/2 1/2\ | 2 117 5 (n2 + 1) 10 n2 1192 42 5 | 3 2 / |(n2 + 1) - ----------------- - ----- - ---- - -------| (55 n2 - 40 n2 - 41 n2 - 18) binomial(3 n2, n2) / ((n2 + 1) (2 n2 + 1) (2 n2 + 3) \ 605 121 6655 1331 / / \ | 4 3 2 4 3 2 | / / (1331 (n2 + 1) + 2442 (n2 + 1) + 247 (n2 + 1) - 1008 n2 - 1224) (1331 n2 + 2442 n2 + 247 n2 - 1008 n2 - 216))| / | | / \ | / \\ || / 2 117 1/2 10 642 42 1/2\ 1/2 2 1/2 \|| |n1 - --- 5 n1 - --- n1 - ---- - ---- 5 | (-1287 5 (n1 + 1) + 6655 (n1 + 1) - 210 5 - 550 n1 - 1192)|||} \ 605 121 6655 1331 / /|| || // "A374866" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { {{ (-n + 1) 2 3 n 3 n , { 2 ((n/2 - 1/2)!) (3 n - 2) (3 n + 2) binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) n::odd { 2 2 { n 2 /3 n \ 3 n { 2 ((n/2)!) |--- + 1| binomial(n, n/2) binomial(---, n/2) n::even { \ 2 / 2 } { { 0 n::odd "A374882" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A374980" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 ) n! BesselI(n + 1/2, 2 ), (-2 ) n! BesselK(n + 1/2, -2 )} "A375021" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375022" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375026" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" / { 0 n1::even\ | { | | { / n1 \ | | { |---- - 1/2| | | { \ 2 / n1 | | { 2 (-1) binomial(n1 - 1, ---- - 1/2) | |n - 1 { 2 | |----- { ----------------------------------------------- n1::odd | n n | \ { n1 + 1 | {(-3/2) , (-3/2) | ) -----------------------------------------------------------------|, | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / / { / n1 \ \ | { |----| | | { \ 2 / | | { 2 (-16) | | { ------------------------------ n1::even| | { n1 | | { (n1 + 1) n1 binomial(n1, ----) | |n - 1 { 2 | |----- { | n | \ { 0 n1::odd | (-3/2) | ) ------------------------------------------------|} | / (n1 + 1) | |----- (-3/2) | \n1 = 0 / "A375086" n n binomial(2 n, n) {2 , ------------------} 2 n - 1 "A375181" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 5 _Z + 5 _Z + 5 _Z - 7 "A375218" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A375222" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 n 1/2 {(-2 ) n! (n + 1) BesselI(n + 1/2, 2 ), (-2 ) n! (n + 1) BesselK(n + 1/2, -2 )} "A375223" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SearchTable: "SearchTable successful" n 2 1/2 n 1/2 1/2 1/2 {(n + 1) (2 n + 1) (1/2) (n!) binomial(2 n, n), -(-2 ) n! (n + 1) ((2 n + 1) 2 BesselI(n + 1/2, 2 ) - 2 BesselI(n - 1/2, 2 )), 1/2 n 1/2 1/2 1/2 -(-2 ) n! (n + 1) ((2 n + 1) 2 BesselK(n + 1/2, -2 ) - 2 BesselK(n - 1/2, -2 ))} "A375248" LREtools/SearchTable: "SearchTable successful" n {(-1) ((-6 n - 15) hypergeom([1/2, -n], [1], 4) + (4 n + 9) hypergeom([1/2, -n - 1], [1], 4)) (n + 1) (n + 5)} "A375253" LREtools/SearchTable: "SearchTable successful" n {(n + 4) (n + 3) (-1) (n + 1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4))} "A375259" LREtools/SearchTable: "SearchTable successful" n {(n + 5) (n + 4) (n + 3) (-1) (n + 1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4))} "A375260" LREtools/SearchTable: "SearchTable successful" n {(n + 3) (n + 2) (n + 1) (-1) hypergeom([1/2, -n], [1], 4)} "A375276" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375292" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A375293" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A375409" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ I HermiteH(n1 + 1, 1/2 I)| {n! | ) ---------------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A375424" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ -I I (HermiteH(n1 + 1, 1/2 I) - 2 I (n1 + 1) HermiteH(n1, 1/2 I))| {n! n, n! n | ) -------------------------------------------------------------------|} | / (n1 + 1)! (n1 + 1) n1 | |----- | \n1 = 0 / "A375425" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | 2 2 | \ I ((2 n1 + 5) HermiteH(n1 + 1, 1/2 I) - 6 I (n1 + 1) HermiteH(n1, 1/2 I))| {n! (n + n + 3), n! (n + n + 3) | ) ---------------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + n1 + 4) (n1 + n1 + 3) | \n1 = 0 / "A375434" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375435" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375436" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375437" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A375470" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A375565" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A375691" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A375990" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1)| n! | ) ------------| | / n1 (n1 + 1)!| |----- | n! \n1 = 0 / {----, ------------------------} n n "A376159" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A376174" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376175" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376269" 2 {n! (n + 2) , n - 1} "A376277" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) } 4 3 2 %1 := _Z - 9 _Z + 26 _Z - 24 _Z - 1 "A376282" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { { /22235661\(n/3 - 2/3) 10 16 19 13 {{ |--------| GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 4/21) GAMMA(n/3 + --) GAMMA(n/3 + 1/21) , { \ 256 / 21 21 21 21 { ------------------------------------------------------------------------------------------------------------------------- irem(n, 3) = 2 { 13 { GAMMA(n/3 + 5/6) GAMMA(n/3 + --) GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) GAMMA(n/3 + 7/12) { 12 { 0 irem(n, 3) = 0 { { /22235661\(n/3 - 1/3) 13 16 19 10 { |--------| GAMMA(n/3 + 4/21) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 1/21) GAMMA(n/3 + --) { \ 256 / 21 21 21 21 { ------------------------------------------------------------------------------------------------------------------------- irem(n, 3) = 1 { 13 { GAMMA(n/3 + 2/3) GAMMA(n/3 + 7/12) GAMMA(n/3 + 1) GAMMA(n/3 + --) GAMMA(n/3 + 1/3) GAMMA(n/3 + 5/6) { 12 { { 0 irem(n, 3) = 2 { /22235661\(n/3) 10 13 19 16 { |--------| GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 4/21) GAMMA(n/3 + --) GAMMA(n/3 + 1/21) GAMMA(n/3 + --) { \ 256 / 21 21 21 21 { ------------------------------------------------------------------------------------------------------------------- irem(n, 3) = 0 { 13 , { GAMMA(n/3 + 1/3) GAMMA(n/3 + --) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) GAMMA(n/3 + 7/12) GAMMA(n/3 + 5/6) } { 12 { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A376317" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {1, (-1) } "A376388" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" / 1/2\n / 1/2 \n | 5 | |5 | {|1/2 - ----| , |---- + 1/2| } \ 2 / \ 2 / "A376395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" 1/2 n 1/2 n {(1 - 2 ) , (1 + 2 ) } "A376489" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376512" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376547" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A376568" LREtools/SearchTable: "SearchTable successful" / 1/2 \n 1/2 | 3 13 | 3 13 {|- ------- + 9/2| hypergeom([1/3, - 1/3 - n], [2/3], 13/2 + -------)} \ 2 / 2 "A376574" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A376721" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A376722" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A376783" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A376784" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A376791" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A376802" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376803" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376804" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376806" memory used=291626.5MB, alloc=3671.5MB, time=2221.05 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A376809" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376810" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376835" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A376836" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A376884" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377011" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(16/3) , (16/3) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (16/3) | \n1 = 0 / "A377013" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ 1 | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A377065" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 3 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A377145" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377148" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377150" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A377152" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377186" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377189" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377190" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377194" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377195" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377196" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377197" LREtools/SearchTable: "SearchTable successful" 2 3 2 (n + 1) (64 n + 40 n - 1) hypergeom([-1/2, -n - 1], [1], -4) + (-64 n - 136 n - 79 n - 1) hypergeom([-1/2, -n], [1], -4) {---------------------------------------------------------------------------------------------------------------------------} n "A377198" LREtools/SearchTable: "SearchTable successful" 2 2 (n - 1) (n + 1) LegendreQ(n + 1, 3) + (-7 n + 2 n + 3) LegendreQ(n, 3) (7 n - 2 n - 3) LegendreP(n, 3) - (n - 1) (n + 1) LegendreP(n + 1, 3) {-----------------------------------------------------------------------, - ----------------------------------------------------------------------} n n "A377199" LREtools/SearchTable: "SearchTable successful" 3 2 {((n + 1) (512 n + 1472 n + 592 n - 3) hypergeom([-1/2, -n - 1], [1], -4) 4 3 2 + (-512 n - 2240 n - 2960 n - 1265 n - 3) hypergeom([-1/2, -n], [1], -4))/n} "A377200" LREtools/SearchTable: "SearchTable successful" 4 3 2 {((n + 1) (4096 n + 27136 n + 44096 n + 14736 n - 15) hypergeom([-1/2, -n - 1], [1], -4) 5 4 3 2 + (-4096 n - 33280 n - 86080 n - 88720 n - 32049 n - 15) hypergeom([-1/2, -n], [1], -4))/n} "A377204" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377213" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377215" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377216" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377233" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (27 n - 1) hypergeom([-1/3, -n - 1], [1], -9) + (-27 n - 35 n - 2) hypergeom([-1/3, -n], [1], -9) {----------------------------------------------------------------------------------------------------------} n "A377234" LREtools/SearchTable: "SearchTable successful" 2 (n + 1) (27 n + 1) hypergeom([1/3, -n - 1], [1], -9) + (-27 n - 19 n - 4) hypergeom([1/3, -n], [1], -9) {--------------------------------------------------------------------------------------------------------} n "A377235" LREtools/SearchTable: "SearchTable successful" 2 3 2 (n + 1) (729 n + 567 n - 2) hypergeom([-1/3, -n - 1], [1], -9) + (-729 n - 1539 n - 844 n - 4) hypergeom([-1/3, -n], [1], -9) {--------------------------------------------------------------------------------------------------------------------------------} n "A377239" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377241" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377243" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377260" LREtools/SearchTable: "SearchTable successful" / 1/2 \n | 3 13 | {|- ------- + 9/2| \ 2 / / 1/2 1/2 \ | 3 13 1/2 3 13 | |(9 n + 9) hypergeom([2/3, - 2/3 - n], [4/3], 13/2 + -------) - (3 n + 2) (3 + 13 ) hypergeom([2/3, 1/3 - n], [4/3], 13/2 + -------)| \ 2 2 / 1/2 (13 - 3)} "A377261" LREtools/SearchTable: "SearchTable successful" / 1/2 \n | 3 13 | {|- ------- + 9/2| \ 2 / / 1/2 1/2 \ | 3 13 1/2 3 13 | |(9 n + 9) hypergeom([1/3, -n - 4/3], [2/3], 13/2 + -------) - (3 + 13 ) (3 n + 4) hypergeom([1/3, - 1/3 - n], [2/3], 13/2 + -------)| \ 2 2 / 1/2 (13 - 3)} "A377267" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { {{ (n/3 - 2/3) 14 11 , { (84375/4) GAMMA(n/3 + 2/15) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 8/15) { 15 15 { ---------------------------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 5/6) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) 14 11 { (84375/4) GAMMA(n/3 + 2/15) GAMMA(n/3 + 8/15) GAMMA(n/3 + --) GAMMA(n/3 + --) { 15 15 , { ---------------------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/3) GAMMA(n/3 + 1) GAMMA(n/3 + 5/6) { { 0 irem(n, 3) = 2 { (n/3) 14 11 { (84375/4) GAMMA(n/3 + 8/15) GAMMA(n/3 + 2/15) GAMMA(n/3 + --) GAMMA(n/3 + --) { 15 15 { ---------------------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 1) GAMMA(n/3 + 4/3) GAMMA(n/3 + 2/3) GAMMA(n/3 + 5/6) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A377394" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \ \ | |----- | | | | \ n2! (n2 + 1)| | | n1! | ) ------------| (n1 + 1)| /n - 1 \ |n - 1 | / n2 (n2 + 1)!| | |----- | |----- |----- | | | \ n1! (n1 + 1)| | \ \n2 = 0 / | n! | ) ------------| n! | ) ----------------------------------| | / n1 (n1 + 1)!| | / n1 (n1 + 1)! | |----- | |----- | n! \n1 = 0 / \n1 = 0 / {----, ------------------------, ----------------------------------------------} n n n "A377395" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n2 - 1 \ \ \ | | |----- | | | | | | \ n3! (n3 + 1)| | | | | n2! | ) ------------| (n2 + 1)| | / /n1 - 1 \ \ | |n1 - 1 | / n3 (n3 + 1)!| | | | |----- | | | |----- |----- | | | | | \ n2! (n2 + 1)| | | | \ \n3 = 0 / | | | n1! | ) ------------| (n1 + 1)| | n1! | ) ----------------------------------| (n1 + 1)| /n - 1 \ |n - 1 | / n2 (n2 + 1)!| | |n - 1 | / n2 (n2 + 1)! | | |----- | |----- |----- | | |----- |----- | | | \ n1! (n1 + 1)| | \ \n2 = 0 / | | \ \n2 = 0 / | n! | ) ------------| n! | ) ----------------------------------| n! | ) --------------------------------------------------------| | / n1 (n1 + 1)!| | / n1 (n1 + 1)! | | / n1 (n1 + 1)! | |----- | |----- | |----- | n! \n1 = 0 / \n1 = 0 / \n1 = 0 / {----, ------------------------, ----------------------------------------------, -------------------------------------------------------------------- n n n n } "A377461" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) 2 4 { 1/4 2 (n + 2) ((n/2)!) binomial(n, n/2) n::even { {{ (-n - 1) 4 , { 2 ((n/2 + 1/2)!) (n + 3) binomial(n + 1, n/2 + 1/2) { ------------------------------------------------------------ n::odd { n + 1 { (-n) 2 3 4 { 4 8 (n + 1) binomial(n, n/2) ((n/2)!) (n + 3) n::even { } { (-3 n + 3) 3 2 3 4 { 2 n (n + 2) binomial(n - 1, n/2 - 1/2) ((n/2 - 1/2)!) n::odd "A377586" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A377659" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {2 , -------------------------------------------------------------------------} n + 2 "A377825" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 1, 2 LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n n | 2 | |2 | {n! (n + 1), (n + 1) (-1) n!, (n + 1) |- ----| n!, (n + 1) |----| n!} \ 2 / \ 2 / "A377954" LREtools/SearchTable: "SearchTable successful" n {(-I) (3 HermiteH(n + 1, 1/2 I) + (2 n + 1) HermiteH(n, 1/2 I) I)} "A377955" LREtools/SearchTable: "SearchTable successful" n {(-I) ((2 n + 5) HermiteH(n + 1, 1/2 I) + (8 n + 3) HermiteH(n, 1/2 I) I)} "A377956" LREtools/SearchTable: "SearchTable successful" n 2 {(-I) ((12 n + 5) HermiteH(n + 1, 1/2 I) + (4 n + 14 n + 11) HermiteH(n, 1/2 I) I)} "A377958" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | | \ HermiteH(n1 + 1, 1/2)| {n! | ) ---------------------|, n!} | / (n1 + 1)! | |----- | \n1 = 0 / "A377959" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | | \ (2 n1 + 2) HermiteH(n1, 1/2) + HermiteH(n1 + 1, 1/2)| {n! (n + 2), n! (n + 2) | ) ----------------------------------------------------|} | / (n1 + 1)! (n1 + 3) (n1 + 2) | |----- | \n1 = 0 / "A377960" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- | 2 2 | \ (-6 n1 - 6) HermiteH(n1, 1/2) + (2 n1 + 3) HermiteH(n1 + 1, 1/2)| {n! (n + 5 n + 3), n! (n + 5 n + 3) | ) ----------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! ((n1 + 1) + 5 n1 + 8) (n1 + 5 n1 + 3) | \n1 = 0 / "A377963" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378060" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 2 2 { 4 16 { 4 binomial(n, n/2) n { -------------------------- n::even { ---------------------- n::even { 2 2 { 2 { (n + 1) binomial(n, n/2) { (n + 2) {{ , { } { (4 n + 4) 2 { 2 2 { 2 n { 16 binomial(n - 1, n/2 - 1/2) n { --------------------------------------------- n::odd { --------------------------------- n::odd { 2 2 2 { 2 { (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) { (n + 1) "A378070" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 16 (n - 1) { ---------------------------- n::even { 2 { 2 2 { 4 binomial(n, n/2) n::even { n (n + 1) binomial(n, n/2) { {{ , { 2 } { (4 n - 4) { binomial(n + 1, n/2 + 1/2) (n - 1) (n + 1) { 4 2 { ------------------------------------------- n::odd { ------------------------------ n::odd { 2 { 2 2 { n { n binomial(n - 1, n/2 - 1/2) "A378078" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (4 LegendreP(n, 1/2 I 2 ) + 2 (3 n + 4) LegendreP(n + 1, 1/2 I 2 ) I) (n + 1) {---------------------------------------------------------------------------------------------------------, (n + 3) (n + 2) n 1/2 n 1/2 1/2 1/2 1/2 -I (-2 I 2 ) 2 (4 LegendreQ(n, 1/2 I 2 ) + 2 (3 n + 4) LegendreQ(n + 1, 1/2 I 2 ) I) (n + 1) ---------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n "A378079" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378100" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 { 0 n::even { (- n/2) {(-1/2 I 2 ) HermiteH(n, 1/2 I 2 ), { , { 2 binomial(n, n/2) (n/2)! n::even} { (n/2 - 1/2) { { 2 (n/2 - 1/2)! n::odd { 0 n::odd "A378153" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" n {(n + 5) (n + 4) (-1) } "A378382" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 3 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 \ |----- n1 | |----- / n1 \| n n | \ (-I) HermiteH(n1, I) | n | \ | (-1) || {(n + 1) (-1) n!, (n + 1) (-1) n! | ) -------------------------------|, (n + 1) (-1) n! | ) |- ------------------||} | / (n1 + 1) | | / \ (n1 + 2) (n1 + 1)!/| |----- (n1 + 2) (-1) (n1 + 1)!| |----- | \n1 = 0 / \n1 = 0 / "A378402" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378406" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A378407" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A378411" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378425" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A378426" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378465" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378483" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1)| {(27/4) , (27/4) | ) ----------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (27/4) | \n1 = 0 / "A378484" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1) (24 n1 + 19) | {|---| (6 n + 7), |---| (6 n + 7) | ) -----------------------------------------------------------------------|} \27 / \27 / | / /256\(n1 + 1) | |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) |---| (6 n1 + 13) (162 n1 + 189)| \n1 = 0 \27 / / "A378503" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ (4 n1 + 1) (2 n1 + 1) binomial(4 n1, n1) (24 n1 + 17)| {|---| , |---| | ) -----------------------------------------------------|} \27 / \27 / | / /256\(n1 + 1) | |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A378504" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) | {(27/4) (n + 1), (27/4) (n + 1) | ) ---------------------------------------------|} | / (n1 + 1) | |----- (2 n1 + 1) (27/4) (n1 + 2) (4 n1 + 4)| \n1 = 0 / "A378611" LREtools/SearchTable: "SearchTable successful" 2 4 (n + 1) (2 n + 1) hypergeom([2 n + 3, -n - 1], [1], -1) + (-61 n - 65 n - 16) hypergeom([-n, 2 n + 1], [1], -1) {------------------------------------------------------------------------------------------------------------------} (17 n + 11) n "A378670" LREtools/SearchTable: "SearchTable successful" 3 (n + 1) (4 n + 1) hypergeom([-n - 1, 3 n + 4], [1], -1) - (3 n + 1) (19 n + 15) hypergeom([-n, 3 n + 1], [1], -1) {-------------------------------------------------------------------------------------------------------------------} 2 (238 n + 289 n + 83) n "A378801" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378816" LREtools/SearchTable: "SearchTable successful" 2 2 (4 n + 6 n - 1) hypergeom([1/2, -n - 1], [1], 4) - (2 n + 1) hypergeom([1/2, -n], [1], 4) {-------------------------------------------------------------------------------------------} n (n + 2) "A378849" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378850" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A378939" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" LegendreQ(n + 1, 3) - 3 LegendreQ(n, 3) 3 LegendreP(n, 3) - LegendreP(n + 1, 3) {---------------------------------------, - ---------------------------------------, n n { 2 ((n + 3) LegendreP(n/2 + 3/2, 3) + (-7 n - 13) LegendreP(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::even { n + 1 { , { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::even { n + 1 { , { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::odd { n { 2 (n + 2) (LegendreP(n/2 + 1, 3) - 3 LegendreP(n/2, 3)) { ------------------------------------------------------- n::even { n { , { 2 ((n + 3) LegendreP(n/2 + 3/2, 3) + (-7 n - 13) LegendreP(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::odd { n + 1 { 2 (n + 2) (LegendreQ(n/2 + 1, 3) - 3 LegendreQ(n/2, 3)) { ------------------------------------------------------- n::even { n { } { 2 ((n + 3) LegendreQ(n/2 + 3/2, 3) + (-7 n - 13) LegendreQ(n/2 + 1/2, 3)) { - ------------------------------------------------------------------------- n::odd { n + 1 "A378941" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 4 LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n (-1) (hypergeom([1/2, -n - 1], [1], 4) - 3 hypergeom([1/2, -n], [1], 4)) {-------------------------------------------------------------------------, n + 2 { (n/2) { (-1) (3 hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) n::even { { (n/2 + 1/2) , { 2 (-1) ((n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + (- n/2 + 1/2) hypergeom([1/2, - n/2 - 1/2], [1], 4)) { --------------------------------------------------------------------------------------------------------------------------- n::odd { n + 1 { (n/2) { 2 (-1) ((n/2 + 3/2) hypergeom([1/2, - n/2 - 3/2], [1], 4) + (- n/2 + 1/2) hypergeom([1/2, - n/2 - 1/2], [1], 4)) { --------------------------------------------------------------------------------------------------------------------- n::even { n + 1 } { { (n/2 - 1/2) { (-1) (3 hypergeom([1/2, - n/2 - 1], [1], 4) - 3 hypergeom([1/2, - n/2], [1], 4)) n::odd "A378947" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((4 n + 10 n + 3) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 8 n + 3) hypergeom([1/2, -n], [1], 4)) {1, -----------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n "A379025" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379034" LREtools/SolveLRE: "Reduced the order of" (n+6)^2*(n+7)*E^3-(n+3)*(7*n+34)*(n+6)*E^2-3*(n+4)*(n+2)*(7*n+27)*E+27*(n+1)*(n+2)*(n+4) "to two: Symmetric square" (n+2)*E^2+(-2*n-3)*E-3*n-3 LREtools/SearchTable: "SearchTable successful" {- (n hypergeom([1/2, -n - 1], [1], 4) + (3 n + 12) hypergeom([1/2, -n], [1], 4)) 2 2 / 2 2 ((n + 16 n + 27) hypergeom([1/2, -n - 1], [1], 4) + 3 (n + 1) hypergeom([1/2, -n], [1], 4)) / ((n + 2) (n + 3) (n + 4))} / "A379035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=293149.7MB, alloc=3671.5MB, time=2232.10 LREtools/SolveLRE: "Reduced the order of" (n+6)*(19170270*n^11+572405211*n^10+7355259513*n^9+53286499674*n^8+238232649528*n^7+670400664795*n^6+ 1136812716237*n^5+949132078416*n^4-106883052508*n^3-942957767136*n^2-738408538880*n-200037676800)*(n+7)^2/(343*n^3+5670*n^2+25469*n+24822)*E^3-(n+6)* (230139091350*n^16+12990606358275*n^15+332174741724378*n^14+5098856679634914*n^13+52449702696670482*n^12+381891066557762916*n^11+2024223920036825658* n^10+7897957199493245850*n^9+22602261543944263624*n^8+46417071704841579081*n^7+64634666298601834068*n^6+51456642160260340116*n^5+3029968567660489224* n^4-42052293756975369792*n^3-46331851861475667424*n^2-22512581993834428800*n-4434145141662028800)/(343*n^3+5670*n^2+25469*n+24822)/(343*n^3+4641*n^2+ 15158*n+4680)*E^2+(230139091350*n^17+13194443839185*n^16+346020706443132*n^15+5502680941437162*n^14+59287920028196454*n^13+457776992897058744*n^12+ 2611177599603668874*n^11+11168591262273682962*n^10+35937113001160442188*n^9+86269541402729820583*n^8+150866352367820298402*n^7+182245276012662945636* n^6+132317405403966030768*n^5+25551036555887539248*n^4-47777590233428230048*n^3-45235053407006315200*n^2-14538858918465772800*n-987451303299840000)/( 343*n^3+4641*n^2+15158*n+4680)/(343*n^3+3612*n^2+6905*n-6180)*E-(19170270*n^13+821618721*n^12+15719403105*n^11+177455795463*n^10+1315272808953*n^9+ 6734017532457*n^8+24384250128207*n^7+62839675691517*n^6+114325072405469*n^5+143407861484362*n^4+118485405917148*n^3+59157507576680*n^2+14859124274368 *n+1067524408320)*n/(343*n^3+3612*n^2+6905*n-6180) "to two: Symmetric square" (n+2)*E^2+(-6*n-9)*E+n+1 LREtools/SearchTable: "SearchTable successful" /n - 1 \ /n - 1 |----- (-n1) (n1 + 1) 3 2 | |----- n n | \ 9 2 (343 n1 + 2583 n1 + 710 n1 - 9816) binomial(2 n1, n1)| n | \ (-n1) (n1 + 1) {(1/2) , (1/2) | ) ------------------------------------------------------------------------|, (1/2) | ) 9 2 | / (n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4) | | / |----- | |----- \n1 = 0 / \n1 = 0 /n1 - 1 |----- 3 2 | \ (2 n2 + 2) (343 n1 + 2583 n1 + 710 n1 - 9816) | ) 3 (( | / |----- \n2 = 0 9 8 7 6 5 4 3 2 6860 n2 + 118888 n2 + 16102335 n2 + 199658811 n2 + 789715971 n2 + 841300533 n2 - 1541044934 n2 - 3921770656 n2 - 2242552128 n2 - 5339520) 2 LegendreP(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (-41160 n2 - 692748 n2 - 9549606 n2 - 76137582 n2 - 243433830 n2 - 158963574 n2 + 604538820 n2 + 1082563584 n2 + 494050176 n2 + 17798400) LegendreP(n2, 3) LegendreP(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (6860 n2 + 112028 n2 + 1114523 n2 + 6942359 n2 + 18413207 n2 + 5471237 n2 - 49863510 n2 - 66842208 n2 - 27383616 n2 - 5339520) 2 / 3 2 3 2 LegendreP(n2, 3) ) / (n2 (n2 + 2) (n2 + 3) (n2 + 4) (343 (n2 + 1) + 2583 (n2 + 1) + 710 n2 - 9106) (343 n2 + 2583 n2 + 710 n2 - 9816) / \ \ /n - 1 | | |----- | | n | \ (-n1) (n1 + 1) binomial(2 n2 + 2, n2 + 1))| binomial(2 n1, n1)/((n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4))|, (1/2) | ) 9 2 | | | / | | |----- / / \n1 = 0 /n1 - 1 |----- 3 2 | \ (2 n2 + 2) (343 n1 + 2583 n1 + 710 n1 - 9816) | ) 3 (( | / |----- \n2 = 0 9 8 7 6 5 4 3 2 6860 n2 + 118888 n2 + 16102335 n2 + 199658811 n2 + 789715971 n2 + 841300533 n2 - 1541044934 n2 - 3921770656 n2 - 2242552128 n2 - 5339520) 2 LegendreQ(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (-41160 n2 - 692748 n2 - 9549606 n2 - 76137582 n2 - 243433830 n2 - 158963574 n2 + 604538820 n2 + 1082563584 n2 + 494050176 n2 + 17798400) LegendreQ(n2, 3) LegendreQ(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (6860 n2 + 112028 n2 + 1114523 n2 + 6942359 n2 + 18413207 n2 + 5471237 n2 - 49863510 n2 - 66842208 n2 - 27383616 n2 - 5339520) 2 / 3 2 3 2 LegendreQ(n2, 3) ) / (n2 (n2 + 2) (n2 + 3) (n2 + 4) (343 (n2 + 1) + 2583 (n2 + 1) + 710 n2 - 9106) (343 n2 + 2583 n2 + 710 n2 - 9816) / \ \ /n - 1 | | |----- | | n | \ (-n1) (n1 + 1) binomial(2 n2 + 2, n2 + 1))| binomial(2 n1, n1)/((n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4))|, (1/2) | ) 9 2 | | | / | | |----- / / \n1 = 0 /n1 - 1 |----- 3 2 | \ (2 n2 + 2) (343 n1 + 2583 n1 + 710 n1 - 9816) | ) 3 (( | / |----- \n2 = 0 9 8 7 6 5 4 3 2 6860 n2 + 118888 n2 + 16102335 n2 + 199658811 n2 + 789715971 n2 + 841300533 n2 - 1541044934 n2 - 3921770656 n2 - 2242552128 n2 - 5339520) LegendreP(n2 + 1, 3) LegendreQ(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (-20580 n2 - 346374 n2 - 4774803 n2 - 38068791 n2 - 121716915 n2 - 79481787 n2 + 302269410 n2 + 541281792 n2 + 247025088 n2 + 8899200) LegendreQ(n2, 3) LegendreP(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (-20580 n2 - 346374 n2 - 4774803 n2 - 38068791 n2 - 121716915 n2 - 79481787 n2 + 302269410 n2 + 541281792 n2 + 247025088 n2 + 8899200) LegendreP(n2, 3) LegendreQ(n2 + 1, 3) + 9 8 7 6 5 4 3 2 (6860 n2 + 112028 n2 + 1114523 n2 + 6942359 n2 + 18413207 n2 + 5471237 n2 - 49863510 n2 - 66842208 n2 - 27383616 n2 - 5339520) / 3 2 LegendreP(n2, 3) LegendreQ(n2, 3)) / (n2 (n2 + 2) (n2 + 3) (n2 + 4) (343 (n2 + 1) + 2583 (n2 + 1) + 710 n2 - 9106) / \ \ | | 3 2 | | (343 n2 + 2583 n2 + 710 n2 - 9816) binomial(2 n2 + 2, n2 + 1))| binomial(2 n1, n1)/((n1 + 1) (n1 + 2) (n1 + 3) (n1 + 4))|} | | | | / / "A379099" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 2 | | \ (2 n1 + 1) binomial(2 n1, n1) | binomial(2 n, n) | ) ------------------------------------| | / 2 | |----- (n1 + 1) binomial(2 n1 + 2, n1 + 1)| binomial(2 n, n) \n1 = 0 / {----------------, --------------------------------------------------------------} n + 1 n + 1 "A379103" LREtools/SearchTable: "SearchTable successful" n n 3 ((21 n + 21) LegendreP(n + 1, 7/3) + (-89 n - 49) LegendreP(n, 7/3)) 3 ((21 n + 21) LegendreQ(n + 1, 7/3) + (-89 n - 49) LegendreQ(n, 7/3)) {-----------------------------------------------------------------------, -----------------------------------------------------------------------} n (n - 1) n (n - 1) "A379173" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379326" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379329" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379382" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379433" LREtools/SearchTable: "SearchTable successful" 2 2 (3 n + 8 n + 9) LegendreP(n + 1, 5) + (-27 n - 76 n - 45) LegendreP(n, 5) {---------------------------------------------------------------------------, (n + 2) n (n - 1) (n - 2) 2 2 (3 n + 8 n + 9) LegendreQ(n + 1, 5) + (-27 n - 76 n - 45) LegendreQ(n, 5) ---------------------------------------------------------------------------} (n + 2) n (n - 1) (n - 2) "A379435" LREtools/SearchTable: "SearchTable successful" n 2 2 2 ((3 n + 7 n + 18) LegendreP(n + 1, 2) + (-9 n - 32 n - 36) LegendreP(n, 2)) {--------------------------------------------------------------------------------, (n + 2) n (n - 1) (n - 2) n 2 2 2 ((3 n + 7 n + 18) LegendreQ(n + 1, 2) + (-9 n - 32 n - 36) LegendreQ(n, 2)) --------------------------------------------------------------------------------} (n + 2) n (n - 1) (n - 2) "A379464" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379809" LREtools/SearchTable: "SearchTable successful" n 36 ((36 n + 27) hypergeom([5/4, -n - 1/4], [3/2], 25/9) + (64 n + 16) hypergeom([5/4, -n + 3/4], [3/2], 25/9)) GAMMA(n + 3/4) {------------------------------------------------------------------------------------------------------------------------------} GAMMA(n + 2) "A379823" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A379824" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 |----- n n n | \ {(-1/2 - 1/2 I) , (-1/2 + 1/2 I) , (-1/2 - 1/2 I) | ) (-1 + I) exp(-1/2 I n1 Pi) | / |----- \n1 = 0 /n1 - 1 \\ |----- / (n2 + 1) \|| | \ | (1 + I) (n2 + 1) (n2 hypergeom([1/2, -n2 - 1], [1], 4) + (3 n2 + 12) hypergeom([1/2, -n2], [1], 4))||| | ) |- -----------------------------------------------------------------------------------------------------------|||} | / \ (n2 + 2) (n2 + 3) /|| |----- || \n2 = 0 // "A380115" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n { 8 8 { n { ------------------ n::even { 2 binomial(n, n/2) n::even { n binomial(n, n/2) {{ , { } { (n - 1) { (3 n + 3) { 4 2 binomial(n - 1, n/2 - 1/2) n::odd { 2 2 { ---------------------------------- n::odd { (n + 1) binomial(n + 1, n/2 + 1/2) "A380208" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380209" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380215" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380260" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380307" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A380308" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380309" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A380310" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A380491" LREtools/SearchTable: "SearchTable successful" 3 2 (2 (2 n + 1) (n - 3) hypergeom([-n - 1], [n + 1], -1) + (n - 3 n + 5 n + 10) hypergeom([-n], [n], -1)) binomial(2 n, n) n! {----------------------------------------------------------------------------------------------------------------------------} (n - 1) (n + 2) (2 n - 1) "A380492" LREtools/SearchTable: "SearchTable successful" 2 ((4 n + 2) hypergeom([-n - 1], [n + 1], -1) + (n - 2 n - 4) hypergeom([-n], [n], -1)) binomial(2 n, n) n! {----------------------------------------------------------------------------------------------------------} (2 n - 1) (n + 2) "A380511" LREtools/SearchTable: "SearchTable not successful" {} "A380591" LREtools/SearchTable: "SearchTable successful" {(2 (2 n + 1) (7 n - 6) hypergeom([- n/2, - n/2 - 1/2], [-2 n - 2], -4) + (7 n + 6) (n + 2) hypergeom([- n/2, - n/2 + 1/2], [-2 n], -4)) binomial(2 n, n)/((n + 1) (n + 2) (3 n + 1))} "A380603" LREtools/SearchTable: "SearchTable successful" n {2 hypergeom([2 n, -n + 1], [], -1/2)} "A380640" LREtools/SearchTable: "SearchTable not successful" {} "A380864" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 binomial(2 n1, n1)| {8 , 8 | ) -------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A381184" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A381482" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A381484" LREtools/SearchTable: "SearchTable successful" n {(-3) n! LaguerreL(n, - 1/9 - n, -1/9)} "A381504" LREtools/SearchTable: "SearchTable successful" n -1 {(-4) n! LaguerreL(n, - 1/16 - n, --)} 16 "A381505" LREtools/SearchTable: "SearchTable successful" n {(-3) n! LaguerreL(n, - 1/9 - n, 2/9)} "A381506" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, - 1/16 - n, 3/16)} "A381559" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { (-n) { 2 (n + 1) binomial(n, n/2) (n/2)! n::even { 1/2 (n/2)! (n + 1) (n + 2) n::even {{ , { } { (-n - 1) { (n/2 + 1/2) (n/2 - 1/2)! n::odd { 2 (n + 1) (n + 2) binomial(n + 1, n/2 + 1/2) (n/2 + 1/2)! n::odd "A381676" LREtools/SearchTable: "SearchTable successful" {hypergeom([-n - 1, -n - 1, -n - 1], [1, 1], -1) - 8 hypergeom([-n, -n, -n], [1, 1], -1)} "A381681" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / /n1 - 1 \\ | |----- n2 || | | \ (-1) || | n1! | ) ---------|| /n - 1 \ |n - 1 | / (n2 + 1)!|| |----- | |----- |----- || | \ n1! | | \ \n2 = 0 /| {n! | ) ---------|, n! | ) ----------------------|, n!} | / (n1 + 1)!| | / (n1 + 1)! | |----- | |----- | \n1 = 0 / \n1 = 0 / "A381744" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable not successful" (6 n + 1) (3 n + 1) (3 n + 2) binomial(6 n, 2 n) {------------------------------------------------} (n + 1) (4 n + 1) (2 n + 1) (4 n + 3) "A381889" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A381984" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382134" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382139" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382274" LREtools/SearchTable: "SearchTable successful" 3 2 (n - 1) (2 n - 3) (n + 1) LegendreP(n + 1, 3) + (-10 n + 29 n - 9) LegendreP(n, 3) {------------------------------------------------------------------------------------, n 3 2 (n - 1) (2 n - 3) (n + 1) LegendreQ(n + 1, 3) + (-10 n + 29 n - 9) LegendreQ(n, 3) ------------------------------------------------------------------------------------} n "A382332" LREtools/SearchTable: "SearchTable successful" 3 2 4 3 2 (n + 1) (2 n - 22 n + 32 n - 15) LegendreP(n + 1, 3) + (-14 n + 104 n - 158 n - 21 n + 45) LegendreP(n, 3) {---------------------------------------------------------------------------------------------------------------, n 3 2 4 3 2 (n + 1) (2 n - 22 n + 32 n - 15) LegendreQ(n + 1, 3) + (-14 n + 104 n - 158 n - 21 n + 45) LegendreQ(n, 3) ---------------------------------------------------------------------------------------------------------------} n "A382405" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382432" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 \\ |----- |----- 1/2 (-n2 - 1) || 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ (2 + 5 ) binomial(2 n2, n2) (3 n2 + 1)|| {(2 - 5 ) , (2 + 5 ) , (2 - 5 ) | ) (2 + 5 ) (2 - 5 ) | ) -------------------------------------------------||} | / | / n2 + 1 || |----- |----- || \n1 = 0 \n2 = 0 // "A382433" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382443" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382514" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 12 _Z + 47 _Z - 64, index = 1) , RootOf(_Z - 12 _Z + 47 _Z - 64, index = 2) , RootOf(_Z - 12 _Z + 47 _Z - 64, index = 3) } "A382515" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 4 3 2 %1 := _Z - 20 _Z + 159 _Z - 640 _Z + 1280 _Z - 1024 "A382536" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) } 5 3 2 %1 := _Z - _Z - 12 _Z - 48 _Z - 64 "A382537" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 7" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n RootOf(%1, index = 7) } 7 5 4 3 2 %1 := _Z - _Z - 20 _Z - 160 _Z - 640 _Z - 1280 _Z - 1024 "A382538" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 9" LREtools/SolveLRE: "Applying recursion to the left-factor" n n n n n n {RootOf(%1, index = 1) , RootOf(%1, index = 2) , RootOf(%1, index = 3) , RootOf(%1, index = 4) , RootOf(%1, index = 5) , RootOf(%1, index = 6) , n n n RootOf(%1, index = 7) , RootOf(%1, index = 8) , RootOf(%1, index = 9) } 9 7 6 5 4 3 2 %1 := _Z - _Z - 28 _Z - 336 _Z - 2240 _Z - 8960 _Z - 21504 _Z - 28672 _Z - 16384 "A382539" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / 1/2\ / 1/2\ 1/2 n | 28 5 | 1/2 n | 28 5 | 1/2 n 1/2 {(2 - 5 ) |n + 13 - -------|, (2 + 5 ) |n + 13 + -------|, (2 - 5 ) (5 n + 65 - 28 5 ) \ 5 / \ 5 / / / / 1/2\ \\ | |n1 - 1 1/2 (-n2 - 1) | 28 5 | || | |----- (2 + 5 ) (2 n2 + 1) |n2 + 14 - -------| binomial(2 n2, n2)|| | 1/2 n1 1/2 (-n1 - 1) 2 | \ \ 5 / || | (2 + 5 ) (2 - 5 ) (n1 + 27 n1 + 14) | ) ---------------------------------------------------------------------|| |n - 1 | / 2 2 || |----- |----- (n2 + 1) ((n2 + 1) + 27 n2 + 41) (n2 + 27 n2 + 14) || | \ \n2 = 0 /| | ) ----------------------------------------------------------------------------------------------------------------------------------|} | / / 1/2\ | |----- | 28 5 | 1/2 | |n1 = 0 |n1 + 13 - -------| (5 n1 + 70 - 28 5 ) | \ \ 5 / / "A382540" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / 1/2\ 1/2\ / / 1/2\ 1/2\ 1/2 n | 2 | 84 5 | 948 5 | 1/2 n | 2 | 84 5 | 948 5 | 1/2 n {(2 - 5 ) |n + |39 - -------| n + 2122/5 - --------|, (2 + 5 ) |n + |39 + -------| n + 2122/5 + --------|, (2 - 5 ) \ \ 5 / 5 / \ \ 5 / 5 / / |n - 1 |----- 1/2 2 1/2 | \ 1/2 n1 1/2 (-n1 - 1) 4 3 2 (-84 5 n + 5 n - 948 5 + 195 n + 2122) | ) (2 + 5 ) (2 - 5 ) (n1 + 80 n1 + 1043 n1 + 1456 n1 + 516) | / |----- \n1 = 0 / / 1/2 1/2\ \ |n1 - 1 1/2 (-n2 - 1) | 2 84 5 (n2 + 1) 948 5 | 2 | |----- (2 + 5 ) (2 n2 + 1) |(n2 + 1) - ---------------- + 39 n2 + 2317/5 - --------| (n2 - 89 n2 - 84) binomial(2 n2, n2)| | \ \ 5 5 / | / | ) -------------------------------------------------------------------------------------------------------------------------------| / ( | / 4 3 2 4 3 2 | / |----- (n2 + 1) ((n2 + 1) + 80 (n2 + 1) + 1043 (n2 + 1) + 1456 n2 + 1972) (n2 + 80 n2 + 1043 n2 + 1456 n2 + 516) | \n2 = 0 / \ | | 2 1/2 1/2 1/2 2 1/2 | (n1 - 84/5 5 n1 + 39 n1 + 2122/5 - 948/5 5 ) (-84 5 (n1 + 1) + 5 (n1 + 1) - 948 5 + 195 n1 + 2317))|} | | / "A382541" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n / 537 2 2154 8389\ {RootOf(%1, index = 1) |n + --- RootOf(%1, index = 1) - ---- RootOf(%1, index = 1) + ----|, \ 214 107 214 / n / 537 2 2154 8389\ RootOf(%1, index = 2) |n + --- RootOf(%1, index = 2) - ---- RootOf(%1, index = 2) + ----|, \ 214 107 214 / n / 537 2 2154 8389\ RootOf(%1, index = 3) |n + --- RootOf(%1, index = 3) - ---- RootOf(%1, index = 3) + ----|} \ 214 107 214 / 3 2 %1 := _Z - 12 _Z + 47 _Z - 64 "A382542" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n {RootOf(%1, index = 1) / 2 /1611 2 6462 25167\ 177 2 3582 31429\ |n + |---- RootOf(%1, index = 1) - ---- RootOf(%1, index = 1) + -----| n + --- RootOf(%1, index = 1) + ---- RootOf(%1, index = 1) - -----|, \ \214 107 214 / 214 107 214 / n RootOf(%1, index = 2) / 2 /1611 2 6462 25167\ 177 2 3582 31429\ |n + |---- RootOf(%1, index = 2) - ---- RootOf(%1, index = 2) + -----| n + --- RootOf(%1, index = 2) + ---- RootOf(%1, index = 2) - -----|, \ \214 107 214 / 214 107 214 / n RootOf(%1, index = 3) / 2 /1611 2 6462 25167\ 177 2 3582 31429\ |n + |---- RootOf(%1, index = 3) - ---- RootOf(%1, index = 3) + -----| n + --- RootOf(%1, index = 3) + ---- RootOf(%1, index = 3) - -----|} \ \214 107 214 / 214 107 214 / 3 2 %1 := _Z - 12 _Z + 47 _Z - 64 "A382543" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n / 1302 2 18 217\ {RootOf(%1, index = 1) |n + ---- RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + ---|, \ 109 109 109/ n / 1302 2 18 217\ RootOf(%1, index = 2) |n + ---- RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + ---|, \ 109 109 109/ n / 1302 2 18 217\ RootOf(%1, index = 3) |n + ---- RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + ---|} \ 109 109 109/ 3 2 %1 := _Z - 9 _Z - 1 "A382544" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 4, 4 LREtools/SolveLRE: "Solutions may be linearly dependent" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n {RootOf(%1, index = 1) / 2 /3906 2 54 651\ 1582758 2 19494 175718\ |n + |---- RootOf(%1, index = 1) + --- RootOf(%1, index = 1) + ---| n + ------- RootOf(%1, index = 1) + ----- RootOf(%1, index = 1) + ------|, \ \109 109 109/ 109 109 109 / n RootOf(%1, index = 2) / 2 /3906 2 54 651\ 1582758 2 19494 175718\ |n + |---- RootOf(%1, index = 2) + --- RootOf(%1, index = 2) + ---| n + ------- RootOf(%1, index = 2) + ----- RootOf(%1, index = 2) + ------|, \ \109 109 109/ 109 109 109 / n RootOf(%1, index = 3) / 2 /3906 2 54 651\ 1582758 2 19494 175718\ |n + |---- RootOf(%1, index = 3) + --- RootOf(%1, index = 3) + ---| n + ------- RootOf(%1, index = 3) + ----- RootOf(%1, index = 3) + ------|} \ \109 109 109/ 109 109 109 / 3 2 %1 := _Z - 9 _Z - 1 "A382644" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382645" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 4" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" n {(-1) } "A382647" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n / 11 3 2 37 3 5179\ {RootOf(_Z - _Z - 4, index = 1) |n - ---- RootOf(_Z - _Z - 4, index = 1) - --- RootOf(_Z - _Z - 4, index = 1) + ----|, \ 3424 856 3424/ 3 n / 11 3 2 37 3 5179\ RootOf(_Z - _Z - 4, index = 2) |n - ---- RootOf(_Z - _Z - 4, index = 2) - --- RootOf(_Z - _Z - 4, index = 2) + ----|, \ 3424 856 3424/ 3 n / 11 3 2 37 3 5179\ RootOf(_Z - _Z - 4, index = 3) |n - ---- RootOf(_Z - _Z - 4, index = 3) - --- RootOf(_Z - _Z - 4, index = 3) + ----|} \ 3424 856 3424/ "A382648" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" n / 2 / 33 2 111 15537\ 69 2 267 15797\ n / 2 / 33 2 111 15537\ 69 2 267 15797\ {%3 |n + |- ---- %3 - --- %3 + -----| n - ---- %3 - --- %3 + -----|, %2 |n + |- ---- %2 - --- %2 + -----| n - ---- %2 - --- %2 + -----|, \ \ 3424 856 3424 / 3424 856 3424 / \ \ 3424 856 3424 / 3424 856 3424 / n / 2 / 33 2 111 15537\ 69 2 267 15797\ %1 |n + |- ---- %1 - --- %1 + -----| n - ---- %1 - --- %1 + -----|} \ \ 3424 856 3424 / 3424 856 3424 / 3 %1 := RootOf(_Z - _Z - 4, index = 3) 3 %2 := RootOf(_Z - _Z - 4, index = 2) 3 %3 := RootOf(_Z - _Z - 4, index = 1) "A382649" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 5" LREtools/SolveLRE: "Applying recursion to the left-factor" n {RootOf(%1, index = 1) / 985111 13113 4 14073 3 250455 2 192417 \ |n + ------ + ------- RootOf(%1, index = 1) - ------- RootOf(%1, index = 1) + ------- RootOf(%1, index = 1) - ------ RootOf(%1, index = 1)|, \ 399136 6386176 1596544 6386176 798272 / n RootOf(%1, index = 2) / 985111 13113 4 14073 3 250455 2 192417 \ |n + ------ + ------- RootOf(%1, index = 2) - ------- RootOf(%1, index = 2) + ------- RootOf(%1, index = 2) - ------ RootOf(%1, index = 2)|, \ 399136 6386176 1596544 6386176 798272 / n RootOf(%1, index = 3) / 985111 13113 4 14073 3 250455 2 192417 \ |n + ------ + ------- RootOf(%1, index = 3) - ------- RootOf(%1, index = 3) + ------- RootOf(%1, index = 3) - ------ RootOf(%1, index = 3)|, \ 399136 6386176 1596544 6386176 798272 / n RootOf(%1, index = 4) / 985111 13113 4 14073 3 250455 2 192417 \ |n + ------ + ------- RootOf(%1, index = 4) - ------- RootOf(%1, index = 4) + ------- RootOf(%1, index = 4) - ------ RootOf(%1, index = 4)|, \ 399136 6386176 1596544 6386176 798272 / n RootOf(%1, index = 5) / 985111 13113 4 14073 3 250455 2 192417 \ |n + ------ + ------- RootOf(%1, index = 5) - ------- RootOf(%1, index = 5) + ------- RootOf(%1, index = 5) - ------ RootOf(%1, index = 5)|} \ 399136 6386176 1596544 6386176 798272 / 5 3 2 %1 := _Z - _Z - 12 _Z - 48 _Z - 64 "A382841" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382848" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A382987" LREtools/SearchTable: "SearchTable successful" n 2 2 (-1) ((n + 2 n - 1) hypergeom([1/2, -n - 1], [1], 4) + (-n + 7) hypergeom([1/2, -n], [1], 4)) {------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A382988" LREtools/SearchTable: "SearchTable successful" n (-1) ((2 n + 1) hypergeom([1/2, -n - 1], [1], 4) + (-4 n + 1) hypergeom([1/2, -n], [1], 4)) {--------------------------------------------------------------------------------------------} n + 2 "A382989" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 3 2 n (-1) ((n + 17 n + 52 n + 42) hypergeom([1/2, -n - 1], [1], 4) - 6 (2 n + 5) (n + 1) hypergeom([1/2, -n], [1], 4)) {3 , --------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A382990" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SearchTable: "SearchTable successful" n 2 3 2 n (-1) ((n + 7) (3 n + 10 n + 6) hypergeom([1/2, -n - 1], [1], 4) + (-4 n - 36 n - 62 n + 18) hypergeom([1/2, -n], [1], 4)) {3 , -----------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A383118" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383231" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 1/5)\| {5 GAMMA(n + 1/5), 5 GAMMA(n + 1/5) | ) |1/5 ---------------||} | / \ GAMMA(n1 + 6/5)/| |----- | \n1 = 0 / "A383232" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 2/5)\| {5 GAMMA(n + 2/5), 5 GAMMA(n + 2/5) | ) |1/5 ---------------||} | / \ GAMMA(n1 + 7/5)/| |----- | \n1 = 0 / "A383233" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ / GAMMA(n1 + 3/5)\| {5 GAMMA(n + 3/5), 5 GAMMA(n + 3/5) | ) |1/5 ---------------||} | / \ GAMMA(n1 + 8/5)/| |----- | \n1 = 0 / "A383234" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" memory used=294628.7MB, alloc=3671.5MB, time=2242.90 /n - 1 \ |----- | n n | \ / GAMMA(n1 + 4/5)\| {5 GAMMA(n + 4/5), 5 GAMMA(n + 4/5) | ) |1/5 ---------------||} | / \ GAMMA(n1 + 9/5)/| |----- | \n1 = 0 / "A383251" binomial(2 n, n) {1, ----------------} n + 1 "A383254" LREtools/SearchTable: "SearchTable successful" {((8 n + 3) hypergeom([-1/2, -n - 1], [1], -4) + (-8 n - 7) hypergeom([-1/2, -n], [1], -4)) (n + 1)} "A383274" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383280" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) (4 (n + 1) (n - 2) LaguerreL(n + 1, -n + 1/2, 3/2) + (6 n - 3) LaguerreL(n, -n + 3/2, 3/2))} "A383281" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) (4 (n + 1) n LaguerreL(n + 1, -n - 1/2, 1/2) + (2 n + 1) LaguerreL(n, -n + 1/2, 1/2))} "A383282" LREtools/SearchTable: "SearchTable successful" n 2 {(-1) (n!) ((2 n + 2) LaguerreL(n + 1, -n - 3/2, -1/2) - LaguerreL(n, -n - 1/2, -1/2))} "A383313" LREtools/SearchTable: "SearchTable successful" n {(-2) n! LaguerreL(n, - 1/4 - n, -1/4)} "A383314" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, - 1/8 - n, -1/8)} "A383315" LREtools/SearchTable: "SearchTable successful" n -1 {(-6) n! LaguerreL(n, - 1/12 - n, --)} 12 "A383316" LREtools/SearchTable: "SearchTable successful" n {(-4) n! LaguerreL(n, - 1/8 - n, 1/8)} "A383317" LREtools/SearchTable: "SearchTable successful" n {(-6) n! LaguerreL(n, - 1/12 - n, 1/12)} "A383326" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- n1 \ (3 n1 + 1) 2 binomial(3 n1, n1) (5 n1 + 3) {1, ) --------------------------------------------} / (n1 + 1) (2 n1 + 1) ----- n1 = 0 "A383344" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 3 2 | \ (-4) | {n! (n + 18 n + 95 n + 142), n! (n + 18 n + 95 n + 142) | ) -------------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 18 (n1 + 1) + 95 n1 + 237) (n1 + 18 n1 + 95 n1 + 142)| \n1 = 0 / "A383355" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A383378" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 2 2 | \ (-3) | {n! (n + 2) (n + 13 n + 39), n! (n + 2) (n + 13 n + 39) | ) -----------------------------------------------------------------------|} | / 2 2 | |----- (n1 + 1)! (n1 + 3) ((n1 + 1) + 13 n1 + 52) (n1 + 2) (n1 + 13 n1 + 39)| \n1 = 0 / "A383380" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | 3 2 3 2 | \ (-2) | {n! (n + 12 n + 41 n + 38), n! (n + 12 n + 41 n + 38) | ) -----------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! ((n1 + 1) + 12 (n1 + 1) + 41 n1 + 79) (n1 + 12 n1 + 41 n1 + 38)| \n1 = 0 / "A383381" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 {n! (n + 3) (n + 15 n + 62 n + 56), /n - 1 \ |----- n1 | 3 2 | \ (-2) | n! (n + 3) (n + 15 n + 62 n + 56) | ) ------------------------------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! (n1 + 4) ((n1 + 1) + 15 (n1 + 1) + 62 n1 + 118) (n1 + 3) (n1 + 15 n1 + 62 n1 + 56)| \n1 = 0 / "A383382" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {n! (n + 22 n + 161 n + 452 n + 393), n! (n + 22 n + 161 n + 452 n + 393) /n - 1 \ |----- n1 | | \ (-3) | | ) -----------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 22 (n1 + 1) + 161 (n1 + 1) + 452 n1 + 845) (n1 + 22 n1 + 161 n1 + 452 n1 + 393)| \n1 = 0 / "A383383" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 3 2 {n! (n + 2) (n + 24 n + 179 n + 412), n! (n + 2) (n + 24 n + 179 n + 412) /n - 1 \ |----- n1 | | \ (-4) | | ) ---------------------------------------------------------------------------------------------------|} | / 3 2 3 2 | |----- (n1 + 1)! (n1 + 3) ((n1 + 1) + 24 (n1 + 1) + 179 n1 + 591) (n1 + 2) (n1 + 24 n1 + 179 n1 + 412)| \n1 = 0 / "A383384" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 4 3 2 4 3 2 {n! (n + 30 n + 305 n + 1220 n + 1569), n! (n + 30 n + 305 n + 1220 n + 1569) /n - 1 \ |----- n1 | | \ (-5) | | ) ---------------------------------------------------------------------------------------------------------------|} | / 4 3 2 4 3 2 | |----- (n1 + 1)! ((n1 + 1) + 30 (n1 + 1) + 305 (n1 + 1) + 1220 n1 + 2789) (n1 + 30 n1 + 305 n1 + 1220 n1 + 1569)| \n1 = 0 / "A383436" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A383437" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | 2 2 | \ 1 | {(n + 1) n!, (n + 1) n! | ) ----------------------------|} | / 2 | |----- (n1 + 2) (n1 + 1)! (n1 + 1)| \n1 = 0 / "A383478" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383479" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383480" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383481" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383499" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383503" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383522" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383524" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383527" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" n - 1 ----- \ n1 {1, ) (-1) (hypergeom([1/2, -n1 - 1], [1], 4) - hypergeom([1/2, -n1], [1], 4))} / ----- n1 = 0 "A383539" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A383568" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 10" {} "A383571" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A383572" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A383573" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383581" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A383582" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A383583" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A383584" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A383597" LREtools/SearchTable: "SearchTable successful" {hypergeom([1/3, -n], [1], -9)} "A383598" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383599" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A383600" LREtools/SearchTable: "SearchTable successful" {hypergeom([1/4, -n], [1], -8)} "A383601" LREtools/SearchTable: "SearchTable successful" {(3 n + 3) hypergeom([-1/3, -n - 1], [1], -9) + (-3 n - 4) hypergeom([-1/3, -n], [1], -9)} "A383602" LREtools/SearchTable: "SearchTable successful" {(4 n + 4) hypergeom([-1/4, -n - 1], [1], -8) + (-4 n - 5) hypergeom([-1/4, -n], [1], -8)} "A383603" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383604" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383605" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383606" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383610" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383615" 2 2 2 (2 n + 3) (2 n + 1) binomial(2 n, n) (2 n + 3) (2 n + 1) binomial(2 n, n) {---------------------------------------, ------------------------------------} 2 2 2 (n + 3) (n + 2) (n + 1) (n + 2) (n + 1) (n + 3) "A383616" 2 binomial(2 n, n) binomial(2 n, n) {-----------------, ----------------} 2 n + 1 (n + 1) "A383624" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383627" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383628" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383629" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A383702" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \\\ | | |----- ||| | | | \ / GAMMA(n2 + 3/4)\||| | | GAMMA(n1 + 3/4) | ) |1/4 ---------------|||| /n - 1 \ |n - 1 | | / \ GAMMA(n2 + 7/4)/||| |----- | |----- | |----- ||| n n | \ / GAMMA(n1 + 3/4)\| n | \ | \n2 = 0 /|| {4 GAMMA(n + 3/4), 4 GAMMA(n + 3/4) | ) |1/4 ---------------||, 4 GAMMA(n + 3/4) | ) |1/4 ----------------------------------------------|| | / \ GAMMA(n1 + 7/4)/| | / \ GAMMA(n1 + 7/4) /| |----- | |----- | \n1 = 0 / \n1 = 0 / } "A383703" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" / / /n1 - 1 \\\ | | |----- ||| | | | \ / GAMMA(n2 + 3/4)\||| | | GAMMA(n1 + 3/4) | ) |1/4 ---------------|||| /n - 1 \ |n - 1 | | / \ GAMMA(n2 + 7/4)/||| |----- | |----- | |----- ||| n n | \ / GAMMA(n1 + 3/4)\| n | \ | \n2 = 0 /|| {4 GAMMA(n + 3/4), 4 GAMMA(n + 3/4) | ) |1/4 ---------------||, 4 GAMMA(n + 3/4) | ) |1/4 ----------------------------------------------||, | / \ GAMMA(n1 + 7/4)/| | / \ GAMMA(n1 + 7/4) /| |----- | |----- | \n1 = 0 / \n1 = 0 / / / / / /n2 - 1 \\\\\ | | | | |----- ||||| | | | | | \ / GAMMA(n3 + 3/4)\||||| | | | | GAMMA(n2 + 3/4) | ) |1/4 ---------------|||||| | | |n1 - 1 | | / \ GAMMA(n3 + 7/4)/||||| | | |----- | |----- ||||| | | | \ | \n3 = 0 /|||| | | GAMMA(n1 + 3/4) | ) |1/4 ----------------------------------------------|||| |n - 1 | | / \ GAMMA(n2 + 7/4) /||| |----- | |----- ||| n | \ | \n2 = 0 /|| 4 GAMMA(n + 3/4) | ) |1/4 -----------------------------------------------------------------------------||} | / \ GAMMA(n1 + 7/4) /| |----- | \n1 = 0 / "A383832" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (2 n1 + 1) | {(16/3) (n + 1), (16/3) (n + 1) | ) ----------------------------------|} | / (n1 + 1) | |----- (16/3) (n1 + 2) (3 n1 + 3)| \n1 = 0 / "A383884" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 3 binomial(2 n1, n1)| {(-4) , (-4) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4) | \n1 = 0 / "A383888" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (5 n1 + 3) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- ------------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A383897" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ (-1) 2 n1!| {2 n!, 2 n! | ) ---------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A383922" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 n - 1 /n1 - 1 \ ----- ----- |----- | \ \ | \ 2 n2 + 5 | {1, ) n1! (n1 + 2), ) n1! (n1 + 2) | ) ---------------------------|} / / | / (n2 + 1)! (n2 + 3) (n2 + 2)| ----- ----- |----- | n1 = 0 n1 = 0 \n2 = 0 / "A383935" LREtools/SearchTable: "SearchTable successful" n {3 ((3 n + 3) hypergeom([-1/3, -n - 2/3], [4/3], -1) + (-3 n - 4) hypergeom([-1/3, 1/3 - n], [4/3], -1))} "A383936" LREtools/SearchTable: "SearchTable successful" n {(-3) hypergeom([1/3, -n], [1], 3)} "A383937" LREtools/SearchTable: "SearchTable successful" n {3 ((3 n + 3) hypergeom([1/3, - 1/3 - n], [5/3], -1) + (-3 n - 1) hypergeom([1/3, -n + 2/3], [5/3], -1))} "A383944" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 7 ) 7 (7 LegendreP(n + 1, 3/7 I 7 ) I - LegendreP(n, 3/7 I 7 )) (n + 1), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreQ(n + 1, 3/7 I 7 ) I - LegendreQ(n, 3/7 I 7 )) (n + 1)} "A383945" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 {(-I 7 ) ((-8 n - 9) LegendreP(n, 3/7 I 7 ) + 7 (8 n + 13) LegendreP(n + 1, 3/7 I 7 ) I) (n + 1), 1/2 n 1/2 1/2 1/2 (-I 7 ) ((-8 n - 9) LegendreQ(n, 3/7 I 7 ) + 7 (8 n + 13) LegendreQ(n + 1, 3/7 I 7 ) I) (n + 1)} "A383946" LREtools/SearchTable: "SearchTable successful" {((16 n + 7) hypergeom([-1/2, -n - 1], [1], -8) + (-16 n - 15) hypergeom([-1/2, -n], [1], -8)) (n + 1)} "A383947" LREtools/SearchTable: "SearchTable successful" 1/2 n 1/2 1/2 1/2 1/2 {-I (-I 7 ) 7 (7 LegendreP(n + 1, 3/7 I 7 ) I + 7 LegendreP(n, 3/7 I 7 )) (n + 1), 1/2 n 1/2 1/2 1/2 1/2 -I (-I 7 ) 7 (7 LegendreQ(n + 1, 3/7 I 7 ) I + 7 LegendreQ(n, 3/7 I 7 )) (n + 1)} "A383948" LREtools/SearchTable: "SearchTable successful" n {3 (n + 1) (hypergeom([-1/2, -n - 1], [1], -4/3) - hypergeom([-1/2, -n], [1], -4/3))} "A383949" LREtools/SearchTable: "SearchTable successful" {(n + 1) (hypergeom([-1/2, -n - 1], [1], -4) - hypergeom([-1/2, -n], [1], -4))} "A383950" LREtools/SearchTable: "SearchTable successful" n {2 (n + 1) (hypergeom([-1/2, -n - 1], [1], -2) - hypergeom([-1/2, -n], [1], -2))} "A383951" LREtools/SearchTable: "SearchTable successful" {(n + 1) (hypergeom([-1/2, -n - 1], [1], -12) - hypergeom([-1/2, -n], [1], -12))} "A383952" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A383958" 2 binomial(2 n, n) {1, -----------------} 2 (n + 1) "A383990" LREtools/SearchTable: "SearchTable successful" n (-1) ((4 n + 2) hypergeom([-n - 1], [n + 2], 1) + (-3 n - 1) hypergeom([-n], [n + 1], 1)) binomial(2 n, n) n! {--------------------------------------------------------------------------------------------------------------} 2 n "A383991" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383993" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383994" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A383995" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A384024" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! binomial(2 n1, n1) (3 n1 + 2) | {n! binomial(2 n, n) n, n! binomial(2 n, n) n | ) ------------------------------------------------|} | / (n1 + 1)! binomial(2 n1 + 2, n1 + 1) (n1 + 1) n1| |----- | \n1 = 0 / "A384163" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 {{ , { (n/3 - 2/3) { (84375/4) GAMMA(n/3 + 1/5) GAMMA(n/3 + 3/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 2/5) { ---------------------------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 1/2) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) { (84375/4) GAMMA(n/3 + 1/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 3/5) GAMMA(n/3 + 2/5) , { ---------------------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 1/2) { { 0 irem(n, 3) = 2 { (n/3) { (84375/4) GAMMA(n/3 + 3/5) GAMMA(n/3 + 1/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 2/5) { ---------------------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 1/2) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A384165" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /12500\(n/2 - 1/2) {{ |-----| GAMMA(n/2 + 2/5) GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 1/5) , { \ 27 / { -------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/3) { /12500\(n/2) { |-----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 1/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) { \ 27 / { -------------------------------------------------------------------------------- n::even} { GAMMA(n/2 + 1/3) GAMMA(n/2 + 2/3) { { 0 n::odd "A384166" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 1 (Liouvillian solutions)" { { {{ { 0 , irem(n, 4) = 0 { 0 , irem(n, 4) = 1 0 , irem(n, 4) = 2 /210827008\(n/4 - 3/4) { |---------| GAMMA(n/4 + 1/7) GAMMA(n/4 + 5/7) GAMMA(n/4 + 2/7) GAMMA(n/4 + 4/7) GAMMA(n/4 + 3/7) GAMMA(n/4 + 6/7) { \ 27 / , { ---------------------------------------------------------------------------------------------------------------------------- , irem(n, 4) = 3 { GAMMA(n/4 + 1/3) GAMMA(n/4 + 2/3) { 0 , irem(n, 4) = 0 0 , irem(n, 4) = 1 /210827008\(n/4 - 1/2) |---------| GAMMA(n/4 + 2/7) GAMMA(n/4 + 6/7) GAMMA(n/4 + 3/7) GAMMA(n/4 + 1/7) GAMMA(n/4 + 5/7) GAMMA(n/4 + 4/7) \ 27 / ---------------------------------------------------------------------------------------------------------------------------- , irem(n, 4) = 2 GAMMA(n/4 + 2/3) GAMMA(n/4 + 1/3) { { , { 0 , irem(n, 4) = 3 { 0 , irem(n, 4) = 0 { /210827008\(n/4 - 1/4) |---------| GAMMA(n/4 + 3/7) GAMMA(n/4 + 4/7) GAMMA(n/4 + 1/7) GAMMA(n/4 + 5/7) GAMMA(n/4 + 6/7) GAMMA(n/4 + 2/7) \ 27 / ---------------------------------------------------------------------------------------------------------------------------- , irem(n, 4) = 1 GAMMA(n/4 + 2/3) GAMMA(n/4 + 1/3) 0 , irem(n, 4) = 2 , 0 , irem(n, 4) = 3 { /210827008\(n/4) { |---------| GAMMA(n/4 + 3/7) GAMMA(n/4 + 2/7) GAMMA(n/4 + 6/7) GAMMA(n/4 + 4/7) GAMMA(n/4 + 5/7) GAMMA(n/4 + 1/7) { \ 27 / { ---------------------------------------------------------------------------------------------------------------------- irem(n, 4) = 0 { GAMMA(n/4 + 1/3) GAMMA(n/4 + 2/3) { } { 0 irem(n, 4) = 1 { { 0 irem(n, 4) = 2 { { 0 irem(n, 4) = 3 "A384186" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A384199" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-n1 - 1) | n n | \ (-1) 3 n1!| {3 n!, 3 n! | ) ---------------------|} | / (n1 + 1)! | |----- | \n1 = 0 / "A384200" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ 3 n1! | {n! | ) ---------|, n!} | / (n1 + 1)!| |----- | \n1 = 0 / "A384201" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 n1!|| {(-1) n!, (-1) n! | ) |- --------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A384202" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 3 n1!|| {(-1) n!, (-1) n! | ) |- --------------||} | / \ (n1 + 1)! /| |----- | \n1 = 0 / "A384253" n n! (-1) n! {1, -----, --------} n + 2 n + 2 "A384266" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" 1/2 n 1/2 n {(4 - 3 2 ) , (4 + 3 2 ) , /n - 1 /n1 - 1 \\ |----- |----- n2 1/2 (-n2 - 1) || 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ 2 (4 + 3 2 ) (2 n2 + 1) binomial(2 n2, n2)|| (4 - 3 2 ) | ) (4 + 3 2 ) (4 - 3 2 ) | ) -------------------------------------------------------||} | / | / (n2 + 1) (n2 + 2) || |----- |----- || \n1 = 0 \n2 = 0 // "A384338" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 3 2 (3 n1 + 1) (7 n1 + 2) binomial(3 n1, n1)| {8 , 8 | ) ---------------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A384365" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (2 n1 - 1) | {(16/3) (2 n + 3), (16/3) (2 n + 3) | ) ---------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (16/3) (2 n1 + 5) (6 n1 + 9)| \n1 = 0 / "A384561" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A384929" (2 n + 3) (2 n + 1) n! binomial(2 n, n) {(n + 1) n!, ---------------------------------------} n + 2 "A384950" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 \| n n | \ |4 12 (40 n1 + 41 n1 + 9) binomial(3 n1, n1)|| {(1/4) , (1/4) | ) |----------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) /| |----- | \n1 = 0 / "A385004" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(3 n1, n1) (15 n1 + 7) | {(27/4) , (27/4) | ) ----------------------------------|} | / (n1 + 1)| |----- (n1 + 1) (2 n1 + 1) (27/4) | \n1 = 0 / "A385065" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n n {(-I) , I } "A385119" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 {{ , { (n/3 - 2/3) { (84375/4) GAMMA(n/3 + 1/5) GAMMA(n/3 + 3/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 2/5) { ---------------------------------------------------------------------------------------- irem(n, 3) = 2 { GAMMA(n/3 + 3/2) GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) { 0 irem(n, 3) = 0 { { (n/3 - 1/3) { (84375/4) GAMMA(n/3 + 1/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 3/5) GAMMA(n/3 + 2/5) , { ---------------------------------------------------------------------------------------- irem(n, 3) = 1 { GAMMA(n/3 + 2/3) GAMMA(n/3 + 3/2) GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) { { 0 irem(n, 3) = 2 { (n/3) { (84375/4) GAMMA(n/3 + 3/5) GAMMA(n/3 + 1/5) GAMMA(n/3 + 4/5) GAMMA(n/3 + 2/5) { ---------------------------------------------------------------------------------- irem(n, 3) = 0 { GAMMA(n/3 + 3/2) GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A385250" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 3 2 || | \ (1/2 + 1/2 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (91 n2 + 483 n2 + 828 n2 + 460)|| | ) ------------------------------------------------------------------------------------------------------||} | / (n2 + 1) (n2 + 2) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) || |----- || \n2 = 0 // "A385251" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ binomial(4 n1, n1) (8 n1 + 5) | {|---| , |---| | ) -----------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (3 n1 + 1) (4 n1 - 1) |---| | \n1 = 0 \27 / / "A385252" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 2 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n n | \ 2 3 (n1 + 2) binomial(2 n1, n1)| {4 , 9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A385299" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" 2 5 n + 6 n + 2 2 {--------------------------------------, (5 n + 6 n + 2) (3 n + 1) (3 n + 2) n binomial(3 n, n) /n - 1 \ |----- 3 2 | | \ (3 n1 + 1) (3 n1 + 2) binomial(2 n1, n1) binomial(3 n1, n1) (145 n1 + 341 n1 + 248 n1 + 60) binomial(3 n1 + 3, n1 + 1)| | ) ------------------------------------------------------------------------------------------------------------------------|/((3 n + 1) | / 2 2 | |----- (n1 + 1) (5 (n1 + 1) + 6 n1 + 8) (5 n1 + 6 n1 + 2) | \n1 = 0 / (3 n + 2) n binomial(3 n, n))} "A385317" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A385319" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 3 binomial(2 n1, n1) (4 n1 + 3)| {(-4) , (-4) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4) | \n1 = 0 / "A385320" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 3 2 n1 (7 n1 + 5) binomial(3 n1, n1)| {8 , 8 | ) -------------------------------------------------|} | / (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A385438" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 3 2 \| n n | \ | 8 (-1) 24 (187 n1 + 284 n1 + 131 n1 + 18) binomial(4 n1, n1)|| {(-1/8) , (-1/8) | ) |- ------------------------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 1) (3 n1 + 2) /| |----- | \n1 = 0 / "A385440" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { (n - 1) 2 3 n {{ 2 ((n/2 - 1/2)!) (3 n - 1) binomial(n - 1, n/2 - 1/2) binomial(--- - 3/2, n/2 - 1/2) , { 2 { -------------------------------------------------------------------------------------------- n::odd { n + 1 { (-n) 2 3 n 3 n { 2 ((n/2)!) binomial(3 n, ---) binomial(---, n/2) { 2 2 { ----------------------------------------------------- n::even} { n + 1 { { 0 n::odd "A385474" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ |4 12 (3 n1 + 1) (3 n1 + 2) (5 n1 + 6) binomial(3 n1, n1)|| {(1/4) , (1/4) | ) |----------------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (2 n1 + 1) (2 n1 + 3) /| |----- | \n1 = 0 / "A385497" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- n n | \ (-n1) n1 4 3 2 {64 , 64 | ) (1/64 3125 729 (2396 n1 + 6654 n1 + 6631 n1 + 2799 n1 + 420) GAMMA(n1 + 1/2) GAMMA(n1 + 1/3) GAMMA(n1 + 2/3) | / |----- \n1 = 0 \ | | GAMMA(n1 + 5/6) GAMMA(n1 + 7/6)/(GAMMA(n1 + 2) GAMMA(n1 + 6/5) GAMMA(n1 + 7/5) GAMMA(n1 + 8/5) GAMMA(n1 + 9/5)))|} | | / "A385514" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) binomial(2 n1, n1)|| {(-1/2) , (-1/2) | ) |- ------------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A385563" LREtools/SearchTable: "SearchTable successful" {((4 n + 1) hypergeom([-1/2, -n - 1], [1], -4) + (-4 n - 3) hypergeom([-1/2, -n], [1], -4)) (n + 1)} "A385572" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 1, 4 LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A385639" LREtools/SearchTable: "SearchTable successful" n 2 {(-8) (24 (3 n + 4) (3 n + 2) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) + (-283 n - 566 n - 240) hypergeom([-n, n + 1], [-3 n - 1], -1)) / 2 binomial(3 n, n) (3 n + 1) / ((2 n + 1) (17 n + 24))} / "A385641" memory used=296185.5MB, alloc=3703.5MB, time=2253.92 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SearchTable: "SearchTable successful" /n - 1 \ |----- n1 | | \ (-1) ((5 n1 + 9) hypergeom([1/2, -n1 - 1], [1], 4) + (-3 n1 - 3) hypergeom([1/2, -n1], [1], 4))| {(n + 2) | ) -------------------------------------------------------------------------------------------------|, n + 2} | / (n1 + 3) (n1 + 2) | |----- | \n1 = 0 / "A385667" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ |4 12 (3 n1 + 1) (20 n1 + 13) binomial(3 n1, n1)|| {(1/4) , (1/4) | ) |-------------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) /| |----- | \n1 = 0 / "A385668" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 2 \| n n | \ | 8 (-1) 24 (4 n1 + 1) (187 n1 + 233 n1 + 70) binomial(4 n1, n1)|| {(-1/8) , (-1/8) | ) |- -------------------------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 1) (3 n1 + 2) /| |----- | \n1 = 0 / "A385669" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 5 binomial(2 n1, n1)| {(-4/3) , (-4/3) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4/3) | \n1 = 0 / "A385670" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (3 n1 + 1) 5 binomial(3 n1, n1) (91 n1 + 58)| {(8/9) , (8/9) | ) ----------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (8/9) | \n1 = 0 / "A385716" LREtools/SearchTable: "SearchTable successful" {((12 n + 5) hypergeom([-1/2, -n - 1], [1], -12) + (-12 n - 11) hypergeom([-1/2, -n], [1], -12)) (n + 1)} "A385719" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A385728" LREtools/SearchTable: "SearchTable successful" n {2 (2 n hypergeom([-1/2, -n - 1], [1], -2) + (-2 n - 1) hypergeom([-1/2, -n], [1], -2)) (n + 1)} "A385813" LREtools/SearchTable: "SearchTable successful" n {3 ((4 n - 1) hypergeom([-1/2, -n - 1], [1], -4/3) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)} "A385823" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 (2 n1 - 1) binomial(3 n1, n1) (5 n1 + 11 n1 - 14)| {8 , 8 | ) ---------------------------------------------------------------|} | / (n1 + 1) (3 n1 - 2) (3 n1 - 1) | |----- | \n1 = 0 / "A385975" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A386006" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 binomial(3 n1, n1) (5 n1 + 11 n1 - 6)| {8 , 8 | ) ---------------------------------------------------|} | / (n1 + 1) (3 n1 - 1) | |----- | \n1 = 0 / "A386291" 2 binomial(2 n, n) binomial(2 n, n) {-----------------, ----------------} 2 n + 1 (n + 1) "A386301" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" n - 1 ----- n \ {1, (-1) , ) / ----- n1 = 0 / / /{ 0 n2::even\\\ |n1 - 1 | |{ ||| |----- | |{ / n2 \ ||| n1 | \ | n2 |{ |---- - 1/2| ||| (-1) | ) |-(-1) |{ \ 2 / |||, | / | |{ 1/2 / 1/2 n2 1/2 n2 1/2 \ ||| |----- | |{ 2 (3 I) |3 LegendreP(---- + 1/2, 3 I) I - LegendreP(---- - 1/2, 3 I)| ||| |n2 = 0 | |{ \ 2 2 / ||| | | |{ --------------------------------------------------------------------------------------------- n2::odd ||| \ \ \{ n2 + 3 /// n - 1 ----- \ ) / ----- n1 = 0 / / /{ 0 n2::even\\\ |n1 - 1 | |{ ||| |----- | |{ / n2 \ ||| n1 | \ | n2 |{ |---- - 1/2| ||| (-1) | ) |-(-1) |{ \ 2 / |||, | / | |{ 1/2 / 1/2 n2 1/2 n2 1/2 \ ||| |----- | |{ 2 (3 I) |3 LegendreQ(---- + 1/2, 3 I) I - LegendreQ(---- - 1/2, 3 I)| ||| |n2 = 0 | |{ \ 2 2 / ||| | | |{ --------------------------------------------------------------------------------------------- n2::odd ||| \ \ \{ n2 + 3 /// / / /{ / n2 \ \\\ n - 1 |n1 - 1 | |{ |----| ||| ----- |----- | |{ \ 2 / ||| \ n1 | \ | n2 |{ 1/2 / 1/2 n2 1/2 n2 1/2 \ ||| ) (-1) | ) |-(-1) |{ (3 I) |3 LegendreP(---- + 1/2, 3 I) I - LegendreP(---- - 1/2, 3 I)| |||, / | / | |{ \ 2 2 / ||| ----- |----- | |{ ------------------------------------------------------------------------------------- n2::even||| n1 = 0 |n2 = 0 | |{ n2 + 3 ||| | | |{ ||| \ \ \{ 0 n2::odd /// / / /{ / n2 \ \\\ n - 1 |n1 - 1 | |{ |----| ||| ----- |----- | |{ \ 2 / ||| \ n1 | \ | n2 |{ 1/2 / 1/2 n2 1/2 n2 1/2 \ ||| ) (-1) | ) |-(-1) |{ (3 I) |3 LegendreQ(---- + 1/2, 3 I) I - LegendreQ(---- - 1/2, 3 I)| |||} / | / | |{ \ 2 2 / ||| ----- |----- | |{ ------------------------------------------------------------------------------------- n2::even||| n1 = 0 |n2 = 0 | |{ n2 + 3 ||| | | |{ ||| \ \ \{ 0 n2::odd /// "A386362" LREtools/SearchTable: "SearchTable successful" (12 n + 5) hypergeom([-1/2, -n - 1], [1], -12) + (-12 n - 11) hypergeom([-1/2, -n], [1], -12) {---------------------------------------------------------------------------------------------} n + 2 "A386365" LREtools/SearchTable: "SearchTable not successful" {} "A386371" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /3125\n1 3 2 \ |----- |----| GAMMA(n1 + 3/5) GAMMA(n1 + 4/5) GAMMA(n1 + 2/5) GAMMA(n1 + 6/5) (37331 n1 + 65343 n1 + 36718 n1 + 6600)| /-32\n /-32\n | \ \256 / | {|---| , |---| | ) ------------------------------------------------------------------------------------------------------------------|} \81 / \81 / | / /-32\(n1 + 1) | |----- GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 5/4) |---| | \n1 = 0 \81 / / "A386387" LREtools/SearchTable: "SearchTable successful" {hypergeom([-1/2, -n - 1], [1], -16) - hypergeom([-1/2, -n], [1], -16)} "A386389" LREtools/SearchTable: "SearchTable successful" (16 n + 7) hypergeom([-1/2, -n - 1], [1], -16) + (-16 n - 15) hypergeom([-1/2, -n], [1], -16) {---------------------------------------------------------------------------------------------} n + 2 "A386398" LREtools/SearchTable: "SearchTable successful" n n {4 (LegendreP(n + 1, 3/2) + LegendreP(n, 3/2)), 4 (LegendreQ(n + 1, 3/2) + LegendreQ(n, 3/2))} "A386413" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { 0 irem(n, 3) = 0 { { 0 irem(n, 3) = 1 { {{ (n/3 - 2/3) 11 , { 6912 GAMMA(n/3 + 5/12) GAMMA(n/3 + --) GAMMA(1/6 + n/3) { 12 { ------------------------------------------------------------------ irem(n, 3) = 2 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(5/3 + n/3) { 0 irem(n, 3) = 0 { { (2 n - 2) 4 n { 3 (4 n - 1) binomial(--- - 4/3, n/3 - 1/3) { 3 , { --------------------------------------------------- irem(n, 3) = 1 { n (n + 2) { { 0 irem(n, 3) = 2 { (n/3) 11 { 6912 GAMMA(n/3 + 5/12) GAMMA(n/3 + --) GAMMA(1/6 + n/3) { 12 { ------------------------------------------------------------ irem(n, 3) = 0 { GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(5/3 + n/3) } { { 0 irem(n, 3) = 1 { { 0 irem(n, 3) = 2 "A386415" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { { { {{ 0 , irem(n, 3) = 0 { { { 0 , irem(n, 3) = 1 /452984832\(n/3 - 2/3) 17 11 23 |---------| GAMMA(n/3 + 5/6) GAMMA(n/3 + 5/24) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 1/12) GAMMA(n/3 + --) GAMMA(n/3 + 7/12) \ 3125 / 24 24 24 --------------------------------------------------------------------------------------------------------------------------------------------- , 17 14 11 GAMMA(n/3 + 1/3) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 1) GAMMA(n/3 + 2/3) GAMMA(n/3 + --) GAMMA(n/3 + 8/15) 15 15 15 { { { irem(n, 3) = 2, { 0 , irem(n, 3) = 0 { { { /452984832\(n/3 - 1/3) 23 17 11 |---------| GAMMA(n/3 + 1/12) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) GAMMA(n/3 + 5/24) GAMMA(n/3 + 5/6) GAMMA(n/3 + --) GAMMA(n/3 + --) \ 3125 / 24 24 24 --------------------------------------------------------------------------------------------------------------------------------------------- , 14 17 11 GAMMA(n/3 + 2/3) GAMMA(n/3 + 8/15) GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) 15 15 15 irem(n, 3) = 1 { { { { 0 , irem(n, 3) = 2, { { { { { / 5 n\ |- ---| n \ 3 / 11 17 23 768 5 GAMMA(n/3 + 1/12) GAMMA(n/3 + 5/6) GAMMA(n/3 + 5/24) GAMMA(n/3 + 7/12) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) 24 24 24 ------------------------------------------------------------------------------------------------------------------------------------ , 11 14 17 GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) GAMMA(n/3 + 8/15) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) 15 15 15 irem(n, 3) = 0 0 , irem(n, 3) = 1 } 0 , irem(n, 3) = 2 "A386416" LREtools/SolveLRE: "Absolute Factorization reduced the order from 3 to 1 (Liouvillian solutions)" { { { {{ 0 , irem(n, 3) = 0 { { { 0 , irem(n, 3) = 1 /452984832\(n/3 - 2/3) 13 19 11 |---------| GAMMA(n/3 + 1/24) GAMMA(n/3 + --) GAMMA(n/3 + 5/12) GAMMA(n/3 + --) GAMMA(n/3 + 7/24) GAMMA(n/3 + --) GAMMA(1/6 + n/3) \ 3125 / 24 24 12 --------------------------------------------------------------------------------------------------------------------------------------------- , 13 16 GAMMA(n/3 + --) GAMMA(n/3 + 7/15) GAMMA(n/3 + 1/3) GAMMA(n/3 + 1) GAMMA(n/3 + --) GAMMA(n/3 + 4/15) GAMMA(n/3 + 2/3) 15 15 { { { irem(n, 3) = 2, { 0 , irem(n, 3) = 0 { { { /452984832\(n/3 - 1/3) 13 19 11 |---------| GAMMA(n/3 + 1/24) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + 5/12) GAMMA(1/6 + n/3) GAMMA(n/3 + 7/24) GAMMA(n/3 + --) \ 3125 / 24 24 12 --------------------------------------------------------------------------------------------------------------------------------------------- , 16 13 GAMMA(n/3 + 4/15) GAMMA(n/3 + 2/3) GAMMA(n/3 + 7/15) GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) GAMMA(n/3 + --) GAMMA(n/3 + --) 15 15 irem(n, 3) = 1 { { { { 0 , irem(n, 3) = 2, { { { { { / 5 n\ |- ---| n \ 3 / 11 13 19 768 5 GAMMA(1/6 + n/3) GAMMA(n/3 + 1/24) GAMMA(n/3 + 5/12) GAMMA(n/3 + 7/24) GAMMA(n/3 + --) GAMMA(n/3 + --) GAMMA(n/3 + --) 12 24 24 ------------------------------------------------------------------------------------------------------------------------------------ , 13 16 GAMMA(n/3 + 1) GAMMA(n/3 + 1/3) GAMMA(n/3 + 2/3) GAMMA(n/3 + 4/15) GAMMA(n/3 + 7/15) GAMMA(n/3 + --) GAMMA(n/3 + --) 15 15 irem(n, 3) = 0 0 , irem(n, 3) = 1 } 0 , irem(n, 3) = 2 "A386496" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A386542" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A386548" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A386565" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | /256\n /256\n | \ binomial(4 n1, n1) (80 n1 + 98 n1 + 27) | {|---| , |---| | ) --------------------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (n1 + 1) (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A386566" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / 3 2 \| /3125\n /3125\n | \ |256 (5000 n1 + 8775 n1 + 4735 n1 + 768) GAMMA(n1 + 1/5) GAMMA(n1 + 2/5) GAMMA(n1 + 3/5) GAMMA(n1 + 4/5)|| {|----| , |----| | ) |---- -----------------------------------------------------------------------------------------------------||} \256 / \256 / | / \3125 GAMMA(n1 + 2) GAMMA(n1 + 3/2) GAMMA(n1 + 5/4) GAMMA(n1 + 7/4) /| |----- | \n1 = 0 / "A386567" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- /46656\n /46656\n | \ {|-----| , |-----| | ) \3125 / \3125 / | / |----- \n1 = 0 \ / 4 3 2 \| |3125 (94500 n1 + 214830 n1 + 171573 n1 + 56243 n1 + 6250) GAMMA(1/6 + n1) GAMMA(n1 + 1/2) GAMMA(n1 + 1/3) GAMMA(n1 + 2/3) GAMMA(n1 + 5/6)|| |----- ---------------------------------------------------------------------------------------------------------------------------------------||} \46656 GAMMA(n1 + 2) GAMMA(n1 + 6/5) GAMMA(n1 + 7/5) GAMMA(n1 + 8/5) GAMMA(n1 + 9/5) /| | / "A386611" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | /256\n /256\n | \ (4 n1 + 1) (4 n1 + 3) binomial(4 n1, n1) (8 n1 + 9) | {|---| , |---| | ) -------------------------------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A386612" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | /256\n /256\n | \ (4 n1 + 1) (2 n1 + 1) binomial(4 n1, n1) (264 n1 + 661 n1 + 405) | {|---| , |---| | ) ------------------------------------------------------------------|} \27 / \27 / | / /256\(n1 + 1)| |----- (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) |---| | \n1 = 0 \27 / / "A386617" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 8) | {(27/4) , (27/4) | ) ------------------------------------------------------|} | / (n1 + 1)| |----- (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) (27/4) | \n1 = 0 / "A386670" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 - 1) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A386700" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ binomial(3 n1, n1) (55 n1 + 45 n1 + 8)| {(-8/9) , (-8/9) | ) ---------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (2 n1 + 1) (-8/9) | \n1 = 0 / "A386722" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-3 n1 - 3) | n n | \ 3 2 n1 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1)| {8 , 8 | ) ------------------------------------------------------------|} | / (n1 + 1) (n1 + 2) (2 n1 + 1) (2 n1 + 3) | |----- | \n1 = 0 / "A386763" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 5 binomial(2 n1, n1) (7 n1 + 5)| {(-9/2) , (-9/2) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-9/2) | \n1 = 0 / "A386766" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ 5 binomial(2 n1, n1) (8 n1 + 5)| {(-4/3) , (-4/3) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4/3) | \n1 = 0 / "A386769" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 5 binomial(2 n1, n1)| {(-9/2) , (-9/2) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-9/2) | \n1 = 0 / "A386770" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-3 n1 - 3) n1 2 \| n n | \ |4 3 20 (3 n1 + 2) (3 n1 + 1) binomial(3 n1, n1)|| {(27/4) , (27/4) | ) |-------------------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (2 n1 + 1) (2 n1 + 3) /| |----- | \n1 = 0 / "A386772" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 5 binomial(2 n1, n1)| {(-4/3) , (-4/3) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (n1 + 2) (-4/3) | \n1 = 0 / "A386773" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (3 n1 + 1) (3 n1 + 2) 5 binomial(3 n1, n1) (7 n1 + 8)| {(8/9) , (8/9) | ) -------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) (8/9) | \n1 = 0 / "A386811" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 (4 n1 + 1) binomial(4 n1, n1) (22 n1 + 38 n1 + 15)| {16 , 16 | ) ----------------------------------------------------------------|} | / (n1 + 1) (3 n1 + 1) (3 n1 + 2) | |----- | \n1 = 0 / "A386812" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-13 n1 - 5) 3 2 | n n | \ 3125 2 (1689 n1 + 3805 n1 + 2698 n1 + 600) GAMMA(n1 + 2/5) GAMMA(n1 + 3/5) GAMMA(n1 + 4/5) GAMMA(n1 + 6/5)| {32 , 32 | ) --------------------------------------------------------------------------------------------------------------------------|} | / GAMMA(n1 + 2) GAMMA(n1 + 3/2) GAMMA(n1 + 5/4) GAMMA(n1 + 7/4) | |----- | \n1 = 0 / "A386821" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- 2 | n n | \ (4 n1 + 1) (2 n1 + 1) binomial(4 n1, n1) (415 n1 + 841 n1 + 396)| {(-1/8) , (-1/8) | ) -----------------------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) (-1/8) | \n1 = 0 / "A386826" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-4 n1 - 4) | n n | \ 3 2 (2 n1 + 3) binomial(2 n1, n1)| {16 , 16 | ) ----------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A386829" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | n n | \ (2 n1 + 1) 5 binomial(2 n1, n1)| {(-9/2) , (-9/2) | ) ---------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-9/2) | \n1 = 0 / "A386830" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-3 n1 - 3) n1 \| n n | \ |4 3 20 (3 n1 + 1) binomial(3 n1, n1)|| {(27/4) , (27/4) | ) |-------------------------------------------------||} | / \ n1 + 1 /| |----- | \n1 = 0 / "A386833" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (3 n1 + 1) 2 binomial(3 n1, n1) n1 (5 n1 + 4)| {(5 n + 1) | ) -----------------------------------------------|, 5 n + 1} | / (n1 + 1) (2 n1 + 1) (5 n1 + 6) (5 n1 + 1) | |----- | \n1 = 0 / "A386835" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) (3 n1 + 2) binomial(2 n1, n1)|| {(-1) (n + 1) (9 n + 2), (-1) (n + 1) (9 n + 2) | ) |- ---------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (9 n1 + 2) (9 n1 + 11) /| |----- | \n1 = 0 / "A386836" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 2 | 2 | \ (3 n1 + 1) (3 n1 + 2) 2 binomial(3 n1, n1) (5 n1 + 3 n1 - 1) | 2 {(25 n + 19 n + 2) | ) --------------------------------------------------------------------|, 25 n + 19 n + 2} | / 2 2 | |----- (n1 + 1) (2 n1 + 1) (25 (n1 + 1) + 19 n1 + 21) (25 n1 + 19 n1 + 2)| \n1 = 0 / "A386839" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 6 _Z - 5 _Z - 1, index = 1) , RootOf(_Z - 6 _Z - 5 _Z - 1, index = 2) , RootOf(_Z - 6 _Z - 5 _Z - 1, index = 3) } "A386843" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | (-1) 2 (2 n1 + 1) binomial(2 n1, n1)|| {(-1) (n + 1), (-1) (n + 1) | ) |- ----------------------------------------||} | / \ (n1 + 1) (n1 + 2) /| |----- | \n1 = 0 / "A386844" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (3 n1 + 1) (3 n1 + 2) 2 binomial(3 n1, n1) (5 n1 + 2)| {(5 n + 3) | ) -------------------------------------------------------|, 5 n + 3} | / (n1 + 1) (2 n1 + 1) (5 n1 + 8) (5 n1 + 3) | |----- | \n1 = 0 / "A386862" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (5 n1 + 3) binomial(2 n1, n1)|| {(-1/2) (5 n + 2), (-1/2) (5 n + 2) | ) |- -----------------------------------------------------||} | / \ (n1 + 1) (5 n1 + 7) (10 n1 + 4) /| |----- | \n1 = 0 / "A386863" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ |4 12 (3 n1 + 1) (4 n1 + 3) (5 n1 + 1) binomial(3 n1, n1)|| {(1/4) (4 n + 1), (1/4) (4 n + 1) | ) |----------------------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) (4 n1 + 5) (16 n1 + 4) /| |----- | \n1 = 0 / "A386865" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) (5 n1 + 3) binomial(2 n1, n1)|| {(-1/2) (n + 1) (25 n + 8), (-1/2) (n + 1) (25 n + 8) | ) |- -----------------------------------------------------||} | / \ (n1 + 1) (n1 + 2) (25 n1 + 8) (25 n1 + 33) /| |----- | \n1 = 0 / "A386866" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 2 \| n 2 n 2 | \ | 4 12 (3 n1 + 1) (3 n1 + 2) (40 n1 + 35 n1 + 3) binomial(3 n1, n1) || {(1/4) (32 n + 27 n + 4), (1/4) (32 n + 27 n + 4) | ) |-----------------------------------------------------------------------||} | / | 2 2 || |----- \(n1 + 1) (2 n1 + 1) (32 (n1 + 1) + 27 n1 + 31) (128 n1 + 108 n1 + 16)/| \n1 = 0 / "A386868" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 \| n n | \ | 2 (-1) 6 (2 n1 + 1) binomial(2 n1, n1)|| {(-1/2) (n + 1), (-1/2) (n + 1) | ) |- ------------------------------------------||} | / \ (n1 + 2) (2 n1 + 2) /| |----- | \n1 = 0 / "A386869" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 \| n n | \ |4 12 (3 n1 + 1) (3 n1 + 2) (40 n1 + 23) binomial(3 n1, n1)|| {(1/4) (8 n + 5), (1/4) (8 n + 5) | ) |------------------------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) (8 n1 + 13) (32 n1 + 20) /| |----- | \n1 = 0 / "A386870" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / n1 n1 2 \| n n | \ | 8 (-1) 24 (2 n1 + 1) (4 n1 + 1) (187 n1 + 223 n1 + 64) binomial(4 n1, n1)|| {(-1/8) (11 n + 5), (-1/8) (11 n + 5) | ) |- ------------------------------------------------------------------------------||} | / \ (n1 + 1) (3 n1 + 1) (3 n1 + 2) (11 n1 + 16) (88 n1 + 40) /| |----- | \n1 = 0 / "A386882" LREtools/SearchTable: "SearchTable successful" 2 3 2 (2 n + 1) (3 n + 7 n + 7) LegendreP(n + 1, 3) + (-n - 2 n - n + 3) LegendreP(n, 3) {-------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) 2 3 2 (2 n + 1) (3 n + 7 n + 7) LegendreQ(n + 1, 3) + (-n - 2 n - n + 3) LegendreQ(n, 3) -------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A386897" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { { /12500\(n/2 - 1/2) {{ |-----| GAMMA(n/2 + 2/5) GAMMA(n/2 + 3/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 1/5) , { \ 27 / { -------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 1/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/3) { /12500\(n/2) { |-----| GAMMA(n/2 + 3/5) GAMMA(n/2 + 1/5) GAMMA(n/2 + 4/5) GAMMA(n/2 + 2/5) { \ 27 / { -------------------------------------------------------------------------------- n::even} { GAMMA(n/2 + 1/3) GAMMA(n/2 + 1) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/2) { { 0 n::odd "A386899" LREtools/SearchTable: "SearchTable successful" / 1/2\n 1/2 | 19 19 | 13718 1064 19 |-14 + --------| GAMMA(n + 4/3) hypergeom([5/6, 1/3 - n], [5/3], ----- + ----------) \ 2 / 6075 6075 {-------------------------------------------------------------------------------------} GAMMA(n + 1) "A386900" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- / (-3 n1 - 3) n1 \| n n | \ |4 5 12 (3 n1 + 1) (22 n1 + 23) binomial(3 n1, n1)|| {(125/4) , (125/4) | ) |--------------------------------------------------------------||} | / \ (n1 + 1) (2 n1 + 1) /| |----- | \n1 = 0 / "A386919" LREtools/SearchTable: "SearchTable successful" n {(-8) (24 (3 n + 2) (3 n + 4) (3 n + 1) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) 3 2 / 2 + (-577 n - 1325 n - 834 n - 144) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1) / ((2 n + 1) (4 n + 1) (17 n + 24))} / "A386920" n {4 binomial(2 n, n), binomial(4 n, 2 n)} "A386937" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { n 3 n { 3 4 n binomial(---, n/2) { 2 { - ------------------------- n::even {{ n + 1 , { { (2 n + 2) 3 n { 1/2 2 binomial(--- + 3/2, n/2 + 1/2) n::odd { 2 { 3 n 3 n { 2 binomial(3 n, ---) binomial(---, n/2) (3 n + 1) { 2 2 { ------------------------------------------------- n::even { binomial(n, n/2) (n + 1) { } { 3 n 3 n { 12 binomial(3 n - 3, --- - 3/2) binomial(--- - 3/2, n/2 - 1/2) (3 n - 2) { 2 2 { - ------------------------------------------------------------------------ n::odd { binomial(n - 1, n/2 - 1/2) (n + 1) "A386939" n (4 n + 1) binomial(4 n, 2 n) 4 binomial(2 n, n) n {----------------------------, ---------------------} n + 1 n + 1 "A386940" LREtools/SearchTable: "SearchTable successful" {binomial(2 n, n) hypergeom([n, -n], [-n + 1/2], 1/4)} "A386941" LREtools/SearchTable: "SearchTable successful" ((-4 n - 1) hypergeom([n, -n], [-n + 1/2], 1/4) + 4 n hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4)) binomial(2 n, n) (2 n + 1) {-----------------------------------------------------------------------------------------------------------------------------} 2 n - 1 "A386942" LREtools/SearchTable: "SearchTable successful" (-(4 n + 1) (4 n + 3) hypergeom([n, -n], [-n + 1/2], 1/4) + (2 n + 1) (8 n + 1) hypergeom([n + 1, -n - 1], [-n - 1/2], 1/4)) binomial(2 n, n) {---------------------------------------------------------------------------------------------------------------------------------------------} 2 n - 1 "A386955" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (n1 - 1) | {(9/2) (n + 2), (9/2) (n + 2) | ) ------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (9/2) (n1 + 3) (2 n1 + 4)| \n1 = 0 / "A386956" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) (7 n1 - 1) | {(81/8) (7 n + 8), (81/8) (7 n + 8) | ) ------------------------------------------------|} | / (n1 + 1) | |----- (n1 + 1) (81/8) (7 n1 + 15) (56 n1 + 64)| \n1 = 0 / "A386957" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(81/8) , (81/8) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (81/8) | \n1 = 0 / "A386958" LREtools/SearchTable: "SearchTable successful" n / -49 -49 \ 4 |(192 n + 128) hypergeom([7/6, -n - 1/6], [3/2], ---) + (-486 n - 81) hypergeom([7/6, -n + 5/6], [3/2], ---)| GAMMA(n + 2/3) \ 32 32 / {-------------------------------------------------------------------------------------------------------------------------------} GAMMA(n + 1) "A386959" LREtools/SearchTable: "SearchTable successful" n / -49 -49 \ 4 |(96 n + 64) hypergeom([7/6, -n - 1/6], [3/2], ---) + (-243 n - 9) hypergeom([7/6, -n + 5/6], [3/2], ---)| GAMMA(n + 2/3) \ 32 32 / {----------------------------------------------------------------------------------------------------------------------------} GAMMA(n + 1) "A386960" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ binomial(2 n1, n1) (7 n1 - 1)| {(81/8) , (81/8) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (81/8) | \n1 = 0 / "A386961" LREtools/SearchTable: "SearchTable successful" {((2 n + 2) LaguerreL(n + 1, -1) + (-3 n - 4) LaguerreL(n, -1)) n!} "A386986" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1) | {(81/8) (n + 1), (81/8) (n + 1) | ) ----------------------------------|} | / (n1 + 1) | |----- (81/8) (n1 + 2) (8 n1 + 8)| \n1 = 0 / "A387006" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387007" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 6)| {8 , 8 | ) ----------------------------------------------------------------|} | / (n1 + 1) (2 n1 + 3) (2 n1 + 1) | |----- | \n1 = 0 / "A387008" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) 2 | n n | \ 2 (3 n1 + 1) (3 n1 + 2) binomial(3 n1, n1) (5 n1 + 11 n1 + 4)| {8 , 8 | ) -------------------------------------------------------------------------|} | / (n1 + 2) (2 n1 + 3) (n1 + 1) (2 n1 + 1) | |----- | \n1 = 0 / "A387009" memory used=297876.9MB, alloc=3735.5MB, time=2265.22 LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) | n n | \ 2 (4 n1 + 1) (2 n1 + 1) binomial(4 n1, n1) (22 n1 + 27)| {16 , 16 | ) ------------------------------------------------------------------|} | / (3 n1 + 4) (3 n1 + 1) (3 n1 + 2) | |----- | \n1 = 0 / "A387010" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 (2 n1 + 1) (4 n1 + 3) (4 n1 + 1) binomial(4 n1, n1) (11 n1 + 30 n1 + 20)| {16 , 16 | ) --------------------------------------------------------------------------------------|} | / (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) | |----- | \n1 = 0 / "A387011" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 (4 n1 + 3) (2 n1 + 3) (2 n1 + 1) (4 n1 + 1) binomial(4 n1, n1) (22 n1 + 60 n1 + 35)| {16 , 16 | ) -------------------------------------------------------------------------------------------------|} | / (n1 + 2) (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) | |----- | \n1 = 0 / "A387012" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 (-2 n1 - 2) | n n | \ 2 3 (n1 + 2) binomial(2 n1, n1)| {9 , 9 | ) --------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A387033" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-3 n1 - 3) | n n | \ 2 binomial(3 n1, n1) (5 n1 + 11)| {8 , 8 | ) -------------------------------------------|} | / n1 + 1 | |----- | \n1 = 0 / "A387034" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 3 2 | n n | \ 2 (3 n1 - 1) binomial(4 n1, n1) (44 n1 + 10 n1 - 94 n1 + 45)| {16 , 16 | ) -------------------------------------------------------------------------|} | / (n1 + 1) (4 n1 - 3) (2 n1 - 1) (4 n1 - 1) | |----- | \n1 = 0 / "A387035" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 3 2 | n n | \ 2 binomial(4 n1, n1) (22 n1 + 16 n1 - 29 n1 + 7)| {16 , 16 | ) -------------------------------------------------------------|} | / (n1 + 1) (2 n1 - 1) (4 n1 - 1) | |----- | \n1 = 0 / "A387036" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 binomial(4 n1, n1) (22 n1 + 27 n1 - 10)| {16 , 16 | ) -----------------------------------------------------|} | / (n1 + 1) (4 n1 - 1) | |----- | \n1 = 0 / "A387037" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- (-4 n1 - 4) 2 | n n | \ 2 binomial(4 n1, n1) (11 n1 + 19 n1 + 5)| {16 , 16 | ) ----------------------------------------------------|} | / (3 n1 + 1) (n1 + 1) | |----- | \n1 = 0 / "A387063" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387083" LREtools/SolveLRE: "Applying recursion to a right-factor defined over an algebraic extension" / 1/2\n / 1/2\n | 2 | | 2 | {|1 - ----| n!, |1 + ----| n!} \ 2 / \ 2 / "A387085" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | n n | \ (2 n1 + 1) binomial(2 n1, n1)| {(-4/3) , (-4/3) | ) -----------------------------|} | / (n1 + 1) | |----- (n1 + 1) (-4/3) | \n1 = 0 / "A387086" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387105" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387125" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387200" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387202" LREtools/SearchTable: "SearchTable successful" 2 ((4 n - 2 n + 1) hypergeom([-1/2, -n - 1], [1], -4) - (2 n - 1) (2 n + 1) hypergeom([-1/2, -n], [1], -4)) (2 n + 3) {--------------------------------------------------------------------------------------------------------------------} (n + 3) (n + 2) n "A387205" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- | | \ n1! (n1 + 1)| n! | ) ------------| | / n1 (n1 + 1)!| |----- | n! \n1 = 0 / {----, ------------------------} n n "A387208" LREtools/SearchTable: "SearchTable successful" 2 {16 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-32 n - 64 n - 33) hypergeom([-1/2, -n], [1], -8)} "A387210" LREtools/SearchTable: "SearchTable successful" 2 {24 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -12) + (-48 n - 96 n - 49) hypergeom([-1/2, -n], [1], -12)} "A387211" LREtools/SearchTable: "SearchTable successful" n 2 {2 (4 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -2) + (-8 n - 16 n - 9) hypergeom([-1/2, -n], [1], -2))} "A387212" LREtools/SearchTable: "SearchTable successful" n 2 {3 (8 (n + 1) (2 n + 1) hypergeom([-1/2, -n - 1], [1], -4/3) + (-16 n - 32 n - 19) hypergeom([-1/2, -n], [1], -4/3))} "A387224" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387228" LREtools/SearchTable: "SearchTable successful" 2 2 {((64 n + 136 n + 47) hypergeom([-1/2, -n - 1], [1], -4) + (-64 n - 168 n - 103) hypergeom([-1/2, -n], [1], -4)) (n + 1)} "A387229" LREtools/SearchTable: "SearchTable successful" 2 2 {((256 n + 528 n + 191) hypergeom([-1/2, -n - 1], [1], -8) + (-256 n - 656 n - 399) hypergeom([-1/2, -n], [1], -8)) (n + 1)} "A387230" LREtools/SearchTable: "SearchTable successful" 2 2 {((576 n + 1176 n + 431) hypergeom([-1/2, -n - 1], [1], -12) + (-576 n - 1464 n - 887) hypergeom([-1/2, -n], [1], -12)) (n + 1)} "A387233" LREtools/SearchTable: "SearchTable successful" n 2 2 {2 ((16 n + 36 n + 11) hypergeom([-1/2, -n - 1], [1], -2) + (-16 n - 44 n - 27) hypergeom([-1/2, -n], [1], -2)) (n + 1)} "A387234" LREtools/SearchTable: "SearchTable successful" n 2 2 {3 ((64 n + 152 n + 39) hypergeom([-1/2, -n - 1], [1], -4/3) + (-64 n - 184 n - 111) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)} "A387237" LREtools/SearchTable: "SearchTable successful" 2 2 {((20 n + 20 n + 9) hypergeom([-1/2, -n - 1], [1], -4) + (-20 n - 30 n - 15) hypergeom([-1/2, -n], [1], -4)) (n + 1)} "A387238" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 {((100 n + 225 n + 311 n + 60) hypergeom([-1/2, -n - 1], [1], -4) + (-100 n - 275 n - 405 n - 180) hypergeom([-1/2, -n], [1], -4)) (n + 1)} "A387239" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 ((100 n + 225 n + 311 n + 60) hypergeom([-1/2, -n - 1], [1], -4) + (-100 n - 275 n - 405 n - 180) hypergeom([-1/2, -n], [1], -4)) (n + 1) {---------------------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A387244" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387272" LREtools/SearchTable: "SearchTable successful" n 2 2 2 ((6 n + 4) hypergeom([-1/2, -n - 1], [1], -2) + (-6 n - 3 n - 3) hypergeom([-1/2, -n], [1], -2)) (n + 1) {-------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A387273" LREtools/SearchTable: "SearchTable successful" n 3 3 2 2 ((18 n + 80 n - 28) hypergeom([-1/2, -n - 1], [1], -2) + (-18 n - 9 n - 78 n - 15) hypergeom([-1/2, -n], [1], -2)) (n + 1) {--------------------------------------------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A387274" LREtools/SearchTable: "SearchTable successful" n 4 2 4 3 2 {2 ((54 n + 712 n - 804 n + 668) hypergeom([-1/2, -n - 1], [1], -2) + (-54 n - 27 n - 705 n + 459 n - 105) hypergeom([-1/2, -n], [1], -2)) (n + 1)/((n + 8) (n + 7) (n + 6) (n + 5))} "A387275" LREtools/SearchTable: "SearchTable successful" n 2 2 3 ((28 n - 28 n + 39) hypergeom([-1/2, -n - 1], [1], -4/3) + (-28 n + 14 n - 21) hypergeom([-1/2, -n], [1], -4/3)) (n + 1) {-----------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A387276" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 {3 ((196 n - 441 n + 1913 n - 1500) hypergeom([-1/2, -n - 1], [1], -4/3) + (-196 n + 343 n - 1687 n + 420) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)/((n + 6) (n + 5) (n + 4))} "A387277" LREtools/SearchTable: "SearchTable successful" n 4 3 2 {3 ((686 n - 2744 n + 20626 n - 54184 n + 49035) hypergeom([-1/2, -n - 1], [1], -4/3) 4 3 2 + (-686 n + 2401 n - 19201 n + 44156 n - 16905) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)/((n + 8) (n + 7) (n + 6) (n + 5))} "A387278" LREtools/SearchTable: "SearchTable successful" n 3 ((4 n - 1) hypergeom([-1/2, -n - 1], [1], -4/3) + (-4 n - 1) hypergeom([-1/2, -n], [1], -4/3)) (n + 1) {---------------------------------------------------------------------------------------------------------} n + 2 "A387280" LREtools/SearchTable: "SearchTable successful" n 2 2 {2 ((6 n + 4) hypergeom([-1/2, -n - 1], [1], -2) + (-6 n - 3 n - 3) hypergeom([-1/2, -n], [1], -2)) (n + 1)} "A387281" LREtools/SearchTable: "SearchTable successful" n 3 3 2 {2 ((18 n + 80 n - 28) hypergeom([-1/2, -n - 1], [1], -2) + (-18 n - 9 n - 78 n - 15) hypergeom([-1/2, -n], [1], -2)) (n + 1)} "A387282" LREtools/SearchTable: "SearchTable successful" n 4 2 4 3 2 {2 ((54 n + 712 n - 804 n + 668) hypergeom([-1/2, -n - 1], [1], -2) + (-54 n - 27 n - 705 n + 459 n - 105) hypergeom([-1/2, -n], [1], -2)) (n + 1)} "A387283" LREtools/SearchTable: "SearchTable successful" n 2 2 {3 ((28 n - 28 n + 39) hypergeom([-1/2, -n - 1], [1], -4/3) + (-28 n + 14 n - 21) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)} "A387284" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 {3 ((196 n - 441 n + 1913 n - 1500) hypergeom([-1/2, -n - 1], [1], -4/3) + (-196 n + 343 n - 1687 n + 420) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)} "A387285" LREtools/SearchTable: "SearchTable successful" n 4 3 2 {3 ((686 n - 2744 n + 20626 n - 54184 n + 49035) hypergeom([-1/2, -n - 1], [1], -4/3) 4 3 2 + (-686 n + 2401 n - 19201 n + 44156 n - 16905) hypergeom([-1/2, -n], [1], -4/3)) (n + 1)} "A387288" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387301" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387307" LREtools/SearchTable: "SearchTable successful" 2 ((12 n + 11) (12 n + 7) hypergeom([-1/2, -n - 1], [1], -8) + (-144 n - 288 n - 153) hypergeom([-1/2, -n], [1], -8)) (n + 1) {----------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A387308" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 {((1296 n + 4374 n + 5098 n + 1495) hypergeom([-1/2, -n - 1], [1], -8) + (-1296 n - 5022 n - 7002 n - 3195) hypergeom([-1/2, -n], [1], -8)) (n + 1)/((n + 6) (n + 5) (n + 4))} "A387309" LREtools/SearchTable: "SearchTable successful" ((12 n + 5) hypergeom([-1/2, -n - 1], [1], -12) + (-12 n - 11) hypergeom([-1/2, -n], [1], -12)) (n + 1) {-------------------------------------------------------------------------------------------------------} n + 2 "A387310" LREtools/SearchTable: "SearchTable successful" 2 2 ((468 n + 780 n + 277) hypergeom([-1/2, -n - 1], [1], -12) + (-468 n - 1014 n - 559) hypergeom([-1/2, -n], [1], -12)) (n + 1) {-------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) "A387311" LREtools/SearchTable: "SearchTable successful" 3 2 {((18252 n + 68445 n + 80835 n + 24748) hypergeom([-1/2, -n - 1], [1], -12) 3 2 + (-18252 n - 77571 n - 110877 n - 51220) hypergeom([-1/2, -n], [1], -12)) (n + 1)/((n + 6) (n + 5) (n + 4))} "A387313" LREtools/SearchTable: "SearchTable successful" 2 {((12 n + 11) (12 n + 7) hypergeom([-1/2, -n - 1], [1], -8) + (-144 n - 288 n - 153) hypergeom([-1/2, -n], [1], -8)) (n + 1)} "A387314" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 {((1296 n + 4374 n + 5098 n + 1495) hypergeom([-1/2, -n - 1], [1], -8) + (-1296 n - 5022 n - 7002 n - 3195) hypergeom([-1/2, -n], [1], -8)) (n + 1)} "A387315" LREtools/SearchTable: "SearchTable successful" 2 2 {((468 n + 780 n + 277) hypergeom([-1/2, -n - 1], [1], -12) + (-468 n - 1014 n - 559) hypergeom([-1/2, -n], [1], -12)) (n + 1)} "A387316" LREtools/SearchTable: "SearchTable successful" 3 2 {((18252 n + 68445 n + 80835 n + 24748) hypergeom([-1/2, -n - 1], [1], -12) 3 2 + (-18252 n - 77571 n - 110877 n - 51220) hypergeom([-1/2, -n], [1], -12)) (n + 1)} "A387337" LREtools/SearchTable: "SearchTable successful" (n + 1) ((24 n + 33) LegendreP(n + 1, 3) + (-4 n - 3) LegendreP(n, 3)) (n + 1) ((24 n + 33) LegendreQ(n + 1, 3) + (-4 n - 3) LegendreQ(n, 3)) {----------------------------------------------------------------------, ----------------------------------------------------------------------} (n + 4) (n + 3) (n + 4) (n + 3) "A387338" LREtools/SearchTable: "SearchTable successful" 2 2 (n + 1) ((99 n + 369 n + 333) LegendreP(n + 1, 3) + (-17 n - 55 n - 39) LegendreP(n, 3)) {------------------------------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) 2 2 (n + 1) ((99 n + 369 n + 333) LegendreQ(n + 1, 3) + (-17 n - 55 n - 39) LegendreQ(n, 3)) ------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A387339" LREtools/SearchTable: "SearchTable successful" n n 2 (n + 1) ((12 n + 14) LegendreP(n + 1, 2) + (-3 n - 1) LegendreP(n, 2)) 2 (n + 1) ((12 n + 14) LegendreQ(n + 1, 2) + (-3 n - 1) LegendreQ(n, 2)) {-------------------------------------------------------------------------, ------------------------------------------------------------------------- (n + 4) (n + 3) (n + 4) (n + 3) } "A387340" LREtools/SearchTable: "SearchTable successful" n 2 2 2 (n + 1) ((78 n + 258 n + 226) LegendreP(n + 1, 2) + (-21 n - 60 n - 47) LegendreP(n, 2)) {---------------------------------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) n 2 2 2 (n + 1) ((78 n + 258 n + 226) LegendreQ(n + 1, 2) + (-21 n - 60 n - 47) LegendreQ(n, 2)) ---------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A387341" LREtools/SearchTable: "SearchTable successful" {(n + 1) ((24 n + 33) LegendreP(n + 1, 3) + (-4 n - 3) LegendreP(n, 3)), (n + 1) ((24 n + 33) LegendreQ(n + 1, 3) + (-4 n - 3) LegendreQ(n, 3))} "A387342" LREtools/SearchTable: "SearchTable successful" 2 2 {(n + 1) ((99 n + 369 n + 333) LegendreP(n + 1, 3) + (-17 n - 55 n - 39) LegendreP(n, 3)), 2 2 (n + 1) ((99 n + 369 n + 333) LegendreQ(n + 1, 3) + (-17 n - 55 n - 39) LegendreQ(n, 3))} "A387343" LREtools/SearchTable: "SearchTable successful" n n {2 (n + 1) ((12 n + 14) LegendreP(n + 1, 2) + (-3 n - 1) LegendreP(n, 2)), 2 (n + 1) ((12 n + 14) LegendreQ(n + 1, 2) + (-3 n - 1) LegendreQ(n, 2)) } "A387344" LREtools/SearchTable: "SearchTable successful" n 2 2 {2 (n + 1) ((78 n + 258 n + 226) LegendreP(n + 1, 2) + (-21 n - 60 n - 47) LegendreP(n, 2)), n 2 2 2 (n + 1) ((78 n + 258 n + 226) LegendreQ(n + 1, 2) + (-21 n - 60 n - 47) LegendreQ(n, 2))} "A387345" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387353" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387358" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387366" LREtools/SearchTable: "SearchTable successful" {(n + 1) (5 LegendreP(n + 1, 5) - LegendreP(n, 5)), (n + 1) (5 LegendreQ(n + 1, 5) - LegendreQ(n, 5))} "A387367" LREtools/SearchTable: "SearchTable successful" {(n + 1) ((120 n + 175) LegendreP(n + 1, 5) + (-12 n - 11) LegendreP(n, 5)), (n + 1) ((120 n + 175) LegendreQ(n + 1, 5) + (-12 n - 11) LegendreQ(n, 5))} "A387368" LREtools/SearchTable: "SearchTable successful" (n + 1) (5 LegendreP(n + 1, 5) - LegendreP(n, 5)) (n + 1) (5 LegendreQ(n + 1, 5) - LegendreQ(n, 5)) {-------------------------------------------------, -------------------------------------------------} n + 2 n + 2 "A387369" LREtools/SearchTable: "SearchTable successful" (n + 1) ((120 n + 175) LegendreP(n + 1, 5) + (-12 n - 11) LegendreP(n, 5)) {--------------------------------------------------------------------------, (n + 4) (n + 3) (n + 1) ((120 n + 175) LegendreQ(n + 1, 5) + (-12 n - 11) LegendreQ(n, 5)) --------------------------------------------------------------------------} (n + 4) (n + 3) "A387401" LREtools/SearchTable: "SearchTable successful" n n (-2 I) (-LegendreP(n + 1, I) I + LegendreP(n, I)) (n + 1) (-2 I) (-LegendreQ(n + 1, I) I + LegendreQ(n, I)) (n + 1) {- ----------------------------------------------------------, - ----------------------------------------------------------} n + 2 n + 2 "A387402" LREtools/SearchTable: "SearchTable successful" n n (-2 I) (-2 I LegendreP(n + 1, I) + LegendreP(n, I)) (n + 1) (n + 2) (-2 I) (-2 I LegendreQ(n + 1, I) + LegendreQ(n, I)) (n + 1) (n + 2) {- --------------------------------------------------------------------, - --------------------------------------------------------------------} (n + 4) (n + 3) (n + 4) (n + 3) "A387403" LREtools/SearchTable: "SearchTable successful" n 2 2 (-2 I) ((3 n + 15 n + 16) LegendreP(n, I) - (7 n + 37 n + 44) LegendreP(n + 1, I) I) (n + 1) {- -----------------------------------------------------------------------------------------------, (n + 6) (n + 5) (n + 4) n 2 2 (-2 I) ((3 n + 15 n + 16) LegendreQ(n, I) - (7 n + 37 n + 44) LegendreQ(n + 1, I) I) (n + 1) - -----------------------------------------------------------------------------------------------} (n + 6) (n + 5) (n + 4) "A387427" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387428" LREtools/SearchTable: "SearchTable successful" n n {(-2 I) LegendreP(n, 3 I), (-2 I) LegendreQ(n, 3 I)} "A387432" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387452" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387476" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387477" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387478" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387479" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387480" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387481" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387482" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387483" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387484" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387485" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387496" memory used=299461.9MB, alloc=3735.5MB, time=2276.93 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387497" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387507" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387508" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387509" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387510" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387511" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387512" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387513" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387514" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387515" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387516" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A387599" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387666" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387667" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387668" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A387669" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387734" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387756" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /3125\n1 3 2 \ |----- |----| GAMMA(n1 + 3/5) GAMMA(n1 + 4/5) GAMMA(n1 + 7/5) GAMMA(n1 + 6/5) (2094 n1 + 5473 n1 + 4556 n1 + 1215)| /-1\n /-1\n | \ \256 / | {|--| , |--| | ) ---------------------------------------------------------------------------------------------------------------|} \16/ \16/ | / /-1\(n1 + 1) | |----- GAMMA(n1 + 3/2) GAMMA(n1 + 2) GAMMA(n1 + 7/4) GAMMA(n1 + 9/4) |--| | \n1 = 0 \16/ / "A387913" 2 (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) {(n + 3) , ----------------------------------------------} (n + 3) (n + 2) (n + 1) "A387914" (2 n + 5) (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {----------------------------------------------, n + 7 n + 14} (n + 5) (n + 4) (n + 2) (n + 1) "A387922" (2 n + 3) (2 n + 1) binomial(2 n, n) 2 {------------------------------------, n + 4 n + 7} (n + 1) (n + 2) "A387928" LREtools/SearchTable: "SearchTable successful" 3 2 2 (2 n + 5) (n + 2) (13 n + 8) hypergeom([-n - 1, -2 n - 4], [2], 3) + (-895 n - 3285 n - 3836 n - 1392) hypergeom([-n, -2 n - 2], [2], 3) {--------------------------------------------------------------------------------------------------------------------------------------------} (11 n + 17) (2 n + 1) "A387929" LREtools/SearchTable: "SearchTable successful" n {3 hypergeom([-2 n, -n], [1], 1/3)} "A387930" LREtools/SearchTable: "SearchTable successful" {hypergeom([-3 n, -n], [1], 3)} "A387931" LREtools/SearchTable: "SearchTable successful" n {2 hypergeom([-n, 3 n + 1], [1], -1/2)} "A387932" LREtools/SearchTable: "SearchTable successful" n 2 2 {2 (128 (4 n + 5) (4 n + 1) (2 n + 1) (4 n + 3) hypergeom([-n - 1, 4 n + 5], [3 n + 5], -1/2) 3 2 / - 3 n (3 n + 4) (3 n + 2) (601 n + 1083 n + 668 n + 144) hypergeom([-n, 4 n + 1], [3 n + 2], -1/2)) binomial(4 n, n) / ((3 n + 1) (3 n + 2) / 3 2 (3 n + 4) (731 n + 1275 n + 706 n + 120))} "A387933" LREtools/SearchTable: "SearchTable successful" n {2 hypergeom([-n, 4 n + 1], [1], -1/2)} "A387945" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/6 - 1/6 I 3 ) , (1/6 + 1/6 I 3 ) , (1/6 - 1/6 I 3 ) | ) (1/6 + 1/6 I 3 ) (1/6 - 1/6 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) 3 2 || | \ (1/6 + 1/6 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (481 n2 + 2593 n2 + 4508 n2 + 2540)|| | ) ----------------------------------------------------------------------------------------------------------||} | / (n2 + 1) (n2 + 2) (2 n2 + 5) (2 n2 + 3) (2 n2 + 1) || |----- || \n2 = 0 // "A387968" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387969" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A387970" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A387982" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 n 3 n 3 n {RootOf(_Z + _Z + 1, index = 1) , RootOf(_Z + _Z + 1, index = 2) , RootOf(_Z + _Z + 1, index = 3) } "A387990" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 /n1 - 1 |----- |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) | \ {(1/6 - 1/18 I 3 ) , (1/6 + 1/18 I 3 ) , (1/6 - 1/18 I 3 ) | ) (1/6 + 1/18 I 3 ) (1/6 - 1/18 I 3 ) | ) | / | / |----- |----- \n1 = 0 \n2 = 0 \\ 1/2 (-n2 - 1) 4 3 2 || (1/6 + 1/18 I 3 ) (2 n2 + 1) (4 n2 + 3) (4 n2 + 1) binomial(4 n2, n2) (27111 n2 + 183762 n2 + 458877 n2 + 500930 n2 + 201880)|| -------------------------------------------------------------------------------------------------------------------------------------------||} (n2 + 2) (n2 + 1) (3 n2 + 7) (3 n2 + 4) (3 n2 + 1) (3 n2 + 5) (3 n2 + 2) || || // "A388041" n {2 n! (n - 2), n!} "A388043" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 |----- 1/2 n 1/2 n 1/2 n | \ 1/2 n1 1/2 (-n1 - 1) {(1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) , (1/2 - 1/2 I 3 ) | ) (1/2 + 1/2 I 3 ) (1/2 - 1/2 I 3 ) | / |----- \n1 = 0 /n1 - 1 \\ |----- 1/2 (-n2 - 1) || | \ (1/2 + 1/2 I 3 ) (3 n2 + 1) (3 n2 + 2) binomial(3 n2, n2) (13 n2 + 17) (7 n2 + 12)|| | ) ---------------------------------------------------------------------------------------------||} | / (n2 + 2) (2 n2 + 3) (n2 + 1) (2 n2 + 1) || |----- || \n2 = 0 // "A388045" LREtools/SearchTable: "SearchTable successful" 3 (n + 1) (3 n + 4) (3 n + 2) hypergeom([-n, -3 n - 3], [2], 2) + (19 n + 14) (3 n + 1) n hypergeom([-3 n, -n + 1], [2], 2) {---------------------------------------------------------------------------------------------------------------------------} 2 35 n + 42 n + 12 "A388046" LREtools/SearchTable: "SearchTable successful" 2 2 {(16 (3 n + 2) (4 n + 7) (4 n + 3) (4 n + 5) (2 n + 3) hypergeom([-n - 1, 4 n + 8], [3 n + 6], -1) 5 4 3 2 - 3 (3964 n + 22396 n + 49248 n + 52538 n + 27119 n + 5412) (3 n + 5) (3 n + 4) hypergeom([-n, 4 n + 4], [3 n + 3], -1)) binomial(4 n, n) / 3 2 (4 n + 1) / ((n + 1) (3 n + 1) (3 n + 2) (3 n + 4) (3 n + 5) (170 n + 720 n + 1009 n + 468))} / "A388047" LREtools/SearchTable: "SearchTable successful" n n {2 (LegendreP(n + 1, 2) - LegendreP(n, 2)), 2 (LegendreQ(n + 1, 2) - LegendreQ(n, 2))} "A388048" LREtools/SearchTable: "SearchTable successful" 2 10 (2 n + 5) (n + 2) hypergeom([-n - 1, -2 n - 4], [2], 3) + (-263 n - 773 n - 564) hypergeom([-n, -2 n - 2], [2], 3) {----------------------------------------------------------------------------------------------------------------------} 11 n + 17 "A388049" LREtools/SearchTable: "SearchTable successful" -4 (9 n + 7) (3 n + 1) hypergeom([-3 n, -n], [1], 3) + 3 (n + 1) (3 n + 2) hypergeom([-n - 1, -3 n - 3], [1], 3) {----------------------------------------------------------------------------------------------------------------} 2 (3 n + 2) "A388050" LREtools/SearchTable: "SearchTable successful" n 2 2 {2 (64 (4 n + 5) (2 n + 1) (4 n + 3) hypergeom([-n - 1, 4 n + 5], [3 n + 5], -1/2) 2 / + 3 (3 n + 2) (199 n + 293 n + 108) (3 n + 4) n hypergeom([-n, 4 n + 1], [3 n + 2], -1/2)) binomial(4 n, n) (4 n + 1) / ((3 n + 1) (3 n + 2) / 3 2 (3 n + 4) (731 n + 1275 n + 706 n + 120))} "A388060" LREtools/SearchTable: "SearchTable successful" {(4 n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-4 n - 3) hypergeom([-1/2, -n], [1], -8)} "A388061" LREtools/SearchTable: "SearchTable successful" n n {4 (LegendreP(n + 1, 3/2) - LegendreP(n, 3/2)), 4 (LegendreQ(n + 1, 3/2) - LegendreQ(n, 3/2))} "A388129" LREtools/SearchTable: "SearchTable successful" (2 n + 2) hypergeom([2 n + 3, -n - 1], [1], -1) + (6 n + 3) hypergeom([-n, 2 n + 1], [1], -1) {---------------------------------------------------------------------------------------------} 17 n + 11 "A388130" LREtools/SearchTable: "SearchTable successful" 6 (n + 1) (3 n + 2) hypergeom([-n - 1, 3 n + 4], [1], -1) + (3 n + 1) (31 n + 23) hypergeom([-n, 3 n + 1], [1], -1) {-------------------------------------------------------------------------------------------------------------------} 2 238 n + 289 n + 83 "A388131" LREtools/SearchTable: "SearchTable successful" 2 32 (2 n + 1) (4 n + 3) (n + 1) hypergeom([-n - 1, 4 n + 5], [1], -1) + (477 n + 659 n + 222) (4 n + 1) hypergeom([-n, 4 n + 1], [1], -1) {-----------------------------------------------------------------------------------------------------------------------------------------} 3 2 4633 n + 8136 n + 4541 n + 798 "A388133" LREtools/SearchTable: "SearchTable successful" n 3 ((2 n + 1) hypergeom([-2 n, -n], [1], 1/3) + (n + 1) hypergeom([-2 n - 2, -n - 1], [1], 1/3)) {------------------------------------------------------------------------------------------------} 25 n + 16 "A388134" LREtools/SearchTable: "SearchTable successful" n 2 (2 (3 n + 1) (23 n + 17) hypergeom([-n, 3 n + 1], [1], -1/2) + 9 (n + 1) (3 n + 2) hypergeom([-n - 1, 3 n + 4], [1], -1/2)) {------------------------------------------------------------------------------------------------------------------------------} 2 65 n + 78 n + 22 "A388135" LREtools/SearchTable: "SearchTable successful" n 2 {2 ((1053 n + 1447 n + 484) (4 n + 1) hypergeom([-n, 4 n + 1], [1], -1/2) + 72 (2 n + 1) (4 n + 3) (n + 1) hypergeom([-n - 1, 4 n + 5], [1], -1/2)) / 3 2 / (3021 n + 5247 n + 2884 n + 496)} / "A388202" LREtools/SearchTable: "SearchTable successful" n n 2 (n LegendreP(n + 1, 2) + LegendreP(n, 2)) 2 (n LegendreQ(n + 1, 2) + LegendreQ(n, 2)) {--------------------------------------------, --------------------------------------------} n + 2 n + 2 "A388203" LREtools/SearchTable: "SearchTable successful" n 2 2 2 ((3 n + 3 n + 4) LegendreP(n + 1, 2) + (-n - 2 n - 5) LegendreP(n, 2)) {---------------------------------------------------------------------------, (n + 2) (n + 3) n 2 2 2 ((3 n + 3 n + 4) LegendreQ(n + 1, 2) + (-n - 2 n - 5) LegendreQ(n, 2)) ---------------------------------------------------------------------------} (n + 2) (n + 3) "A388204" LREtools/SearchTable: "SearchTable successful" n 3 2 3 2 ((4 n + 6 n + 10 n - 4) LegendreP(n + 1, 2) + (-n + 3 n + 14) LegendreP(n, 2)) {------------------------------------------------------------------------------------, (n + 4) (n + 3) (n + 2) n 3 2 3 2 ((4 n + 6 n + 10 n - 4) LegendreQ(n + 1, 2) + (-n + 3 n + 14) LegendreQ(n, 2)) ------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A388205" LREtools/SearchTable: "SearchTable successful" 2 2 (4 n + 2 n + 1) hypergeom([-1/2, -n - 1], [1], -8) - (2 n + 1) hypergeom([-1/2, -n], [1], -8) {-----------------------------------------------------------------------------------------------} n + 2 "A388206" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (12 n + 9 n + 16 n + 1) hypergeom([-1/2, -n - 1], [1], -8) + (-12 n - 15 n - 18 n - 9) hypergeom([-1/2, -n], [1], -8) {-------------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A388207" LREtools/SearchTable: "SearchTable successful" 4 3 2 4 3 2 (18 n + 18 n + 67 n - 3 n + 11) hypergeom([-1/2, -n - 1], [1], -8) + (-18 n - 27 n - 72 n - 27 n + 9) hypergeom([-1/2, -n], [1], -8) {------------------------------------------------------------------------------------------------------------------------------------------} (n + 4) (n + 3) (n + 2) "A388409" LREtools/SearchTable: "SearchTable successful" n n 2 ((3 n + 4) LegendreP(n + 1, 2) + LegendreP(n, 2)) 2 ((3 n + 4) LegendreQ(n + 1, 2) + LegendreQ(n, 2)) {----------------------------------------------------, ----------------------------------------------------} n + 2 n + 2 "A388410" LREtools/SearchTable: "SearchTable successful" n 2 2 2 ((15 n + 51 n + 40) LegendreP(n + 1, 2) + (-3 n - 6 n + 1) LegendreP(n, 2)) {--------------------------------------------------------------------------------, (n + 2) (n + 3) n 2 2 2 ((15 n + 51 n + 40) LegendreQ(n + 1, 2) + (-3 n - 6 n + 1) LegendreQ(n, 2)) --------------------------------------------------------------------------------} (n + 2) (n + 3) "A388411" LREtools/SearchTable: "SearchTable successful" n n 2 ((12 n + 14) LegendreP(n + 1, 2) + (-3 n - 1) LegendreP(n, 2)) 2 ((12 n + 14) LegendreQ(n + 1, 2) + (-3 n - 1) LegendreQ(n, 2)) {-----------------------------------------------------------------, -----------------------------------------------------------------} n + 4 n + 4 "A388412" LREtools/SearchTable: "SearchTable successful" 2 2 (16 n + 32 n + 13) hypergeom([-1/2, -n - 1], [1], -8) - (4 n + 5) hypergeom([-1/2, -n], [1], -8) {--------------------------------------------------------------------------------------------------} n + 2 "A388413" LREtools/SearchTable: "SearchTable successful" 3 2 3 2 (48 n + 198 n + 236 n + 77) hypergeom([-1/2, -n - 1], [1], -8) + (-48 n - 222 n - 324 n - 153) hypergeom([-1/2, -n], [1], -8) {---------------------------------------------------------------------------------------------------------------------------------} (n + 2) (n + 3) "A388414" LREtools/SearchTable: "SearchTable successful" 4 3 2 {((288 n + 2016 n + 4744 n + 4320 n + 1223) hypergeom([-1/2, -n - 1], [1], -8) 4 3 2 + (-288 n - 2160 n - 5688 n - 6264 n - 2475) hypergeom([-1/2, -n], [1], -8))/((n + 4) (n + 3) (n + 2))} "A388530" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388531" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388532" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388533" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388534" memory used=301119.7MB, alloc=3735.5MB, time=2288.46 LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388535" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388536" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A388537" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A388538" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A388725" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388726" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388727" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388730" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388731" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388734" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A388847" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A388857" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" {} "A388911" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388912" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388913" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388914" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388915" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A388916" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389060" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389061" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389062" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389063" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389064" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389125" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389126" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389127" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389128" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389129" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389130" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389131" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389132" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389151" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389153" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389155" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389157" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389196" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389225" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389245" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389246" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389247" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389249" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A389250" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389251" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389253" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389273" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389284" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 8" {} "A389285" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A389289" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389290" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389291" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389292" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389293" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389294" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389295" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389311" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" /n - 1 \ |----- n1 | | \ (2 n1 + 1) (-1) binomial(2 n1, n1) | n! binomial(2 n, n) (n + 1) | ) ------------------------------------------------------| | / (n1 + 1) (n1 + 1)! binomial(2 n1 + 2, n1 + 1) (n1 + 2)| |----- | n! binomial(2 n, n) (n + 1) \n1 = 0 / {---------------------------, -------------------------------------------------------------------------------------------} n n "A389322" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389324" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389325" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389326" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389327" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot reduce the operator to order two" {1} "A389328" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389329" memory used=302752.4MB, alloc=3767.5MB, time=2299.75 LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389345" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389346" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389347" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389348" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389349" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389350" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389351" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389361" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" / / |n - 1 | / 1/2\n / 1/2\n / 1/2\n |----- | | 5 5 | | 5 5 | | 5 5 | | \ | n1 1/2 (-n1 - 1) 1/2 n1 {|11/2 - ------| , |11/2 + ------| , |11/2 - ------| | ) |-2 (-1) (-11 + 5 5 ) (11 + 5 5 ) \ 2 / \ 2 / \ 2 / | / | |----- | \n1 = 0 \ / / 1/2\(-n2 - 1) \\\ |n1 - 1 | 5 5 | 5 4 3 2 ||| |----- |11/2 + ------| binomial(4 n2, n2) (2245 n2 + 13137 n2 + 25264 n2 + 20586 n2 + 7204 n2 + 864)||| | \ \ 2 / ||| | ) ----------------------------------------------------------------------------------------------------------|||} | / (n2 + 2) (n2 + 1) (3 n2 + 4) (3 n2 + 1) (3 n2 + 2) ||| |----- ||| \n2 = 0 /// "A389375" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389376" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389377" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389402" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389403" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389404" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389405" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389406" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389410" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389411" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389423" LREtools/SolveLRE: "Input is an LCLM of operators of orders" 2, 2 LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { 0 n::even { 0 n::even { { 1/2 n binomial(n, n/2) n::even {{ , { (2 n - 2) , { , { (n - 1) { 2 { 0 n::odd { 2 binomial(n - 1, n/2 - 1/2) n::odd { -------------------------- n::odd { binomial(n - 1, n/2 - 1/2) { n { 2 8 { ------------------ n::even} { n binomial(n, n/2) { { 0 n::odd "A389435" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389436" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389437" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389438" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" {} "A389439" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389440" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389444" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 6" {} "A389471" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 1" LREtools/SolveLRE: "Applying recursion to the left-factor" n - 1 ----- \ (4 n1 + 5) (4 n1 + 1) (2 n1 + 3) (2 n1 + 1) (4 n1 + 3) binomial(4 n1, n1) {1, ) -------------------------------------------------------------------------} / (n1 + 1) (3 n1 + 4) (3 n1 + 1) (3 n1 + 5) (3 n1 + 2) ----- n1 = 0 "A389563" LREtools/SolveLRE: "Absolute Factorization reduced the order from 2 to 1 (Liouvillian solutions)" { /46656\(n/2) { |-----| GAMMA(n/2 + 1/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 4/3) GAMMA(n/2 + 5/6) GAMMA(n/2 + 7/6) (5 n + 4) (5 n + 6) { \3125 / { --------------------------------------------------------------------------------------------------------------------- n::even { GAMMA(n/2 + 1) GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) GAMMA(n/2 + 8/5) (3 n + 1) (3 n + 2) { {{ /46656\(n/2 - 1/2) , { 12 |-----| GAMMA(n/2 + 5/6) GAMMA(1/6 + n/2) GAMMA(n/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 1/3) { \3125 / { ---------------------------------------------------------------------------------------------------- n::odd { 11 { GAMMA(n/2 + 1/2) GAMMA(n/2 + 3/10) GAMMA(n/2 + --) GAMMA(n/2 + 9/10) GAMMA(n/2 + 7/10) { 10 { /46656\(n/2) { 12 |-----| GAMMA(n/2) GAMMA(n/2 + 1/3) GAMMA(1/6 + n/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 5/6) { \3125 / { ---------------------------------------------------------------------------------------------- n::even { 11 { GAMMA(n/2 + 9/10) GAMMA(n/2 + --) GAMMA(n/2 + 1/2) GAMMA(n/2 + 7/10) GAMMA(n/2 + 3/10) { 10 } { { /46656\(n/2 + 1/2) { |-----| GAMMA(n/2 + 1/2) GAMMA(n/2 + 2/3) GAMMA(n/2 + 4/3) GAMMA(n/2 + 5/6) GAMMA(n/2 + 7/6) (5 n + 4) (5 n + 6) { \3125 / { --------------------------------------------------------------------------------------------------------------------------- n::odd { GAMMA(n/2 + 1) GAMMA(n/2 + 4/5) GAMMA(n/2 + 6/5) GAMMA(n/2 + 7/5) GAMMA(n/2 + 8/5) (3 n + 1) (3 n + 2) "A389602" LREtools/SolveLRE: "Absolute Factorization reduced the order from 4 to 2" LREtools/SearchTable: "SearchTable successful" LREtools/SolveLRE: "Solutions may be linearly dependent" memory used=304333.9MB, alloc=3767.5MB, time=2310.33 { / / 9 n \ \ { 6 |(n/2 + 4) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) + |- --- - 27| hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)| {{ \ \ 2 / / { ---------------------------------------------------------------------------------------------------------------------------------------- , n::even { 2 { (n + 5) - 6 ( 2 (1/2 n + 11/2 n + 18) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) - 9/2 (n + 5) (n + 4) hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) { { , { /(n (n + 2) (n + 6)) , n::odd { - 24 ( { { 2 (1/2 n + 11/2 n + 18) hypergeom([1/2, - n/2 - 2, - n/2 - 2], [1, 1], 4) - 9/2 (n + 5) (n + 4) hypergeom([1/2, - n/2 - 1, - n/2 - 1], [1, 1], 4)) /(n (n + 2) (n + 6)) , n::even / / 9 n \ \ 24 |(n/2 + 4) hypergeom([1/2, - n/2 - 3/2, - n/2 - 3/2], [1, 1], 4) + |- --- - 27| hypergeom([1/2, - n/2 - 1/2, - n/2 - 1/2], [1, 1], 4)| \ \ 2 / / ----------------------------------------------------------------------------------------------------------------------------------------- , 2 (n + 5) } n::odd "A389628" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389629" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389630" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389631" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389632" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389658" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389659" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389660" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389661" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389662" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389691" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389692" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389693" LREtools/SearchTable: "SearchTable successful" {(21 (3 n + 4) (3 n + 2) hypergeom([n + 2, -n - 1], [-3 n - 4], -1) - 2 (19 n + 26) (4 n + 3) hypergeom([-n, n + 1], [-3 n - 1], -1)) binomial(3 n, n) (3 n + 1)/((n + 1) (n + 2) (2 n + 1) (17 n + 24))} "A389694" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} "A389697" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" 3 2 n 3 2 n 3 2 n {RootOf(_Z - 4 _Z + 3 _Z - 1, index = 1) , RootOf(_Z - 4 _Z + 3 _Z - 1, index = 2) , RootOf(_Z - 4 _Z + 3 _Z - 1, index = 3) } "A389871" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n n {(-I) (n + 3), I (n + 3)} "A389872" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n n {(-I) (n + 5) (n + 3), I (n + 5) (n + 3)} "A389873" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n 1/2 n 1/2 n {(-1) (n + 4), (1/2 - 1/2 I 3 ) (n + 4), (1/2 + 1/2 I 3 ) (n + 4)} "A389890" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" n 1/2 n 1/2 n {(-1) , (1/2 - 1/2 I 3 ) , (1/2 + 1/2 I 3 ) } "A389893" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {(-1) (n + 3), n + 3} "A389894" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" 1/2 n 1/2 n {(- 1/2 - 1/2 I 3 ) (n + 4), (- 1/2 + 1/2 I 3 ) (n + 4), n + 4} "A389942" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 2" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 5" n {1, (-1) } "A389943" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 9" {} "A389944" LREtools/SolveLRE: "Applying recursion to the largest right-factor, which has order 3" LREtools/SolveLRE: "Applying recursion to the left-factor" LREtools/SolveLRE: "Cannot solve an absolutely irreducible operator of order 7" 1/2 n 1/2 n {1, (- 1/2 - 1/2 I 3 ) , (- 1/2 + 1/2 I 3 ) } "A389992" LREtools/SolveLRE: "Cannot reduce the operator to order two" {} > quit memory used=305350.7MB, alloc=3767.5MB, time=2314.03