#----------------------------------------------------------------------------# # Case 1 # StepSet [[1, 0], [0, 1], [0, -1], [-1, 0]] # A005566 sol[1] := 1+2*t+6*t^2+18*t^3+60*t^4+200*t^5+700*t^6+2450*t^7+8820*t^8+31752*t^9+116424*t^10; SOL_CASE[1] := (16*t^2-1)*hypergeom([3/2, 3/2],[2],16*t^2)+(1/2/t+2)*hypergeom([1/2, 1/2],[1],16*t^2)-1/2/t; if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 2 # StepSet [[-1, 1], [-1, 0], [1, 0], [1, -1]] # A005558 sol[2] := 1+t+3*t^2+6*t^3+20*t^4+50*t^5+175*t^6+490*t^7+1764*t^8+5292*t^9+19404*t^10; SOL_CASE[2] := 1/(4*t^2)*((16*t^2-1)*(hypergeom([1/2, 1/2],[1],16*t^2)+2*t*(4*t-1)*hypergeom([3/2, 3/2],[2],16*t^2)) -2*t+1); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 3 # StepSet [[1, 1], [-1, 1], [1, -1], [-1, -1]] # A018224 sol[3] := 1+t+4*t^2+9*t^3+36*t^4+100*t^5+400*t^6+1225*t^7+4900*t^8+15876*t^9+63504*t^10; SOL_CASE[3] := (1 + 1/4/t)*hypergeom([1/2, 1/2],[1],16*t^2)-1/4/t; if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 4 # StepSet [[1, 1], [0, 1], [-1, 1], [-1, 0], [-1, -1], [0, -1], [1, -1], [1, 0]] # A151331 sol[4] := 1+3*t+18*t^2+105*t^3+684*t^4+4550*t^5+31340*t^6+219555*t^7+1564080*t^8+11271876*t^9+82059768*t^10; SOL_CASE[4] := 1/t*Int(-(16*t^2+24*t-1)/(1+4*t)^5*hypergeom([5/4, 5/4],[2],-2*t/(t+1/4)^4*(t+1)*(t-1/8)),t); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 5 # StepSet [[1, 0], [1, -1], [1, 1], [-1, 0], [-1, 1], [-1, -1]] A151312 sol[5] := 1+2*t+10*t^2+39*t^3+210*t^4+960*t^5+5340*t^6+26250*t^7+148610*t^8+761796*t^9+4360356*t^10; SOL_CASE[5] := 1/t*Int(-(-1+2*t)/(1-6*t)^(7/6)/(6*t+1)^(1/6)*hypergeom([1/6, 1/6],[1], -432*t^4*(-1+2*t)*(2*t+1)/(6*t-1)/(6*t+1))-12*t^3*(-1+12*t^2)*(-1+2*t)*(2*t+1)/(6*t+1) ^(7/6)/(1-6*t)^(13/6)*hypergeom([7/6, 7/6],[2],-432*t^4*(-1+2*t)*(2*t+1)/(6*t-1)/(6*t+1)),t); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 6 # StepSet [[1, 0], [-1, 1], [-1, 0], [-1, -1]] # A151261 sol[6] := 1+t+3*t^2+5*t^3+17*t^4+34*t^5+121*t^6+265*t^7+969*t^8+2246*t^9+8351*t^10; SOL_CASE[6] := 1/4*(-2*t+1)/t^2*(1-1/(1-4*t)^(1/2)*(1+2*Int((-(1+4*t^2)*(8*t^3-16*t^2+1)*hypergeom([1/2, 1/2],[2],16/(1+4*t^2)*t^2)-2*t^2*(24*t^3-36*t ^2+2*t+1)*hypergeom([3/2, 3/2],[3],16/(1+4*t^2)*t^2))/(1-4*t)^(1/2)/(-1+2*t)^2/(1+4*t^2)^(3/2),t))); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 7 # StepSet [[1, 0], [-1, 1], [-1, -1]] # A151255 sol[7] := 1+t+2*t^2+3*t^3+8*t^4+15*t^5+39*t^6+77*t^7+216*t^8+459*t^9+1265*t^10; SOL_CASE[7] := (1-2*t)/4/t^2 * (1 - ((t+1)/(1-3*t))^(1/2) * (1 -2 *Int( ((8*t^2+1)*(4*t+1)*(2*t^2-4*t+1)*hypergeom([1/4, 3/4],[1],64*t^4)+12*(8*t^2+1)*t^3*(8*t^2-1)*(1-7*t+4*t^2)*hypergeom([5 /4, 7/4],[2],64*t^4))/(1-3*t)^(1/2)/(-1+2*t)^2/(t+1)^(3/2) ,t) ) ); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 8 # StepSet [[1, 0], [1, 1], [1, -1], [-1, 0]] A151291 sol[8]:= 1+2*t+7*t^2+23*t^3+84*t^4+301*t^5+1127*t^6+4186*t^7+15891*t^8+60128*t^9+230334*t^10; SOL_CASE[8] := 1/(t-1)*(-1/2+1/t*(-1/4-1/4*(-1+2*t)/(1-4*t)^(1/2)+Int(-1/2*t/(1-4*t)^(3/2)*Int((4*t*(t-1)*(1-4*t)^(1/2)*hypergeom([3/2, 3/2],[2],16/(1+4 *t^2)*t^2)+(1+4*t^2)*(4*t^2+2*t-1)*(1-4*t)^(1/2)*hypergeom([1/2, 1/2],[1],16/(1+4*t^2)*t^2))/(1+4*t^2)^(3/2)/t^2,t),t))); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 9 # StepSet, [[1, 1], [1, -1], [-1, 0]] # A151266 sol[9] := 1+t+3*t^2+7*t^3+19*t^4+49*t^5+139*t^6+379*t^7+1079*t^8+3011*t^9+8681*t^10; SOL_CASE[9] := -1/(t-1)*(1/2+1/t*Int(t^2/(t+1)^(1/2)/(1-3*t)^(3/2)*(13/2+Int(((1-3*t)/(t+1))^(1/2)*((10-1/t^3)*hypergeom([1/4, 3/4],[1],64*t^4)+6*(8*t^2+1)*(10*t^2-3)*hypergeom([5/4, 7/4],[2],64*t^4)),t)),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 10 # StepSet, [[1, 1], [0, 1], [-1, 1], [-1, 0], [1, 0], [0, -1]] # A151326 sol[10] := 1+3*t+15*t^2+74*t^3+392*t^4+2116*t^5+11652*t^6+64967*t^7+365759*t^8+2074574*t^9+11836868*t^10; SOL_CASE[10] := 1/4-3/8/t-3/8*(2-1/t)*((2*t+1)/(1-6*t))^(1/2)-6/t*Int(Int((6*t+1)/(1-6*t)^(5/2)/(2*t+1)^(3/2)*Int(((1-6*t)/(1-8*t^2))^(3/2) *(2*t+1)^(1/2)*((32*t^3-32*t^2-42*t-5)*hypergeom([1/4,3/4],[1],64/(8*t^2-1)^2*(2*t+1)*t^3)+(2+14*t-128*t^2 -832*t^3-1936*t^4-1600*t^5+1152*t^6+2048*t^7)/(1-8*t^2)^2*hypergeom([5/4, 7/4],[2],64/(1-8*t^2)^2*(2*t+1)*t^3))/(-1+4*t+8*t^2)/(6*t+1)^2,t),t),t); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 11 # StepSet, [[1, 0], [0, 1], [0, -1], [-1, 0], [-1, 1], [-1, -1]] # A151297 SOL_CASE[11] := (1-62*t)/t^2/128+(2*t+1)/t^2*(-1/128+Int(t/((2*t+1)*(1-6*t))^(3/2)*(-3-1/2*Int(((1-6*t)/(2*t+1)/(1-8*t^2)^3)^(1/2)*((24*t^3+4*t^2+4*t+1)/t^2*hypergeom ([1/4, 3/4],[1],64/(8*t^2-1)^2*(2*t+1)*t^3)+12*t*(8*t^2+4*t+1)*(1+14*t+16*t^2)/(1-8*t^2)^2*hypergeom([5/4, 7/4],[2],64/(8*t^2-1)^2*(2*t+1)*t^3)),t)),t)); sol[11] := 1+2*t+7*t^2+26*t^3+105*t^4+444*t^5+1944*t^6+8728*t^7+39999*t^8+186266*t^9+879108*t^10; if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 12 # StepSet, [[0, 1], [1, 0], [1, -1], [-1, 0], [-1, -1]] # A151287 sol[12] := 1+2*t+6*t^2+21*t^3+76*t^4+290*t^5+1148*t^6+4627*t^7+19038*t^8+79554*t^9+336112*t^10; SOL_CASE[12] := -1/2/t+(1+t)/t^2*Int(t^2/(1+t)^2/(1+3*t)^(1/2)/(1-5*t)^(3/2)*(-13/2+Int((1+t)*((1-5*t)/(1+3*t)/(1-2*t)^3/(1+2*t)^3)^(1/2)*((4*t^3-2*t-1)/t^3*hypergeom([1 /4, 3/4],[1],64*(1+t)*t^3/(1-2*t)^2/(1+2*t)^2)+6*(-3-2*t+4*t^2)*(12*t^2+4*t+1)/(1+2*t)^2/(1-2*t)^2*hypergeom([5/4, 7/4],[2],64*(1+t)*t^3/(1-2*t)^2/(1+2*t)^2)),t)),t); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 13 # StepSet, [[1, 1], [1, 0], [-1, 0], [-1, 1], [0, -1]] # A151307 sol[13] := 1+2*t+9*t^2+34*t^3+151*t^4+659*t^5+2999*t^6+13714*t^7+63799*t^8+298397*t^9+1408415*t^10; SOL_CASE[13] := 1/(t^2-t)*(-t+Int(Int(t/(1-5*t)^(5/2)/(1+3*t)^(3/2)*(-26-2*Int((1-5*t)^(3/2)*((1+3*t)/(1-4*t^2))^(1/2)*((24*t^4+32*t^3+t^2+12*t+1)* (1-4*t^2)^2*hypergeom([1/4, 3/4],[1],64*(1+t)*t^3/(1-4*t^2)^2)-3*t*(3-10*t-63*t^2-212*t^3-220*t^4-464*t^5-288*t^6+64*t^7)*hypergeom ([5/4, 7/4],[2],64*(1+t)*t^3/(1-4*t^2)^2))/(1+t)/(4*t^2+4*t-1)/(1-4*t^2)^3/t^2,t)),t),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 14 # StepSet, [[1, 1], [-1, 1], [1, -1], [-1, -1], [-1, 0]] # A151275 sol[14] := 1+t+5*t^2+13*t^3+61*t^4+199*t^5+939*t^6+3389*t^7+16129*t^8+61601*t^9+295373*t^10; SOL_CASE[14] := 1/(t^2-t)*(-t+Int(Int(6*(t+1)/(3*t+1)^(3/2)/(1-5*t)^(5/2)*Int(((3*t+1)/(16*t^2+1))^(1/2)*(1-5*t)^(3/2)*( (312*t^6-27*t^5+26*t^4-66*t^3+32*t^2+t-2)*(16*t^2+1)^2*hypergeom([1/4,3/4],[1],64/(16*t^2+1)^2*t^2*(t^2+1)) +(2+3*t-92*t^2-84*t^3+218*t^4-3909*t^5-5976*t^6+30666*t^7+73776*t^8+25488*t^9+8064*t^10)*hypergeom([5/4,7/4],[2],64/(16*t^2+1)^2*t^2*(t^2+1)) )/t/(16*t^2+1)^3/(t^2+1)/(24*t^2-1)/(t+1)^2,t),t),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 15 # StepSet, [[1, 0], [1, -1], [1, 1], [-1, 0], [-1, 1], [-1, -1], [0, 1]] # A151329 sol[15] := 1+3*t+16*t^2+86*t^3+509*t^4+3065*t^5+19088*t^6+120401*t^7+771758*t^8+4991255*t^9+32580974*t^10; SOL_CASE[15] := 1/(2*t^2+t)*(t+Int(Int((2*t+1)/(1-7*t)^(5/2)/(5*t+1)^(3/2)*(10-6*Int((1/(1+12*t^2)*(5*t+1))^(1/2)*(1-7*t)^(3/2)*( (2+3*t-86*t^2-698*t^3-4440*t^4-14752*t^5-10608*t^6+33312*t^7+33792*t^8-39168*t^9-42240*t^10)*hypergeom([5/4, 7/4],[2],64*(t^2+t+1)*t^2/(1+12*t^2)^2) +(800*t^6+1120*t^5+568*t^4-60*t^3-40*t^2-t-2)*(1+12*t^2)^2*hypergeom([1/4,3/4],[1],64*(t^2+t+1)*t^2/(1+12*t^2)^2) )/t/(1+12*t^2)^3/(t^2+t+1)/(20*t^2+4*t-1)/(2*t+1)^2,t)),t),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 16 # StepSet, [[1, 1], [-1, 1], [1, -1], [-1, -1], [0, 1]] # A151302 sol[16] := 1+2*t+8*t^2+29*t^3+129*t^4+535*t^5+2467*t^6+10844*t^7+50982*t^8+231404*t^9+1101030*t^10; SOL_CASE[16] := 1/(t^2+t)*(t+Int(Int(6*(t+1)/(3*t+1)^(3/2)/(1-5*t)^(5/2)*(1-Int((1-5*t)^(3/2)*((3*t+1)/(16*t^2+1))^(1/2)*( 2*(216*t^6+153*t^5+244*t^4-8*t^3+7*t^2-t-1)*(16*t^2+1)^2*hypergeom([1/4, 3/4],[1],64/(16*t^2+1)^2*t^2*(t^2+1)) +(2+3*t-84*t^2-182*t^3-382*t^4-1885*t^5+10080*t^6+28348*t^7-19360*t^8-41952*t^9-20736*t^10)*hypergeom([5/4, 7/4],[2],64/(16*t^2+1)^2*t^2*(t^2+1)) )/t/(16*t^2+1)^3/(t^2+1)/(24*t^2-1)/(t+1)^2,t)),t),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 17 # StepSet, [[1, 1], [0, 1], [-1, 1], [-1, 0], [-1, -1], [0, -1], [1, -1]] # A151314 sol[17] := 1+2*t+11*t^2+49*t^3+277*t^4+1479*t^5+8679*t^6+49974*t^7+301169*t^8+1805861*t^9+11097563*t^10; SOL_CASE[17] := 1/(t-t^2)*(t+Int(Int((2*t+1)/(1-7*t)^(5/2)/(5*t+1)^(3/2)*(2+6*Int((1-7*t)^(3/2)*(1/(1+12*t^2)*(5*t+1))^(1/2)*( (-2-4*t+84*t^2+594*t^3+3402*t^4+18783*t^5+64868*t^6+94192*t^7+5136*t^8-73776*t^9-36480*t^10)*hypergeom([5/4, 7/4],[2],64*(t^2+t+1)*t^2/(1+12*t^2)^2) +(20*t^6+704*t^5+541*t^4+176*t^3+7*t^2-t+2)*(1+12*t^2)^2*hypergeom([1/4, 3/4],[1],64*(t^2+t+1)*t^2/(1+12*t^2)^2) )/t/(1+12*t^2)^3/(t^2+t+1)/(20*t^2+4*t-1)/(2*t+1)^2,t)),t),t)); if convert(series(%%-%,t=0,11),polynom)<>0 then error fi; #----------------------------------------------------------------------------# # Case 18 # StepSet, [[0, 1], [-1, 0], [1, -1]] # A005789, A151334 (same except for offset) SOL_CASE[18] := subs(x=t, 1/30*(1/x-27)*(9*hypergeom([1/3, 2/3],[1],27*x)+(216*x+1)*hypergeom([4/3, 5/3],[2],27*x))-1/3/x ); #----------------------------------------------------------------------------# # Case 19 # StepSet, [[1, 0], [0, 1], [-1, 0], [0, -1], [1, -1], [-1, 1]] # A151366 SOL_CASE[19] := subs(x=t, 11/27*(3*x+1)/x^3-5/27/x^3*(2+3*x)^2-1/6*(2*x+1)*((6*x-1)*(18*x^2+3*x+2)*(3*x+1)^2*hypergeom([1/3, 1/3],[1],27/(3*x+1)^2/(6*x-1)*x^2*(2*x+1))+x*(1+24*x+36*x^2)*hypergeom([4/3, 4/3],[2],27/(3*x+1)^2/(6*x-1)*x^2*(2*x+1)))/(1-6*x)^(4/3)/(3*x+1)^(8/3)/x^3 );