ClearAll;
$DefaultFont = {"Times-Roman",10};
L[0,x_] = 1; L[1,x_] = 1 - x; L[n_,x_]:= Simplify[(2n-1-x)L[n-1,x] - (n-1)^2 L[n-2,x]];
L[4,x]
L[6,x]