OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - Sum_{n>=0} x^(2*n+1) / (2*n+1)!^2).
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2).
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^2 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^2, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 17 2022
STATUS
approved