OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / ((2*n + 1)!)^2 ).
a(0) = 1; a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * (2*k+1) * a(n-2*k-1).
MATHEMATICA
nmax = 20; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[x]] - BesselJ[0, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, 2 k + 1]^2 (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 11 2021
STATUS
approved