OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = exp( Sum_{n>=1} x^(2*n) / (2*n)!^2 ).
Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^2 = exp( (BesselI(0,2*sqrt(x)) + BesselJ(0,2*sqrt(x))) / 2 - 1 ).
MATHEMATICA
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^2 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}]
nmax = 26; Take[CoefficientList[Series[Exp[Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 17 2022
STATUS
approved