OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..245
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = -log((3 - BesselI(0,2*sqrt(2*x))) / 2).
MATHEMATICA
a[0] = 0; a[n_] := a[n] = 2^(n - 1) + (1/n) Sum[Binomial[n, k]^2 2^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(3 - BesselI[0, 2 Sqrt[2 x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
PROG
(SageMath)
@CachedFunction
def a(n): return 0 if (n==0) else 2^(n-1) + (1/n)*sum(binomial(n, k)^2 *2^(k-1)*(n-k)*a(n-k) for k in (1..n-1)) # a= A333981
[a(n) for n in (0..30)] # G. C. Greubel, Jun 09 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 04 2020
STATUS
approved