A337597 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 6^(k-1) * a(n-k).

1, 1, 8, 96, 1896, 55416, 2182752, 111162528, 7088997888, 550749341952, 51058009732608, 5556160183592448, 699989463219105792, 100917906076208203776, 16486415052067886690304, 3026039346413717945757696, 619431153899977856767131648, 140491838894751995366936641536, 35102748598142373142198776889344

Offset: 0

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A005012, A337592, A337593, A337594, A337595.

Programs

Maple

S:= series(exp((BesselI(0,2*sqrt(6*x))-1)/6),x,51):
seq(coeff(S,x,j)*(j!)^2, j=0..50); # Robert Israel, Sep 06 2020

Mathematica

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[6 x]] - 1)/6], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(6*x)) - 1) / 6).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 6^(n-1) * x^n / (n!)^2).