A337594 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 4^(k-1) * a(n-k).

1, 1, 6, 58, 920, 21176, 654960, 26114768, 1298070912, 78359732608, 5630565514496, 473796572027648, 46060380961356800, 5114737212582603776, 642502387594286036992, 90542358999393528670208, 14209873001490130067095552, 2467784343879850163370295296, 471558856613839054976849608704

Offset: 0

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

• nonn

Crossrefs

Cf. A004213, A337592, A337593, A337595, A337597.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 4 Sqrt[x]] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,4*sqrt(x)) - 1) / 4).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 4^(n-1) * x^n / (n!)^2).