A337593 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 3^(k-1) * a(n-k).

1, 1, 5, 42, 573, 11226, 294804, 9946791, 417064365, 21187915362, 1278636342660, 90195692894451, 7338668846348844, 681052861293535251, 71405270562056271741, 8388541745045127600597, 1096298129481068449931085, 158383969954582566159384786, 25153555538082783169267336764

Offset: 0

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

• nonn

Crossrefs

Cf. A004212, A337592, A337594, A337595, A337597.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[3 x]] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(3*x)) - 1) / 3).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 3^(n-1) * x^n / (n!)^2).