A293493 Expansion of e.g.f. exp(x/(1 - x^3)).

1, 1, 1, 1, 25, 121, 361, 5881, 62161, 365905, 5145841, 84831121, 812318761, 11450816521, 243450591385, 3371056121161, 51784222133281, 1275605799044641, 23531635543402081, 410559590525890465, 11089633716053137081, 256375957896260034841, 5161258224477109736521

Offset: 0

Author

Seiichi Manyama, Oct 10 2017

Keywords

• nonn

Crossrefs

Cf. A293494.

Programs

PARI

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1-x^3))))

PARI

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)))))

Formula

E.g.f.: Product_{k>0} exp(x^(3*k-2)).

a(n) ~ exp(2*sqrt(3*n)/3 - n + 1/6) * n^(n-1/4) / (sqrt(2) * 3^(1/4)). - Vaclav Kotesovec, Oct 10 2017

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024