A355252 Expansion of e.g.f. exp(2*(exp(x) - 1) + 3*x).

1, 5, 27, 157, 979, 6517, 46107, 345261, 2726243, 22623525, 196712171, 1787356765, 16929897395, 166808851541, 1706299041211, 18088031239437, 198392625389315, 2248104026019461, 26283054263021963, 316637825898555069, 3926250785070282579, 50056384077880370101

Offset: 0

Author

Vaclav Kotesovec, Jun 26 2022

Keywords

• nonn

Comments

Binomial transform of A355247.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Crossrefs

Cf. A001861, A035009, A355247, A355253.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+3*x], {x, 0, nmax}], x] * Range[0, nmax]!

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) ~ n^(n+3) * exp(n/LambertW(n/2) - n - 2) / (8 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+3)).

a(0) = 1; a(n) = 3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023