A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198

Offset: 0

Author

Vaclav Kotesovec, Jun 25 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

Formula

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022