A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

1, 2, 7, 29, 142, 785, 4813, 32240, 233449, 1812161, 14980768, 131174939, 1211111629, 11745451658, 119255234371, 1264050651953, 13952113296766, 160006824960725, 1902825936046105, 23423342243273696, 297982102750214605, 3911917977005948453, 52926119656555824520

Offset: 0

Author

Vaclav Kotesovec, Jun 26 2022

Keywords

• nonn

Comments

Inverse binomial transform of A027710.

In general, if m >= 1 and e.g.f. = exp(m*exp(x) + r*x + s) then

a(n) ~ n^(n+r) * exp(n/LambertW(n/m) - n + s) / (m^r * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n+r)).

Equivalently, a(n) ~ n! * (n/m)^r * exp(n/LambertW(n/m) + s) / (sqrt(2*Pi*n * (1 + LambertW(n/m))) * LambertW(n/m)^(n+r)).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..545

Crossrefs

Cf. A000296, A027710, A078940, A217924.

Programs

Mathematica

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

PARI

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

Formula

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

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