A355253 Expansion of e.g.f. exp(2*(exp(x) - 1) - 3*x).

1, -1, 3, -5, 19, -29, 171, -69, 2339, 5139, 57563, 303403, 2397011, 17237507, 139011211, 1151110299, 10076637827, 91903924979, 874688607035, 8656097294091, 88932728790195, 946748093175523, 10426787247224043, 118620906668843131, 1392128306377939427, 16833088095308098003

Offset: 0

Author

Vaclav Kotesovec, Jun 26 2022

Keywords

• sign

Comments

Inverse binomial transform of A194689.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..555

Crossrefs

Cf. A001861, A217923, A194689, A355252.

Programs

Mathematica

nmax = 30; 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) ~ 8 * n^(n-3) * exp(n/LambertW(n/2) - n - 2) / (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