A375541 a(n) = 2^n * n! * [x^n] (-1/2 - exp(-x))^n.

1, 2, 20, 318, 7080, 202650, 7089516, 293122998, 13984363728, 756140833458, 45695657262420, 3052241497352718, 223293580147036152, 17756185491727424586, 1524930579202933587132, 140665424413881644688870, 13870450317973173165542304, 1455954744856343617306729314

Offset: 0

Author

Peter Luschny, Sep 01 2024

Keywords

• nonn

Programs

Maple

gf := n -> (-1/2 - exp(-x))^n:
ser := n -> series(gf(n), x, 20):
a := n -> expand(2^n*n!*coeff(ser(n), x, n)):
seq(a(n), n = 0..17);

Mathematica

Table[2^n * n! * SeriesCoefficient[(-1/2 - E^(-x))^n,{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 01 2024 *)

Formula

a(n) ~ n^n / (sqrt(1+LambertW(exp(-1)/2)) * exp(n) * (LambertW(exp(-1)/2))^n). - Vaclav Kotesovec, Sep 01 2024

a(n) = Sum_{k=0..n} k^n*2^k*binomial(n,k). - Ridouane Oudra, Jun 16 2025