A375540 a(n) = 2^n * n! * [x^n] (1/2 - exp(-x))^n.

1, 2, 12, 126, 1880, 36250, 856212, 23928758, 772172592, 28253043378, 1155731972780, 52265163565582, 2589097062756360, 139428505876012106, 8110011431007355716, 506710228437429986790, 33844577422630735656032, 2406541293179536265812834, 181497377154154817667851100

Offset: 0

Author

Peter Luschny, Sep 01 2024

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..362
• Sangchul Lee, Fast and simple recursive algorithm for A375540, answer to question on Mathematics Stack Exchange, Jul 31 2025.

Crossrefs

Cf. A002720, A195979.

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..18);
# second Maple program:
b:= proc(n, k) option remember;
     `if`(n=0, 1, k*(b(n-1, k)+b(n-1, k-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 03 2025

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