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

0,2

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, 2025.

Sangchul Lee, Closed form for A375540, answer to question on Mathematics Stack Exchange, 2025.

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) = f(n,n) where f(n,k) = Sum_{j=0..n} j!*2^j*binomial(n,j)*Stirling2(k,j) = Sum_{j=0..n} binomial(n,j)*j^k*2^j*(-1)^(n-j) (due to Sangchul Lee). - Mikhail Kurkov, Mar 20 2026

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);
# Alternative:
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 *)

Crossrefs

Cf. A002720, A195979.

Main diagonal of A394444.

Sequence in context: A253282 A375899 A385486 * A381418 A201470 A349268

Adjacent sequences: A375537 A375538 A375539 * A375541 A375542 A375543

Keyword

nonn

Author

Peter Luschny, Sep 01 2024

Status

approved