A362862 a(n) = (-1)^n * Sum_{k=0..n} (-n*k)^k * binomial(n,k).

1, 0, 13, 629, 58993, 8998399, 2035844461, 640881617123, 267995012680641, 143734541641235567, 96200314049944377901, 78599287990433271805699, 76993408916168689318057201, 89072357257840197226050646151

Offset: 0

Author

Seiichi Manyama, May 06 2023

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Main diagonal of A362019.

Cf. A290158.

Programs

Mathematica

Table[(-1)^n*(1 + Sum[(-n*k)^k*Binomial[n, k], {k, 1, n}]), {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

PARI

a(n) = (-1)^n * sum(k=0, n, (-n*k)^k*binomial(n, k));

Formula

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

a(n) = [x^n] Sum_{k>=0} (n*k*x)^k / (1 + x)^(k+1).