A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405

Offset: 0

Author

Ilya Gutkovskiy, Apr 23 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..232

Crossrefs

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Programs

Mathematica

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

PARI

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

Formula

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).

a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.

a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021

E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022