A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Author

Ilya Gutkovskiy, Apr 23 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Programs

Mathematica

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

PARI

a(n) = sum(k=0, n, 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} k^n / (n-k)!.

a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021

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