A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*k!*n^k.

1, 0, 5, 116, 4785, 307024, 28435285, 3598112580, 596971515329, 125802906617600, 32834740225688901, 10399056510149276980, 3929349957207906673585, 1746371472945523953503376, 901944505258819679842017365, 535692457387043907059336566724, 362573376628272441934460817960705

Offset: 0

Author

Ilya Gutkovskiy, Sep 18 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..232

Crossrefs

Main diagonal of A320032.

Cf. A000166, A000180, A000354, A001907, A001908, A277452.

Programs

Maple

b:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*b(n-1, k)+(-1)^n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 07 2020

Mathematica

Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! n^k, {k, 0, n}], {n, 16}]]
Table[n! SeriesCoefficient[Exp[-x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
Table[(-1)^n HypergeometricPFQ[{1, -n}, {}, n], {n, 0, 16}]

PARI

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*k!*n^k); \\ Michel Marcus, Sep 18 2018

Formula

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

a(n) = exp(-1/n)*n^n*Gamma(n+1,-1/n) for n > 0, where Gamma(a,x) is the incomplete gamma function.

a(n) ~ n! * n^n. - Vaclav Kotesovec, Jun 09 2019