A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

1, 2, 13, 226, 7889, 458026, 39684637, 4788052298, 766526598721, 157108817646514, 40104442275129101, 12472587843118746322, 4641978487740740993233, 2036813028164774540010266, 1040451608604560812273060189, 612098707457003526384666111226

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2016

Keywords

• nonn

Links

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

Crossrefs

Main diagonal of A320031.

Cf. A086331, A277373, A277423, A277453.

Programs

Maple

a := n -> simplify(hypergeom([1, -n], [], -n)):
seq(a(n), n=0..15); # Peter Luschny, Oct 03 2018
# second Maple program:
b:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*b(n-1, k), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 09 2020

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k]*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

PARI

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

Formula

a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).

a(n) ~ n^n * n!.

a(n) = n! * [x^n] exp(x)/(1 - n*x). - Ilya Gutkovskiy, Sep 18 2018

a(n) = floor(n^n*n!*exp(1/n)), n > 0. - Peter McNair, Dec 20 2021