A368718 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^5 / k!.

0, 1, 30, 153, 412, 1065, 1386, 7105, -24072, 275697, -2656970, 29387721, -352403820, 4581620953, -64142155518, 962133092145, -15394128425744, 261700184657505, -4710603321945522, 89501463119441017, -1790029262385620340, 37590614510102111241

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• sign

Comments

In general, for m >=0, Sum_{k=0..n} (-1)^(n-k) * k^m / k! ~ A000587(m) * (-1)^n * exp(-1). - Vaclav Kotesovec, Jul 18 2025

Links

• Robert Israel, Table of n, a(n) for n = 0..448
• Eric Weisstein's World of Mathematics, Bell Polynomial.
• Wikipedia, Touchard polynomials

Crossrefs

Column k=5 of A368724.

Cf. A048993, A368587, A368719.

Programs

Maple

f:= proc(n) option remember;
  - n*procname(n-1)+n^5
end proc:
f(0):= 0:
seq(f(i),i=0..21); # Robert Israel, May 13 2025

Mathematica

Table[-5*n + 3*n^3 + n^4 - 2*(-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)

PARI

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 5, stirling(5, k, 2)*x^k)*exp(x)/(1+x))))

Formula

a(0) = 0; a(n) = -n*a(n-1) + n^5.

E.g.f.: B_5(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

a(n) ~ -2*(-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025