A368717 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^4 / k!.

0, 1, 14, 39, 100, 125, 546, -1421, 15464, -132615, 1336150, -14683009, 176216844, -2290790411, 32071104170, -481066511925, 7697064256336, -130850092274191, 2355301661040414, -44750731559637545, 895014631192910900, -18795307255050934419

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• sign

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=4 of A368724.

Cf. A048993, A337002, A368586.

Programs

Maple

f:= proc(n) option remember;
  - n*procname(n-1)+n^4
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025

Mathematica

Table[-n + 2*n^2 + n^3 + (-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, 4, stirling(4, k, 2)*x^k)*exp(x)/(1+x))))

Formula

a(0) = 0; a(n) = -n*a(n-1) + n^4.

E.g.f.: B_4(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

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