A368716 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^3 / k!.

0, 1, 6, 9, 28, -15, 306, -1799, 14904, -133407, 1335070, -14684439, 176214996, -2290792751, 32071101258, -481066515495, 7697064252016, -130850092279359, 2355301661034294, -44750731559644727, 895014631192902540, -18795307255050944079

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• sign

Comments

abs(a(n))/n is prime for n = 2, 3, 4, 5, 7, 13, 19, 28, 643 and no others up to n = 2000. - Robert Israel, May 13 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=3 of A368724.

Cf. A048993, A337001, A368585.

Programs

Maple

f:= proc(n) option remember;
  - n*procname(n-1)+n^3
end proc:
f(0):= 0:
seq(f(i),i=0..30); # Robert Israel, May 13 2025

Mathematica

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

Formula

a(0) = 0; a(n) = -n*a(n-1) + n^3.

E.g.f.: B_3(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.

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