A368719 a(n) = n! * Sum_{k=0..n} k^5 / k!.

0, 1, 34, 345, 2404, 15145, 98646, 707329, 5691400, 51281649, 512916490, 5642242441, 67707158124, 880193426905, 12322708514494, 184840628476785, 2957450056677136, 50276650964931169, 904979717370650610, 17194614630044837689, 343892292600899953780

Offset: 0

Author

Seiichi Manyama, Jan 04 2024

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial.
• Wikipedia, Touchard polynomials

Crossrefs

Column k=5 of A337085.

Cf. A048993, A368718.

Programs

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) ~ 52*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024