A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..283

Eric Weisstein's World of Mathematics, Bell Polynomial.

Wikipedia, Touchard polynomials

Formula

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.

a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).

E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022

a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024

Mathematica

Join[{1}, Table[n!*Sum[k^n/k!, {k, 0, n}], {n, 1, 20}]]

Prog

(PARI) a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

Crossrefs

Cf. A031971, A072034, A242446, A337001, A337002, A354436.

Main diagonal of A337085.

Sequence in context: A294511 A242817 A295238 * A361291 A145170 A180255

Adjacent sequences: A256013 A256014 A256015 * A256017 A256018 A256019

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jun 01 2015

Extensions

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

Status

approved