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
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..283
- Eric Weisstein's World of Mathematics, Bell Polynomial.
- Wikipedia, Touchard polynomials
Crossrefs
Programs
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Mathematica
Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]] -
PARI
a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020
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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
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
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025
Extensions
a(0)=1 prepended by Seiichi Manyama, Aug 14 2020