A256016 - OEIS

%I #39 Jan 04 2024 08:56:27

%S 1,1,6,57,796,15145,374526,11669665,447595800,20733553809,

%T 1141067915290,73552752257281,5484203261135028,467864288815609465,

%U 45236104846954021014,4915818294874879570305,596044703812665607374256,80118478395137652912476449,11870487496575403846760198322

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

%H Seiichi Manyama, <a href="/A256016/b256016.txt">Table of n, a(n) for n = 0..283</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>

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

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

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

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

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

%o (PARI) a(n) = n!*sum(k=0, n, k^n/k!); \\ _Michel Marcus_, Aug 15 2020

%o (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

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

%Y Main diagonal of A337085.

%K nonn,easy

%O 0,3

%A _Vaclav Kotesovec_, Jun 01 2015

%E a(0)=1 prepended by _Seiichi Manyama_, Aug 14 2020