This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A256016 #46 Jun 16 2025 19:19:20
%S A256016 1,1,6,57,796,15145,374526,11669665,447595800,20733553809,
%T A256016 1141067915290,73552752257281,5484203261135028,467864288815609465,
%U A256016 45236104846954021014,4915818294874879570305,596044703812665607374256,80118478395137652912476449,11870487496575403846760198322
%N A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.
%H A256016 Seiichi Manyama, <a href="/A256016/b256016.txt">Table of n, a(n) for n = 0..283</a>
%H A256016 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A256016 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A256016 a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
%F A256016 a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
%F A256016 E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - _Seiichi Manyama_, Aug 23 2022
%F A256016 a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - _Seiichi Manyama_, Jan 04 2024
%F A256016 a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - _Ridouane Oudra_, Jun 16 2025
%t A256016 Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]
%o A256016 (PARI) a(n) = n!*sum(k=0, n, k^n/k!); \\ _Michel Marcus_, Aug 15 2020
%o A256016 (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 A256016 Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.
%Y A256016 Main diagonal of A337085.
%K A256016 nonn,easy
%O A256016 0,3
%A A256016 _Vaclav Kotesovec_, Jun 01 2015
%E A256016 a(0)=1 prepended by _Seiichi Manyama_, Aug 14 2020