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 A122020 #8 Feb 16 2025 08:33:02
%S A122020 1,6,66,1140,28280,948570,41173776,2238150600,148570107264,
%T A122020 11804909261310,1104566746764800,120062928157552380,
%U A122020 14986973664751315968,2127288759957421124610,340440417300990616995840
%N A122020 Sum[k=0..n] Eulerian[n,k]*n^k.
%C A122020 n divides a(n). 2^m divides a(n), where m(n) = {0,1,1,2,3,1,4,3,7,1,9,2,10,1,11,4,15,1,17,2,18,1,20,3,22,...}. p^k divides from a(p^k-1), a(p^k), a(p^k+1) for prime p>2 and integer k>0.
%H A122020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian number</a>
%H A122020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F A122020 a(n) = Sum[ Eulerian[n,k]*n^(n-k-1), {k,0,n} ] = n*A122778[n]. a(n) = n(n-1)*A086914[n] for n>1. a(n) = ((n-1)^(n+1)) * PolyLog[ -n, 1/n ] = ((n-1)^(n+1)) * Sum[ k^n/n^k, {k,1,Infinity} ] = ((n-1)^(n+1)) * A121376[n]/A121985[n] for n>1.
%F A122020 a(n) ~ exp(-1) * n! * n^(n+1) / log(n)^(n+1). - _Vaclav Kotesovec_, Jun 06 2022
%t A122020 Table[Sum[Eulerian[n,k]*n^k,{k,0,n}],{n,1,25}]
%t A122020 Flatten[{1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n], {n, 2, 20}]}] (* _Vaclav Kotesovec_, Oct 16 2016 *)
%Y A122020 Cf. A122778, A121376, A121985, A086914.
%K A122020 nonn
%O A122020 1,2
%A A122020 _Alexander Adamchuk_, Sep 12 2006, Sep 14 2006