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A122020 Sum[k=0..n] Eulerian[n,k]*n^k. 2

%I #7 Jun 06 2022 11:23:00

%S 1,6,66,1140,28280,948570,41173776,2238150600,148570107264,

%T 11804909261310,1104566746764800,120062928157552380,

%U 14986973664751315968,2127288759957421124610,340440417300990616995840

%N Sum[k=0..n] Eulerian[n,k]*n^k.

%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.

%F 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 a(n) ~ exp(-1) * n! * n^(n+1) / log(n)^(n+1). - _Vaclav Kotesovec_, Jun 06 2022

%t Table[Sum[Eulerian[n,k]*n^k,{k,0,n}],{n,1,25}]

%t Flatten[{1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n], {n, 2, 20}]}] (* _Vaclav Kotesovec_, Oct 16 2016 *)

%Y Cf. A122778, A121376, A121985, A086914.

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Sep 12 2006, Sep 14 2006

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)