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

1, 6, 66, 1140, 28280, 948570, 41173776, 2238150600, 148570107264, 11804909261310, 1104566746764800, 120062928157552380, 14986973664751315968, 2127288759957421124610, 340440417300990616995840

Offset: 1

Author

Alexander Adamchuk, Sep 12 2006, Sep 14 2006

Keywords

• nonn

Comments

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.

Links

• Eric Weisstein's World of Mathematics, Eulerian number
• Eric Weisstein's World of Mathematics, Polylogarithm.

Crossrefs

Cf. A122778, A121376, A121985, A086914.

Programs

Mathematica

Table[Sum[Eulerian[n,k]*n^k,{k,0,n}],{n,1,25}]
Flatten[{1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n], {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 16 2016 *)

Formula

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.

a(n) ~ exp(-1) * n! * n^(n+1) / log(n)^(n+1). - Vaclav Kotesovec, Jun 06 2022