A122778 a(n) = Sum_{k=0..n} A(n,k)*n^k where A(n,k) are Eulerian numbers.

1, 1, 3, 22, 285, 5656, 158095, 5881968, 279768825, 16507789696, 1180490926131, 100415158796800, 10005244013129365, 1152844128057793536, 151949197139815794615, 22696027820066041133056, 3810644613584486281328625

Offset: 0

Author

Max Alekseyev, Sep 11 2006

Keywords

• nonn

Comments

Prime p divides a(p-1) for p>2. - Alexander Adamchuk, Sep 12 2006

Let A_n(x) denote the Eulerian polynomials with coefficients the Eulerian numbers as defined in the DLMF (number of permutations of {1,2,..,n} with k ascents) then a(n) = A_n(n). - Peter Luschny, Aug 09 2010

Links

• Digital Library of Mathematical Functions, Table 26.14.1
• Eric Weisstein's World of Mathematics, Eulerian number at MathWorld
• Eric Weisstein's World of Mathematics, Polylogarithm at MathWorld

Crossrefs

Cf. A008292.

Programs

Maple

A122778 := n -> add(n^k*add((-1)^j*binomial(n+1,j)*(k-j+1)^n,j=0..k),k=0..n); # Peter Luschny, Aug 09 2010
seq(add(combinat:-eulerian1(n,k)*n^k,k=0..n),n=0..16); # Peter Luschny, Oct 19 2016

Mathematica

<< Combinatorica`; Table[Sum[Combinatorica`Eulerian[n, k] If[n == k == 0, 1, n^k], {k, 0, n}], {n, 0, 20}] (* Alexander Adamchuk, Sep 12 2006; corrected by Vladimir Reshetnikov, Oct 15 2016 *)
Flatten[{1, 1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n]/n, {n, 2, 20}]}] (* Vaclav Kotesovec, Oct 16 2016 *)

Formula

a(n) = Sum_{k=0..n} A(n,k) * n^k

a(n) = Sum_{k=0..n} A(n,k) * n^(n-k).

a(n) = ((n-1)^(n+1))/n * Sum_{k>=1} k^n/n^k for n>1.

a(n) = ((n-1)^(n+1))/n * Li_{-n}(1/n) for n>1. - Alexander Adamchuk, Sep 12 2006

a(n) = (n-1)*A086914(n), n>1. - Vladeta Jovovic, Sep 12 2006

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

Extensions

a(0)=1 changed by Max Alekseyev, Nov 28 2011