OFFSET
0,3
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
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Sep 11 2006
EXTENSIONS
a(0)=1 changed by Max Alekseyev, Nov 28 2011
STATUS
approved