%I #36 Jun 06 2022 11:25:56
%S 1,1,3,22,285,5656,158095,5881968,279768825,16507789696,1180490926131,
%T 100415158796800,10005244013129365,1152844128057793536,
%U 151949197139815794615,22696027820066041133056,3810644613584486281328625
%N a(n) = Sum_{k=0..n} A(n,k)*n^k where A(n,k) are Eulerian numbers.
%C Prime p divides a(p-1) for p>2. - _Alexander Adamchuk_, Sep 12 2006
%C 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
%H Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#T1">Table 26.14.1 </a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian number</a> at MathWorld
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a> at MathWorld
%F a(n) = Sum_{k=0..n} A(n,k) * n^k
%F a(n) = Sum_{k=0..n} A(n,k) * n^(n-k).
%F a(n) = ((n-1)^(n+1))/n * Sum_{k>=1} k^n/n^k for n>1.
%F a(n) = ((n-1)^(n+1))/n * Li_{-n}(1/n) for n>1. - _Alexander Adamchuk_, Sep 12 2006
%F a(n) = (n-1)*A086914(n), n>1. - _Vladeta Jovovic_, Sep 12 2006
%F a(n) ~ exp(-1) * n! * n^n / log(n)^(n+1). - _Vaclav Kotesovec_, Jun 06 2022
%p 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
%p seq(add(combinat:-eulerian1(n,k)*n^k,k=0..n),n=0..16); # _Peter Luschny_, Oct 19 2016
%t << 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 *)
%t Flatten[{1, 1, Table[(n-1)^(n+1)*PolyLog[-n, 1/n]/n, {n, 2, 20}]}] (* _Vaclav Kotesovec_, Oct 16 2016 *)
%Y Cf. A008292.
%K nonn
%O 0,3
%A _Max Alekseyev_, Sep 11 2006
%E a(0)=1 changed by _Max Alekseyev_, Nov 28 2011
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