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%I #15 Jun 28 2020 10:25:53
%S 1,1,-1,-2,69,-1104,11395,189232,-21769335,1156775680,-41290278129,
%T -136576564992,234678001445965,-32256618406068224,3018646161081366075,
%U -158289126522080405504,-15471427638848015017455,6998210972374723086974976,-1487059744246923349187974457,223233959091253143036239872000
%N Eulerian polynomials at nonpositive integers, A_{n}(-n).
%C Define the Eulerian numbers A(n,k) as the number of permutations of {1,2,..,n} with k ascents and the Eulerian polynomials A_{0}(x) = 1; A_{n}(x) = sum_{k=0..n-1} A(n,k) x^k for n > 0. Then a(n) = A_{n}(-n) are the values of the Eulerian polynomials for n = 0,-1,-2,-3,...
%H OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.
%F a(n) = -(n+1)^(n+1)*Li_{-n}(n)/n, where Li_{n}(z) denotes the polylogarithm. For n = 0, interpret it as a limit for continuous n -> 0, that gives a(0) = 1. - _Vladimir Reshetnikov_, Oct 15 2016
%F a(n) = n! * [x^n] (n + 1) / (n + exp(-(n + 1)*x)). - _Ilya Gutkovskiy_, Jun 28 2020
%p c := proc(n,m) local k; add((-1)^k*binomial(n+1,k)*(m+1-k)^n,k=0..m) end:
%p a := proc(n) local k; `if`(n=0,1,add(c(n,k)*(-n)^k,k=0..n-1)) end:
%t a[0] = 1; a[n_] := -(n + 1)^(n + 1) PolyLog[-n, -n]/n; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2016 *)
%o (PARI) a(n) = if (n==0, 1, -(n + 1)^(n + 1)*polylog(-n, -n)/n); \\ _Michel Marcus_, May 30 2018
%Y Cf. A122778
%K sign
%O 0,4
%A _Peter Luschny_, Aug 12 2010
%E More terms from _Michel Marcus_, May 30 2018
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