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%I #35 May 17 2022 03:25:02
%S 1,-1,-3,-1,69,455,-1155,-50065,-334155,4107095,112058925,491352575,
%T -24429589275,-535893782425,606194735325,249291355871375,
%U 4380933801391125,-56204145098271625,-4031655689182933875
%N Polylogarithm li(-n,-1/4) multiplied by (5^(n+1))/4.
%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=4.
%H Stanislav Sykora, <a href="/A213127/b213127.txt">Table of n, a(n) for n = 0..100</a>
%H OEIS-Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>
%F See formula in A212846, setting p=1,q=4
%F From _Peter Bala_, Jun 24 2012: (Start)
%F E.g.f.: A(x) = 5/(4 + exp(5*x)) = 1 - x - 3*x^2/2! - x^3/3! + 69*x^4/4! + ....
%F The compositional inverse (A(-x) - 1)^(-1) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + 205*x^5/5 + ... is the logarithmic generating function for A015521.
%F (End)
%F G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 4*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2013
%F a(n) = Sum_{k=0..n} k! * (-1)^k * 5^(n-k) * Stirling2(n,k). - _Seiichi Manyama_, Mar 13 2022
%F a(n) ~ n! * (log(2) * cos(n*arctan(Pi/(2*log(2)))) - Pi * sin(n*arctan(Pi/(2*log(2))))/2) * 5^(n+1) / (Pi^2 + 4*log(2)^2)^(1 + n/2). - _Vaclav Kotesovec_, May 17 2022
%e polylog(-5,-1/4)*5^6/4 = 455
%p seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*4^k, k=0..n),n=0..18); # _Peter Luschny_, Apr 21 2013
%t a[0] = 1; a[n_] := PolyLog[-n, -1/4] * 5^(n+1)/4;
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 29 2018 *)
%o (PARI) /* see A212846; run limnpq(nmax,1,4) */
%o (PARI) x='x+O('x^66); Vec(serlaplace( 5/(4+exp(5*x)) )) \\ _Joerg Arndt_, Apr 21 2013
%o (PARI) a(n) = sum(k=0, n, k!*(-1)^k*5^(n-k)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Mar 13 2022
%Y Cf. A212846, A210246, A212847, A213128 through A213157. A015521.
%K sign
%O 0,3
%A _Stanislav Sykora_, Jun 06 2012
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