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%I #22 Mar 13 2022 11:15:19
%S 1,-1,-8,-46,64,7280,118720,406160,-35578880,-1156775680,-12796467200,
%T 444964083200,27457634713600,594958346547200,-9096689344716800,
%U -1258068242084608000,-45330583283597312000,24150498582339584000,95678058298287259648000,5379182782796767182848000
%N Polylogarithm li(-n,-1/9) multiplied by (10^(n+1))/9.
%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=9.
%H Stanislav Sykora, <a href="/A213132/b213132.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=9.
%F E.g.f.: 10/(9+exp(10*x)). [_Joerg Arndt_, Apr 21 2013]
%F a(n) = Sum_{k=0..n} k! * (-1)^k * 10^(n-k) * Stirling2(n,k). - _Seiichi Manyama_, Mar 13 2022
%e polylog(-5, -1/9)*10^6/9 = 7280.
%p seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*9^k, k=0..n),n=0..17); # _Peter Luschny_, Apr 21 2013
%t Table[If[n == 0, 1, PolyLog[-n, -1/9] 10^(n+1)/9], {n, 0, 19}] (* _Jean-François Alcover_, Jun 27 2019 *)
%o (PARI) /* See A212846; run limnpq(nmax, 1, 9) */
%o (PARI) x='x+O('x^66); Vec(serlaplace( 10/(9+exp(10*x)) )) \\ _Joerg Arndt_, Apr 21 2013
%o (PARI) a(n) = sum(k=0, n, k!*(-1)^k*10^(n-k)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Mar 13 2022
%Y Cf. A212846, A210246, A212847, A213127 to A213131
%Y Cf. A213133 to A213157
%K sign
%O 0,3
%A _Stanislav Sykora_, Jun 06 2012
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