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%I #23 Mar 13 2022 11:15:04
%S 1,-1,-5,-13,115,2099,11395,-177373,-5116685,-40481581,948973795,
%T 36701972867,375364322515,-12090607539661,-580544884927805,
%U -7188739235243293,301374306966657715,17150539711123411859,246564346727945106595,-12988846468460187345853
%N Polylogarithm li(-n,-1/6) multiplied by (7^(n+1))/6.
%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=6.
%H Stanislav Sykora, <a href="/A213129/b213129.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=6.
%F E.g.f.: 7/(6+exp(7*x)). [_Joerg Arndt_, Apr 21 2013]
%F a(n) = Sum_{k=0..n} k! * (-1)^k * 7^(n-k) * Stirling2(n,k). - _Seiichi Manyama_, Mar 13 2022
%e polylog(-5,-1/6)*7^6/6 = 2099.
%p seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*6^k, k=0..n),n=0..17); # _Peter Luschny_, Apr 21 2013
%t Table[If[n == 0, 1, PolyLog[-n, -1/6] 7^(n+1)/6], {n, 0, 19}] (* _Jean-François Alcover_, Jun 27 2019 *)
%o (PARI) /* See A212846; run limnpq(nmax,1,6) */
%o (PARI) x='x+O('x^66); Vec(serlaplace( 7/(6+exp(7*x)) )) \\ _Joerg Arndt_, Apr 21 2013
%o (PARI) a(n) = sum(k=0, n, k!*(-1)^k*7^(n-k)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Mar 13 2022
%Y Cf. A212846, A210246, A212847, A213127, A213128.
%Y Cf. A213130 through A213157
%K sign
%O 0,3
%A _Stanislav Sykora_, Jun 06 2012
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