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%I #22 Mar 13 2022 11:15:12
%S 1,-1,-7,-33,105,5199,64953,-46593,-21769335,-497664081,-1941272487,
%T 256114020447,9566995408425,99966666676239,-6245895772363527,
%U -366865939437422913,-6924777575908002615,259022993102904450159,24387711970312991335833,716398360186298080983327
%N Polylogarithm li(-n,-1/8) multiplied by (9^(n+1))/8.
%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=8.
%H Stanislav Sykora, <a href="/A213131/b213131.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=8.
%F E.g.f.: 9/(8+exp(9*x)). [_Joerg Arndt_, Apr 21 2013]
%F a(n) = Sum_{k=0..n} k! * (-1)^k * 9^(n-k) * Stirling2(n,k). - _Seiichi Manyama_, Mar 13 2022
%e polylog(-5,-1/8)*9^6/8 = 5199.
%p seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*8^k, k=0..n),n=0..17); # _Peter Luschny_, Apr 21 2013
%t Table[If[n == 0, 1, PolyLog[-n, -1/8] 9^(n+1)/8], {n, 0, 19}] (* _Jean-François Alcover_, Jun 27 2019 *)
%o (PARI) /* See A212846; run limnpq(nmax, 1, 8) */
%o (PARI) x='x+O('x^66); Vec(serlaplace( 9/(8+exp(9*x)) )) \\ _Joerg Arndt_, Apr 21 2013
%o (PARI) a(n) = sum(k=0, n, k!*(-1)^k*9^(n-k)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Mar 13 2022
%Y Cf. A212846, A210246, A212847, A213127 to A213130
%Y Cf. A213132 to A213157
%K sign
%O 0,3
%A _Stanislav Sykora_, Jun 06 2012
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