A213128 - OEIS

%I #28 Mar 13 2022 11:14:59

%S 1,-1,-4,-6,96,1104,2016,-112176,-1718784,-642816,437031936,

%T 7656021504,-24274059264,-3939918299136,-72733516959744,

%U 699443277686784,67781787782086656,1236409075147014144,-25430445045847425024

%N Polylogarithm li(-n,-1/5) multiplied by (6^(n+1))/5.

%C See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=5.

%H Stanislav Sykora, <a href="/A213128/b213128.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=5

%F From _Peter Bala_, Jun 24 2012: (Start)

%F E.g.f.: A(x) = 6/(5 + exp(6*x)) = 1 - x - 4*x^2/2! - 6 x^3/3! + 96*x^4/4! + ....

%F The compositional inverse (A(-x) - 1)^(-1) = x + 4*x^2/2 + 21*x^3/3 + 104*x^4/4 + 521*x^5/5 + ... is the logarithmic generating function for A015531.

%F (End)

%F G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 5*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2013

%F a(n) = Sum_{k=0..n} k! * (-1)^k * 6^(n-k) * Stirling2(n,k). - _Seiichi Manyama_, Mar 13 2022

%e polylog(-5,-1/5)*6^6/5 = 1104.

%p seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*5^k, k=0..n),n=0..18); # _Peter Luschny_, Apr 21 2013

%t Table[If[n == 0, 1, PolyLog[-n, -1/5] 6^(n+1)/5], {n, 0, 18}] (* _Jean-François Alcover_, Jun 29 2019 *)

%o (PARI) /*See A212846; run limnpq(nmax,1,5) */

%o (PARI) x='x+O('x^66); Vec(serlaplace( 6/(5+exp(6*x)) )) \\ _Joerg Arndt_, Apr 21 2013

%o (PARI) a(n) = sum(k=0, n, k!*(-1)^k*6^(n-k)*stirling(n, k, 2)); \\ _Seiichi Manyama_, Mar 13 2022

%Y Cf. A212846, A210246, A212847, A213127

%Y Cf. A213129 through A213157.

%Y Cf. A015531.

%K sign

%O 0,3

%A _Stanislav Sykora_, Jun 06 2012