A213128 Polylogarithm li(-n,-1/5) multiplied by (6^(n+1))/5.

1, -1, -4, -6, 96, 1104, 2016, -112176, -1718784, -642816, 437031936, 7656021504, -24274059264, -3939918299136, -72733516959744, 699443277686784, 67781787782086656, 1236409075147014144, -25430445045847425024

Offset

0,3

Comments

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

Links

Stanislav Sykora, Table of n, a(n) for n = 0..100

OEIS-Wiki, Eulerian polynomials

Formula

See formula in A212846, setting p=1,q=5

From Peter Bala, Jun 24 2012: (Start)

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

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.

(End)

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

a(n) = Sum_{k=0..n} k! * (-1)^k * 6^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

Example

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

Maple

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

Mathematica

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

Prog

(PARI) /*See A212846; run limnpq(nmax, 1, 5) */
(PARI) x='x+O('x^66); Vec(serlaplace( 6/(5+exp(6*x)) )) \\ Joerg Arndt, Apr 21 2013
(PARI) a(n) = sum(k=0, n, k!*(-1)^k*6^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

Crossrefs

Cf. A212846, A210246, A212847, A213127

Cf. A213129 through A213157.

Cf. A015531.

Sequence in context: A307101 A087934 A052684 * A105037 A139730 A367880

Adjacent sequences: A213125 A213126 A213127 * A213129 A213130 A213131

Keyword

sign

Author

Stanislav Sykora, Jun 06 2012

Status

approved