A213127 Polylogarithm li(-n,-1/4) multiplied by (5^(n+1))/4.

1, -1, -3, -1, 69, 455, -1155, -50065, -334155, 4107095, 112058925, 491352575, -24429589275, -535893782425, 606194735325, 249291355871375, 4380933801391125, -56204145098271625, -4031655689182933875

Offset: 0

Author

Stanislav Sykora, Jun 06 2012

Keywords

• sign

Comments

See the sequence A212846 which describes the general case of li(-n,-p/q). This sequence is obtained for p=1,q=4.

Examples

			polylog(-5,-1/4)*5^6/4 = 455
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..100
• OEIS-Wiki, Eulerian polynomials

Crossrefs

Cf. A212846, A210246, A212847, A213128 through A213157. A015521.

Programs

Maple

seq(add((-1)^(n-k)*combinat[eulerian1](n,k)*4^k, k=0..n),n=0..18); # Peter Luschny, Apr 21 2013

Mathematica

a[0] = 1; a[n_] := PolyLog[-n, -1/4] * 5^(n+1)/4;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 29 2018 *)

PARI

/* see A212846; run limnpq(nmax,1,4) */

PARI

x='x+O('x^66); Vec(serlaplace( 5/(4+exp(5*x)) )) \\ Joerg Arndt, Apr 21 2013

PARI

a(n) = sum(k=0, n, k!*(-1)^k*5^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

Formula

See formula in A212846, setting p=1,q=4

From Peter Bala, Jun 24 2012: (Start)

E.g.f.: A(x) = 5/(4 + exp(5*x)) = 1 - x - 3*x^2/2! - x^3/3! + 69*x^4/4! + ....

The compositional inverse (A(-x) - 1)^(-1) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + 205*x^5/5 + ... is the logarithmic generating function for A015521.

(End)

G.f.: 1/Q(0), where Q(k) = 1 + x*(k+1)/( 1 - 4*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 17 2013

a(n) = Sum_{k=0..n} k! * (-1)^k * 5^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

a(n) ~ n! * (log(2) * cos(n*arctan(Pi/(2*log(2)))) - Pi * sin(n*arctan(Pi/(2*log(2))))/2) * 5^(n+1) / (Pi^2 + 4*log(2)^2)^(1 + n/2). - Vaclav Kotesovec, May 17 2022