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=4.
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=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
EXAMPLE
polylog(-5,-1/4)*5^6/4 = 455
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Stanislav Sykora, Jun 06 2012
STATUS
approved