A212846 Polylogarithm li(-n,-1/2) multiplied by (3^(n+1))/2.

1, -1, -1, 3, 15, -21, -441, -477, 19935, 101979, -1150281, -14838957, 60479055, 2328851979, 3529587879, -403992301437, -3333935426625, 72778393505979, 1413503392326039, -10851976875907917, -554279405351601105, -713848745428080021

Offset

0,4

Comments

Apart from sign, same as A087674: a(n) = A087674*(-1)^n

Given integers n, p, q, 0<p<q, the value of li(-n,-p/q)=sum(k>=0, ((k^n)/(-p/q)^k) ) = s(n), multiplied by ((p+q)^(n+1))/q is an integer a(n). For this sequence set p=1 and q=2.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..455 (terms 0..100 from Stanislav Sykora)

OEIS-Wiki, Eulerian polynomials

S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), Stan's Library Vol. I, April 2006, updated March 2012. See Eq.(29).

Eric W. Weisstein, MathWorld: Polylogarithm

Formula

General recurrence: s(n+1)=(-p/(p+q))*SUM(C(n+1,i)*s(i)), where i=0,1,2,...,n, C(n,m) are binomial coefficients, and the starting value is s(0)=SUM((-p/q)^k)=q/(p+q). For this sequence set p=1 and q=2.

From Peter Bala, Jun 24 2012: (Start)

E.g.f.: A(x) = 3/(2+exp(3*x)).

The compositional inverse (A(-x) - 1)^(-1) = x + x^2/2 + 3*x^3/3 + 5*x^4/4 + 11*x^5/5 + ... is the logarithmic generating function for A001045.

(End)

a(n+1) = -3*a(n) + 2*sum(k=0..n, binomial(n,k)*a(k)*a(n-k) ), with a(0) = 1. - Peter Bala, Mar 12 2013

Let A(x) be the g.f. of A212846, B(x) the g.f. of A087674, then A(x) = B(-x).

G.f.: 1/Q(0), where Q(k)= 1 + x*(k+1)/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

O.g.f.: Sum_{n>=0} n!*(-x)^n / Product_{k=0..n} (1-3*k*x). - Paul D. Hanna, May 30 2013

For n>0, a(n) = -A179929(n)/2. - Stanislav Sykora, May 15 2014

a(n) = Sum_{k=0..n} k! * (-1)^k * 3^(n-k) * Stirling2(n,k). - Seiichi Manyama, Mar 13 2022

a(n) ~ n! * (log(2) * cos(n*arctan(Pi/log(2))) - Pi * sin(n*arctan(Pi/log(2)))) * 3^(n+1) / (Pi^2 + log(2)^2)^(1 + n/2). - Vaclav Kotesovec, May 17 2022

Example

a(5) = polylog(-5,-1/2)*3^6/2 = -21.
E.g.f.: A(x) = 1 - x - x^2/2! + 3*x^3/3! + 15*x^4/4! - 21*x^5/5! + ...
O.g.f.: G(x) = 1 - x - x^2 + 3*x^3 + 15*x^4 - 21*x^5 - 441*x^6 +...
where G(x) = 1 - x/(1-3*x) + 2!*x^2/((1-3*x)*(1-6*x)) - 3!*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4!*x^4/((1-3*x)*(1-6*x)*(1-9*x)*(1-12*x)) +...

Maple

seq(add((-1)^(n-k)*combinat[eulerian1](n, k)*2^k, k=0..n), n=0..21); # Peter Luschny, Apr 21 2013

Mathematica

f[n_] := PolyLog[-n, -1/2] 3^(n + 1)/2; Array[f, 21] (* Robert G. Wilson v, May 28 2012 *)
a[ n_] := If[ n < 0, 0, n! 3/2 SeriesCoefficient[ 1 / (1 + Exp[3 x] / 2), {x, 0, n}]]; (* Michael Somos, Aug 27 2018 *)

Prog

(PARI) /* for this sequence, run limnpq(nmax, 1, 2) */
limnpq(nmax, p, q) = {
  f=vector(nmax+1); f[1]=q/(p+q); r=-p/(p+q);
  for (i=2, nmax+1, p1=i-1; bc=1; m=p1; s=0;
    for(j=1, i-1, p2=j-1; if (p2, bc=bc*m/p2; m=m-1; );
    s=s+bc*f[j]; ); f[i]=r*s; );
fac=(p+q)/q;
for(i=1, nmax+1, f[i]=f[i]*fac; fac=(p+q)*fac;
  write("outputfile", i-1, " ", f[i]); ); }
(PARI) x='x+O('x^66); Vec(serlaplace(3/(2+exp(3*x)))) \\ Joerg Arndt, Apr 21 2013
(PARI) /* O.g.f.: */
{a(n)=polcoeff(sum(m=0, n, m!*(-x)^m/prod(k=1, m, 1-3*k*x+x*O(x^n))), n)} \\ Paul D. Hanna, May 30 2013
(PARI) a(n) = sum(k=0, n, k!*(-1)^k*3^(n-k)*stirling(n, k, 2)); \\ Seiichi Manyama, Mar 13 2022

Crossrefs

Similar cases: A210246 (p=1,q=3), A212847 (p=2,q=3)

Cf. A210244 (similar).

Cf. A213127 through A213157.

Cf. A001045, A087674, A179929.

Sequence in context: A277585 A318134 A087674 * A276804 A009057 A289712

Adjacent sequences: A212843 A212844 A212845 * A212847 A212848 A212849

Keyword

sign

Author

Stanislav Sykora, May 28 2012

Status

approved