A210244 Numerators of the polylogarithm li(-n,-1/2)/2.

-1, -1, 1, 5, -7, -49, -53, 2215, 1259, -14201, -183197, 248885, 9583753, 14525053, -554173253, -4573299625, 99833187251, 215440236599, -1654012631597, -84480933600305, -36267273557287, 10992430255511053, 117548575473066241, -1380910044674479865

Offset: 1

Author

Stanislav Sykora, Mar 19 2012

Keywords

• sign
• frac

Comments

Given an integer n>0, consider the infinite series s(n) = li(-n,-1/2) = Sum_{k>=1} (-1)^k*k^n/2^k. Then s(n)=2*a(n)/A131137(n+1).

Examples

			s(1)=-2/9, s(2)=-2/27, s(3)=+2/27, s(4)=+10/81.
		

Links

• 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

Crossrefs

Denominators: A131137, offset by 1.

Cf. A212846.

Programs

Mathematica

nn = 30; s[0] = 1; Do[s[n+1] = (-1/3) Sum[Binomial[n+1,i] s[i], {i, 0, n}], {n, 0, nn}]; Numerator[Table[s[n], {n, 0, nn}]] (* T. D. Noe, Mar 20 2012 *)
Table[PolyLog[-n, -1/2]/2, {n, 30}] (* T. D. Noe, Mar 23 2012 *)

PARI

a(n)=numerator(polylog(-n,-1/2)/2) \\ Charles R Greathouse IV, Jul 15 2014

Formula

Recurrence: s(n+1)=(-1/3)*Sum_{i=0..n} binomial(n+1,i)*s(i), with the starting value of s(0)=2/3.