A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).
5, 0, 5, 4, 2, 9, 4, 7, 4, 6, 8, 3, 5, 1, 9, 2, 4, 1, 6, 4, 2, 4, 5, 0, 4, 8, 1, 9, 0, 8, 4, 3, 2, 1, 4, 9, 1, 8, 8, 6, 6, 9, 0, 1, 4, 5, 6, 8, 2, 6, 2, 8, 6, 4, 9, 8, 2, 6, 6, 4, 7, 1, 2, 8, 7, 5, 7, 3, 3, 4, 7, 3, 3, 7, 6, 1, 7, 5, 9, 0, 6, 8, 2, 7, 1, 6, 4, 5, 3, 3, 1, 8, 1, 5, 0, 0, 1, 3, 6, 6, 1, 9, 6
Offset: 0
Examples
0.5054294746835192416424504819084321491886690145682628649826647...
Links
- J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
- Eric Weisstein's MathWorld, Central Binomial Coefficient
Programs
-
Mathematica
S[5] = 2*Pi*Im[PolyLog[4, (-1)^(1/3)]] + (1/9)*Pi^2*Zeta[3] - 19*Zeta[5]/3; RealDigits[S[5], 10, 103] // First
-
PARI
suminf(n=1, 1/(n^5*binomial(2*n,n))) \\ Michel Marcus, Sep 03 2015
Formula
S(5) = 2*Pi*Im(PolyLog(4, (-1)^(1/3))) + (1/9)*Pi^2*zeta(3) -19*zeta(5)/3.
Equals (1/2) 4F3(1,1,1,1; 3/2,2,2; 1/4).
Also equals (1/(2592*sqrt(3)))*(Pi*(PolyGamma(3, 1/6) + PolyGamma(3, 1/3) - PolyGamma(3, 2/3) - PolyGamma(3, 5/6))) + (1/9)*Pi^2*zeta(3) - 19*zeta(5)/3.