OFFSET
0,1
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
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.
EXAMPLE
0.5054294746835192416424504819084321491886690145682628649826647...
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
PROG
(PARI) suminf(n=1, 1/(n^5*binomial(2*n, n))) \\ Michel Marcus, Sep 03 2015
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Sep 03 2015
STATUS
approved