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%I #11 Sep 03 2015 08:46:55
%S 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,
%T 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,
%U 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
%N Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).
%H J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, <a href="http://arxiv.org/abs/hep-th/0004153">Central Binomial Sums, Multiple Clausen Values and Zeta Values</a>, arXiv:hep-th/0004153, 2000.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>
%F S(5) = 2*Pi*Im(PolyLog(4, (-1)^(1/3))) + (1/9)*Pi^2*zeta(3) -19*zeta(5)/3.
%F Equals (1/2) 4F3(1,1,1,1; 3/2,2,2; 1/4).
%F 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.
%e 0.5054294746835192416424504819084321491886690145682628649826647...
%t 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
%o (PARI) suminf(n=1, 1/(n^5*binomial(2*n,n))) \\ _Michel Marcus_, Sep 03 2015
%Y Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)).
%K cons,easy,nonn
%O 0,1
%A _Jean-François Alcover_, Sep 03 2015
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