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Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

5, 0, 1, 3, 2, 5, 8, 7, 2, 6, 8, 8, 1, 7, 8, 8, 0, 9, 4, 0, 2, 2, 9, 6, 7, 1, 0, 5, 5, 2, 7, 4, 9, 4, 4, 3, 7, 2, 6, 8, 7, 8, 3, 2, 9, 8, 5, 8, 0, 4, 5, 6, 8, 1, 5, 3, 6, 4, 5, 1, 2, 1, 7, 3, 3, 8, 8, 8, 7, 4, 1, 5, 8, 4, 5, 0, 6, 0, 6, 5, 3, 3, 0, 9, 0, 3, 1, 1, 3, 8, 8, 9, 7, 9, 4, 3, 9, 8, 9, 6, 1, 8, 1, 9, 4

0,1

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.

Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals -6*Pi*Im(-PolyLog(6, (-1)^(1/3))) + (17*Pi^4*zeta(3))/1620 + (1/3)*Pi^2*zeta(5) - (493*zeta(7))/24.

0.501325872688178809402296710552749443726878329858...

S[7]=-6*Pi*Im[-PolyLog[6, (-1)^(1/3)]] + (17*Pi^4*Zeta[3])/1620 + (1/3)*Pi^2*Zeta[5] - (493*Zeta[7])/24; RealDigits[S[7], 10, 105]//First

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261852 (S(8)).

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nonn,cons,easy

Jean-François Alcover, Sep 03 2015

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