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

0,1

Links

Table of n, a(n) for n=0..102.

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

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)).

Sequence in context: A199382 A254289 A140240 * A091672 A144702 A238192

Adjacent sequences: A261836 A261837 A261838 * A261840 A261841 A261842

Keyword

cons,easy,nonn

Author

Jean-François Alcover, Sep 03 2015

Status

approved