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Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).
5

%I #12 May 27 2014 13:31:03

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

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

%U 7,3,5,7,7,6,4,4,1,3,2,9,9,3,1,5,0,4,2,6,2,1,9,1,3

%N Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).

%C Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.47 gives Pi*sqrt(3)*(psi(2/3)-psi(1/3))/72-Zeta(3)/3 which is negative and therefore not correct.

%C Comment from Mikhail Kalmykov (kalmykov.mikhail(AT)googlemail.com), Jun 01 2009: Analytical results for this sum were also given in Eq. (8) of the Kalmykov and Veretin paper. These results confirm the last comment from _Alois P. Heinz_.

%H J. M. Borwein, R. Girgensohn, <a href="http://locutus.cs.dal.ca:8088/archive/00000131/">Evaluation of Binomial Series</a>, CECM-02-188 (2002).

%H A. I. Davydychev, M. Yu. Kalmykov, <a href="http://dx.doi.org/10.1016/j.nuclphysb.2004.08.020">Massive Feynman diagrams and inverse binomial sums</a>, Nucl. Phys. B 699 (2004), 3-64.

%H M. Yu. Kalmykov and O. Veretin, <a href="http://dx.doi.org/10.1016/S0370-2693(00)00574-8">Single-scale diagrams and multiple binomial sums</a>, Phys. Lett. B 483 (2000) 315-323.

%H R. J. Mathar, <a href="http://arxiv.org/abs/0905.0215">Corrigenda to "Interesting Series involving.."</a>, arXiv:0905.0215 [math.CA]

%F Comment from _Alois P. Heinz_, Feb 08 2009: Maple's answer to this is: a:= sum(1/(n^3*binomial(2*n,n)), n=1..infinity); a:= 1/2 hypergeom([1, 1, 1, 1], [2, 2, 3/2], 1/4); evalf (a, 140); .522946192133335108491185183527303540163044591743977841465941014...

%F Equals A019693*A143298-4*A002117/3 =2*Pi*Cl_2(Pi/3)/3-4*zeta(3)/3. [From _R. J. Mathar_, Feb 09 2009]

%e 0.522946...

%t RealDigits[ N[1/18*(Sqrt[3]* Pi*(-PolyGamma[1, 2/3] + PolyGamma[1, 4/3] + 9) - 24*Zeta[3]), 105]][[1]] (* _Jean-François Alcover_, Nov 08 2012, after _R. J. Mathar_ *)

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Feb 08 2009