A145438 Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).

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, 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, 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

Offset: 0

Author

R. J. Mathar, Feb 08 2009

Keywords

• cons
• easy
• nonn

Comments

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.

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.

Examples

			0.522946...
		

Links

• J. M. Borwein, R. Girgensohn, Evaluation of Binomial Series, CECM-02-188 (2002).
• A. I. Davydychev, M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B 699 (2004), 3-64.
• M. Yu. Kalmykov and O. Veretin, Single-scale diagrams and multiple binomial sums, Phys. Lett. B 483 (2000) 315-323.
• R. J. Mathar, Corrigenda to "Interesting Series involving..", arXiv:0905.0215 [math.CA]

Programs

Mathematica

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 *)

Formula

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...

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