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

0,1

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.

Links

Table of n, a(n) for n=0..92.

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]

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]

Example

0.522946...

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 *)

Crossrefs

Sequence in context: A011411 A201328 A199189 * A354197 A346040 A244290

Adjacent sequences: A145435 A145436 A145437 * A145439 A145440 A145441

Keyword

cons,easy,nonn

Author

R. J. Mathar, Feb 08 2009

Status

approved