[seqfan] Re: any value storing constants of power ratios?

[Prof. Dr. Alois Heinz]

Richard Mathar schrieb:

>Does somebody know the closed form solution to A000020, that would be
>an erratum to the table? It is probably not far off the one I am quoting.
>
>%I A000020
>%S A000020 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,
>%T A000020 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,
>%U A000020 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
>%N A000020 Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).
>%C A000020 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.
>%e A000020 0.522946...
>%K A000020 cons,easy,nonn
>%O A000020 0,1
>%A A000020 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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\
        1442073577644132993150426219131503384066199612068581269307\
        6502579031017748980