[seqfan] Re: Conjectured integral representation for A057718

[Joerg Arndt]

* Vladimir Reshetnikov <v.reshetnikov at gmail.com> [Jan 20. 2011 11:19]:
> Dear Seqfans,
> 
> It appears that Integrate[(16 x)^k EllipticK[x] EllipticK[1 - x]/Pi^3, {x,
> 0, 1}] == A057718[k]. Alas, no proof for that.
> Is it worth submitting to OEIS?
> 

Definitely (as: "It appears that ...")!

Please also give in terms of  hypergeom([1/2,1/2],[1], ? )
This because (IIRC) Mathematica's ellK(x) is
   Pi/2 * hypergeom([1/2,1/2],[1], x )
and not
   Pi/2 * hypergeom([1/2,1/2],[1], x^2 )
as customary.

> --
> Thanks
> Vladimir Reshetnikov
> 
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