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%I #60 Jun 27 2021 07:59:24
%S 7,-37,19899,-235225,268989175,-4985687133,1052143756587,
%T -25075299330081,71491170131441775,-1979286926244381325,
%U 319756423353994489291,-9700423363591011143001,5919065321069316557189503,-189993537046726536185033125
%N Numerators of the fraction (30*n+7) * binomial(2*n,n)^2 * 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.
%C The Gaussian hypergeometric function 2F1() is a polynomial in n because at least one of the "numerators" is a negative integer. 2F1( [(1-n)/2,-n/2], [1], 64) = A098441(n). - _R. J. Mathar_, Jan 09 2013
%H G. C. Greubel, <a href="/A220852/b220852.txt">Table of n, a(n) for n = 0..460</a>
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1102.5649">List of conjectural series for powers of Pi and other constants</a>, arXiv:1102.5649 [math.CA], 2011-2014; Conjecture I1 page 24.
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1101.0600">On sums related to central binomial and trinomial coefficients</a>, arXiv:1101.0600 [math.NT], 2011-2014.
%F Sum_{n>=0} a(n)/A220853(n) = 24/Pi.
%F More directly, Sum_{k>=0} (30*k+7) * binomial(2k,k)^2 * (Hypergeometric2F1[1/2 - k/2, -k/2, 1,64])/(-256)^k = 24/Pi.
%F Another version of this identity is Sum_{k>=0} (30*k+7) * binomial(2k,k)^2 * (Sum_{m=0..k/2} binomial(k-m,m) * binomial(k,m) * 16^m)/(-256)^k.
%p A220852 := proc(n)
%p hypergeom([1/2-n/2,-n/2],[1],64) ;
%p simplify(%) ;
%p (30*n+7)*binomial(2*n,n)^2*%/(-256)^n ;
%p numer(%) ;
%p end proc: # _R. J. Mathar_, Jan 09 2013
%t Numerator[Table[(30*n + 7)*Binomial[2*n, n]^2* Hypergeometric2F1[(1 - n)/2, -n/2, 1, 64]/(-256)^n, {n, 0, 50}]] (* _G. C. Greubel_, Feb 20 2017 *)
%Y Cf. A098441, A132714, A220853.
%K sign,frac
%O 0,1
%A _Alexander R. Povolotsky_, Dec 23 2012
%E _R. J. Mathar_'s comment and data corrected by _G. C. Greubel_, Feb 20 2017
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