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A220853 Denominators 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.

%I A220853 #84 Aug 15 2025 03:50:42
%S A220853 1,64,16384,1048576,1073741824,68719476736,17592186044416,
%T A220853 1125899906842624,4611686018427387904,295147905179352825856,
%U A220853 75557863725914323419136,4835703278458516698824704,4951760157141521099596496896,316912650057057350374175801344
%N A220853 Denominators 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 A220853 From _Alexander R. Povolotsky_, Jan 25 2013: (Start)
%C A220853 Sum_{n>=0} A220852(n)/a(n) = 24/Pi.
%C A220853 In more direct way, Sum_{k>=0} ((30*k+7) * binomial(2*k,k)^2 * (2F1([1/2 - k/2, -k/2], [1], 64))/(-256)^k) = 24/Pi.
%C A220853 Another version of this identity is: Sum_{k>=0} ((30*k+7) * binomial(2*k,k)^2 * (Sum_{m=0..floor(k/2)} (binomial(k-m,m) * binomial(k,m) * 16^m))/(-256)^k) = 24/Pi. (End)
%H A220853 G. C. Greubel, <a href="/A220853/b220853.txt">Table of n, a(n) for n = 0..415</a>
%H A220853 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 A220853 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 A220853 Conjectures from _Alexander R. Povolotsky_, Feb 27 2013: (Start)
%F A220853 a(n) = (A061549(n))^2.
%F A220853 a(n) = 4^A120738(n).
%F A220853 a(n) = 4^(log_2(16^n/((n/2) + (1/2) + (Sum_{k=0..n} (-(-1)^(binomial(n,k)))/2)))). (End)
%p A220853 A220853 := proc(n)
%p A220853     hypergeom([1/2-n/2,-n/2],[1], 64) ;
%p A220853     simplify(%) ;
%p A220853     (30*n+7)*binomial(2*n,n)^2*%/(-256)^n ;
%p A220853     denom(%) ;
%p A220853 end proc: # _R. J. Mathar_, Jan 09 2013
%t A220853 Denominator[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 A220853 Cf. A061549, A220852, A132714, A120738.
%K A220853 nonn,frac
%O A220853 0,2
%A A220853 _Alexander R. Povolotsky_, Dec 23 2012
%E A220853 Wrong conjecture removed by _R. J. Mathar_, Jun 17 2016