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A220853
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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.
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2
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1, 64, 16384, 1048576, 1073741824, 68719476736, 17592186044416, 1125899906842624, 4611686018427387904, 295147905179352825856, 75557863725914323419136, 4835703278458516698824704, 4951760157141521099596496896, 316912650057057350374175801344
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OFFSET
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0,2
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COMMENTS
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In more direct way, Sum_{k>=0} ((30*k+7) * binomial(2k,k)^2 * (2F1([1/2 - k/2, -k/2], [1], 64))/(-256)^k) = 24/Pi.
Another version of this identity is: Sum_{k>=0} ((30*k+7) * binomial(2k,k)^2 * (Sum_{m=0..floor(k/2)} (binomial(k-m,m) * binomial(k,m) * 16^m))/(-256)^k) = 24/Pi. (end)
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LINKS
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FORMULA
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a(n) = 4^(log_2(16^n/((n/2) + (1/2) + (Sum_{k=0..n} (-(-1)^(binomial(n,k)))/2)))). (End)
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MAPLE
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hypergeom([1/2-n/2, -n/2], [1], 64) ;
simplify(%) ;
(30*n+7)*binomial(2*n, n)^2*%/(-256)^n ;
denom(%) ;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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