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.

1, 64, 16384, 1048576, 1073741824, 68719476736, 17592186044416, 1125899906842624, 4611686018427387904, 295147905179352825856, 75557863725914323419136, 4835703278458516698824704, 4951760157141521099596496896, 316912650057057350374175801344

Offset: 0

Author

Alexander R. Povolotsky, Dec 23 2012

Keywords

• nonn
• frac

Comments

From Alexander R. Povolotsky, Jan 25 2013: (Start)

Sum_{n>=0} A220852(n)/a(n) = 24/Pi.

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.

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)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..415
• Zhi-Wei Sun, List of conjectural series for powers of Pi and other constants, arXiv:1102.5649 [math.CA], 2011-2014; Conjecture I1 page 24.
• Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, arXiv:1101.0600 [math.NT], 2011-2014.

Crossrefs

Cf. A061549, A220852, A132714, A120738.

Programs

Maple

A220853 := proc(n)
    hypergeom([1/2-n/2,-n/2],[1], 64) ;
    simplify(%) ;
    (30*n+7)*binomial(2*n,n)^2*%/(-256)^n ;
    denom(%) ;
end proc: # R. J. Mathar, Jan 09 2013

Mathematica

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 *)

Formula

Conjectures from Alexander R. Povolotsky, Feb 27 2013: (Start)

a(n) = (A061549(n))^2.

a(n) = 4^A120738(n).

a(n) = 4^(log_2(16^n/((n/2) + (1/2) + (Sum_{k=0..n} (-(-1)^(binomial(n,k)))/2)))). (End)

Extensions

Wrong conjecture removed by R. J. Mathar, Jun 17 2016