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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. 2
1, 64, 16384, 1048576, 1073741824, 68719476736, 17592186044416, 1125899906842624, 4611686018427387904, 295147905179352825856, 75557863725914323419136, 4835703278458516698824704, 4951760157141521099596496896, 316912650057057350374175801344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Sum_{n>=0} A220852(n)/A220853(n) = 24/Pi.

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,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

NMBRTHRY, Re: A conjectural series for 1/pi of a new type

Zhi-Wei Sun, List of conjectural series for powers of pi, Conjecture I1 page 19.

FORMULA

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

Conjectures:

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

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

a(n) = 4^(log2(16^n/((n/2) + (1/2) + (sum_{k=0,n} (-(-1)^(binomial(n,k)))/2)))). (end)

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 *)

CROSSREFS

Cf. A061549, A220852, A132714, A120738.

Sequence in context: A264058 A123051 A064068 * A013781 A330482 A187407

Adjacent sequences:  A220850 A220851 A220852 * A220854 A220855 A220856

KEYWORD

nonn,frac

AUTHOR

Alexander R. Povolotsky, Dec 23 2012

EXTENSIONS

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

STATUS

approved

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Last modified August 4 20:00 EDT 2020. Contains 336202 sequences. (Running on oeis4.)