A132714 -id:A132714 - cp's OEIS frontend

A220852 Numerators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.

7, -37, 19899, -235225, 268989175, -4985687133, 1052143756587, -25075299330081, 71491170131441775, -1979286926244381325, 319756423353994489291, -9700423363591011143001, 5919065321069316557189503, -189993537046726536185033125

Offset: 0

Alexander R. Povolotsky, Dec 23 2012

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

G. C. Greubel, Table of n, a(n) for n = 0..460
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.

Cf. A098441, A132714, A220853.

A220852 := proc(n)
    hypergeom([1/2-n/2,-n/2],[1],64) ;
    simplify(%) ;
    (30*n+7)*binomial(2*n,n)^2*%/(-256)^n ;
    numer(%) ;
end proc: # R. J. Mathar, Jan 09 2013

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 *)

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

R. J. Mathar's comment and data corrected by G. C. Greubel, Feb 20 2017

A220853 Denominators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.

Original entry on oeis.org

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

Offset: 0

Alexander R. Povolotsky, Dec 23 2012

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)

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.

Cf. A061549, A220852, A132714, A120738.

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

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 *)

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)

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

cp's OEIS Frontend

A220852 Numerators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A220853 Denominators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

cp's OEIS Frontend

A220852 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A220852 Numerators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.

A220853 Denominators of the fraction (30n+7) binomial(2n,n)^2 2F1([1/2 - n/2, -n/2], [1], 64)/(-256)^n, where 2F1 is the hypergeometric function.