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.

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

Offset

0,1

Comments

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

Links

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.

Formula

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.

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A098441, A132714, A220853.

Sequence in context: A078303 A127729 A129736 * A290176 A003352 A346278

Adjacent sequences: A220849 A220850 A220851 * A220853 A220854 A220855

Keyword

sign,frac

Author

Alexander R. Povolotsky, Dec 23 2012

Extensions

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

Status

approved