login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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..floor(k/2)} (binomial(k-m,m) * binomial(k,m) * 16^m))/(-256)^k) = 24/Pi. (end)
LINKS
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
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)
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
Sequence in context: A264058 A123051 A064068 * A013781 A330482 A187407
KEYWORD
nonn,frac
AUTHOR
EXTENSIONS
Wrong conjecture removed by R. J. Mathar, Jun 17 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)