A277170 - OEIS

%I #48 Oct 22 2016 12:41:15

%S 1,-1,1,-1,1,-3,1,-1,25,-1,1,-49,1,-1,9,-3,1,-363,3025,-1,169,-169,1,

%T -3,1,-49,289,-289,7225,-361,361,-361,1,-1,1,-529,529,-529,330625,

%U -148225,3025,-675,9,-3,7569,-2523,142129,-409757907,808201,-961,8649,-2883,1,-147

%N Numerator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

%C Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

%D Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

%H Seiichi Manyama, <a href="/A277170/b277170.txt">Table of n, a(n) for n = 0..1000</a>

%F (s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = a(n) / A277520(n).

%F s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.

%F s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

%t a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Numerator; Table[a[n], {n, 0, 53}] (* _Jean-François Alcover_, Oct 22 2016 *)

%Y Cf. A005810, A066802, A052203, A187364, A277520.

%K sign,frac

%O 0,6

%A _Seiichi Manyama_, Oct 19 2016