Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #33 Nov 05 2016 14:16:49
%S 1,3,25,147,1089,20449,48841,312987,55190041,14322675,100100025,
%T 32065374675,4546130625,29873533563,1859904071089,4089135109921,
%U 9399479144449,22568149425822049,1293753708921104809,2835106739783283,3289668853728536041
%N Denominator 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="/A277520/b277520.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) = A277170(n) / a(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] // Denominator;
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 22 2016 *)
%Y Cf. A005810, A052203, A066802, A187364, A277170 (numerators).
%K nonn,frac
%O 0,2
%A _Seiichi Manyama_, Oct 19 2016