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%I #22 May 13 2022 19:26:16
%S 1,3,45,1008,27225,819819,26509392,901402560,31818681873,
%T 1156122556875,42985853635725,1628541825580800,62667882587091600,
%U 2443473892345873968,96351855806554401600,3836565846094702507776,154071018890153214025473
%N a(n) = binomial(3n,n)^2/(2n+1).
%H Seiichi Manyama, <a href="/A188681/b188681.txt">Table of n, a(n) for n = 0..606</a>
%H Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%F Recurrence: 4*(n+1)^2*(2*n+1)*(2*n+3)*a(n+1)-9*(3*n+1)^2*(3*n+2)^2*a(n)=0.
%F a(n) ~ 3^(6*n+1)/(Pi*2^(4*n+3)*n^2). - _Vaclav Kotesovec_, Aug 16 2013
%t Table[Binomial[3k,k]^2/(2k+1),{k,0,20}}.
%t CoefficientList[Series[HypergeometricPFQ[{1/3,1/3,2/3,2/3}, {1/2,1,3/2}, (729 x)/16],{x,0,20}],x] (* _Harvey P. Dale_, Apr 22 2011 *)
%o (Maxima) makelist(binomial(3*k,k)^2/(2*k+1),k,0,20);
%Y Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188682, A188683, A188684, A188685, A188686, A188687.
%K nonn,easy
%O 0,2
%A _Emanuele Munarini_, Apr 08 2011
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