A188681 a(n) = binomial(3*n,n)^2/(2*n+1).

1, 3, 45, 1008, 27225, 819819, 26509392, 901402560, 31818681873, 1156122556875, 42985853635725, 1628541825580800, 62667882587091600, 2443473892345873968, 96351855806554401600, 3836565846094702507776, 154071018890153214025473

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..606
• Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188682, A188683, A188684, A188685, A188686, A188687.

Programs

Mathematica

Table[Binomial[3k,k]^2/(2k+1),{k,0,20}]
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 *)

Maxima

makelist(binomial(3*k,k)^2/(2*k+1),k,0,20);

Formula

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.

a(n) ~ 3^(6*n+1)/(Pi*2^(4*n+3)*n^2). - Vaclav Kotesovec, Aug 16 2013