%I #14 Aug 24 2022 09:28:07

%S 1,2,8,42,260,1816,13962,116094,1029124,9609144,93569808,942642696,

%T 9763181946,103455616400,1117379189926,12264816349938,136501928050116,

%U 1537591374945704,17503603786398576,201128739609458904,2330480521265639136

%N Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

%H Vincenzo Librandi, <a href="/A188912/b188912.txt">Table of n, a(n) for n = 0..93</a>

%F a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k)/((2*k+1)*(2*n-2*k+1)).

%F E.g.f.: F(1/3,2/3;1,3/2;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.

%F From _Vaclav Kotesovec_, Jun 10 2019: (Start)

%F Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(324*n^7-2187*n^6+4689*n^5-4185*n^4+1464*n^3+122*n^2-223*n+44)*a(n-1) - 18*(n-1)*(3645*n^7-30618*n^6+96066*n^5-144585*n^4+103662*n^3-21834*n^2-10860*n+4480)*a(n-2) + 2187*(n-2)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).

%F a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

%t Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1)Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

%o (Maxima) makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

%Y Cf. A005809, A001764, A005809, A006256, A006013, A045721, A188911, A188913

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 13 2011