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A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

%I A188913 #12 Jun 10 2019 04:51:47
%S A188913 1,4,24,168,1300,10896,97734,928752,9262116,96091440,1029267888,
%T A188913 11311712352,126921365298,1448378629600,16760687848890,
%U A188913 196237061599008,2320532776851972,27676644749022672,332568471941572944,4022574792189178080
%N A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).
%H A188913 Vincenzo Librandi, <a href="/A188913/b188913.txt">Table of n, a(n) for n = 0..105</a>
%F A188913 a(n) = sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k=0..n).
%F A188913 E.g.f.: F(1/3,2/3;1/2,1;27*x/4)*F(1/3,2/3;1,3/2;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series.
%F A188913 Recurrence: 8*n^2 * (2*n+1)^2 * (9*n^3 - 54*n^2 + 84*n - 35)*a(n) = 24*(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*(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)*(n-1)*(3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 3*n + 4)*a(n-3). - _Vaclav Kotesovec_, Feb 25 2014
%F A188913 a(n) ~ 3^(3*n+1) / (Pi*n^2*2^(n+1)). - _Vaclav Kotesovec_, Feb 25 2014
%t A188913 Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]
%o A188913 (Maxima) makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);
%o A188913 (PARI) a(n) = sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1));
%o A188913 vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */
%Y A188913 Cf. A005809, A001764, A006256, A006013, A045721, A188911, A188912.
%K A188913 nonn,easy
%O A188913 0,2
%A A188913 _Emanuele Munarini_, Apr 13 2011