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A188911 Binomial convolution of the binomial coefficients bin(3n,n) (A005809).

%I A188911 #20 Aug 24 2022 09:00:54
%S A188911 1,6,48,438,4356,46056,509106,5814738,68050116,811240872,9810384048,
%T A188911 119990105208,1481115683754,18421300391760,230574816629310,
%U A188911 2901721280735838,36688485233689668,465774244616805624,5934465567864915024
%N A188911 Binomial convolution of the binomial coefficients bin(3n,n) (A005809).
%H A188911 Vincenzo Librandi, <a href="/A188911/b188911.txt">Table of n, a(n) for n = 0..90</a>
%F A188911 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k).
%F A188911 E.g.f.: F(1/3,2/3;1/2,1;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.
%F A188911 Recurrence: 8*n^2 * (2*n-1)^2 * (9*n^3 - 54*n^2 + 102*n - 61)*a(n) = 24*(3*n-1)*(108*n^6 - 855*n^5 + 2628*n^4 - 4059*n^3 + 3380*n^2 - 1470*n + 264)*a(n-1) - 18*(3645*n^7 - 34992*n^6 + 138348*n^5 - 291843*n^4 + 352980*n^3 - 241794*n^2 + 84684*n - 11104)*a(n-2) + 2187*(n-2)^2 * (3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 21*n - 4)*a(n-3). - _Vaclav Kotesovec_, Feb 25 2014
%F A188911 a(n) ~ 3^(3*n+1) / (Pi * n * 2^(n+1)). - _Vaclav Kotesovec_, Feb 25 2014
%t A188911 Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k], {k,0,n}], {n,0,22}]
%o A188911 (Maxima) makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k),k,0,n),n,0,12);
%o A188911 (PARI) a(n)=sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k));
%o A188911 vector(66,n,a(n-1)) /* show terms */ /* _Joerg Arndt_, Apr 13 2011 */
%Y A188911 Cf. A005809, A001764, A006256, A006013, A045721, A188912, A188913.
%K A188911 nonn,easy
%O A188911 0,2
%A A188911 _Emanuele Munarini_, Apr 13 2011