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A386844 a(n) = Sum_{k=0..n} binomial(3*n+2,k) * binomial(3*n-k,n-k).

%I A386844 #14 Aug 08 2025 14:29:51
%S A386844 1,8,83,942,11177,136164,1688031,21187546,268409813,3424751568,
%T A386844 43948343243,566607282118,7333422759873,95225755205564,
%U A386844 1239995365588919,16186010348814258,211729232160358317,2774813844884684712,36425708310248816547,478880147399497482142,6304133921156502650777
%N A386844 a(n) = Sum_{k=0..n} binomial(3*n+2,k) * binomial(3*n-k,n-k).
%H A386844 Vincenzo Librandi, <a href="/A386844/b386844.txt">Table of n, a(n) for n = 0..350</a>
%F A386844 a(n) = [x^n] (1+x)^(3*n+2)/(1-x)^(2*n+1).
%F A386844 a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(2*n+1)).
%F A386844 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(3*n+2,k).
%F A386844 a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(2*n+k,k).
%F A386844 a(n) ~ 3^(3*n + 5/2) / (25 * sqrt(Pi*n) * 2^(n-1)). - _Vaclav Kotesovec_, Aug 07 2025
%t A386844 Table[Sum[Binomial[3*n+2,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 06 2025 *)
%o A386844 (PARI) a(n) = sum(k=0, n, binomial(3*n+2, k)*binomial(3*n-k, n-k));
%o A386844 (Magma) [&+[Binomial(3*n+2,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2025
%Y A386844 Cf. A386843, A386845.
%K A386844 nonn
%O A386844 0,2
%A A386844 _Seiichi Manyama_, Aug 05 2025