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A385667 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

%I A385667 #15 Aug 05 2025 16:19:07
%S A385667 1,10,151,2542,44983,819160,15197404,285653350,5421341311,
%T A385667 103659081034,1993769491591,38532753357064,747680491747876,
%U A385667 14556620712375856,284217498703106224,5563106991308471062,109124768598722692111,2144648671343440349182
%N A385667 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
%H A385667 Vincenzo Librandi, <a href="/A385667/b385667.txt">Table of n, a(n) for n = 0..350</a>
%F A385667 a(n) = [x^n] (1+x)^(3*n+1)/(1-2*x)^(2*n+1).
%F A385667 a(n) = [x^n] 1/((1-x) * (1-3*x)^(2*n+1)).
%F A385667 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+1,k).
%F A385667 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k,k).
%t A385667 Table[Sum[2^(n-k)*Binomial[3*n+1,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 05 2025 *)
%o A385667 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));
%o A385667 (Magma) [&+[2^(n-k) * Binomial(3*n+1,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 05 2025
%Y A385667 Cf. A384950.
%K A385667 nonn
%O A385667 0,2
%A A385667 _Seiichi Manyama_, Aug 04 2025