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A386920 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(3*n-k,n-k).

%I A386920 #19 Aug 21 2025 07:36:57
%S A386920 1,7,83,1102,15395,221402,3244430,48173244,722264355,10910288290,
%T A386920 165788618138,2531447611524,38807906496398,596945491933252,
%U A386920 9208704207465020,142410375212008952,2207122379129757987,34272045530904650610,533075544700619580002,8304126391210396590900
%N A386920 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(3*n-k,n-k).
%H A386920 Vincenzo Librandi, <a href="/A386920/b386920.txt">Table of n, a(n) for n = 0..350</a>
%F A386920 a(n) = [x^n] (1+x)^(4*n)/(1-x)^(2*n+1).
%F A386920 a(n) = [x^n] 1/((1-x)^n * (1-2*x)^(2*n+1)).
%F A386920 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(2*n-k-1,n-k).
%F A386920 a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(2*n-k-1,n-k).
%F A386920 a(n) ~ (2 + sqrt(2)) * 2^(4*n-2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 21 2025
%t A386920 Table[Sum[Binomial[4*n,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 10 2025 *)
%o A386920 (PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(3*n-k, n-k));
%o A386920 (Magma) [&+[Binomial(4*n,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 10 2025
%Y A386920 Cf. A066381, A386918, A386919.
%K A386920 nonn
%O A386920 0,2
%A A386920 _Seiichi Manyama_, Aug 08 2025