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A386867 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+2,k) * binomial(4*n-k-1,n-k).

%I A386867 #8 Aug 06 2025 08:39:30
%S A386867 1,12,249,5842,144636,3690840,96028606,2532467934,67454242092,
%T A386867 1810467982144,48887478311673,1326582594222918,36143786784056716,
%U A386867 988134308856642048,27093384379207568028,744735869371387679158,20516019688758402141372,566266846186568482197840
%N A386867 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+2,k) * binomial(4*n-k-1,n-k).
%F A386867 a(n) = [x^n] (1+x)^(4*n+2)/(1-2*x)^(3*n).
%F A386867 a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^(3*n)).
%F A386867 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+2,k) * binomial(n-k+2,n-k).
%F A386867 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k) * binomial(n-k+2,n-k).
%o A386867 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+2, k)*binomial(4*n-k-1, n-k));
%Y A386867 Cf. A385438, A386864.
%Y A386867 Cf. A386837.
%K A386867 nonn
%O A386867 0,2
%A A386867 _Seiichi Manyama_, Aug 06 2025