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A386864 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k-1,n-k).

%I A386864 #8 Aug 06 2025 08:39:36
%S A386864 1,11,228,5350,132476,3380955,87974188,2320223552,61804459260,
%T A386864 1658904186124,44796539697968,1215611557398534,33121179085639252,
%U A386864 905520072985022570,24828701435772435528,682496748843439692868,18801742541632193099996,518957827806105486222372
%N A386864 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k-1,n-k).
%F A386864 a(n) = [x^n] (1+x)^(4*n+1)/(1-2*x)^(3*n).
%F A386864 a(n) = [x^n] 1/((1-x)^2 * (1-3*x)^(3*n)).
%F A386864 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * (n-k+1) * binomial(4*n+1,k).
%F A386864 a(n) = Sum_{k=0..n} 3^k * (n-k+1) * binomial(3*n+k-1,k).
%o A386864 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k-1, n-k));
%Y A386864 Cf. A385438, A386867.
%K A386864 nonn
%O A386864 0,2
%A A386864 _Seiichi Manyama_, Aug 06 2025