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A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

%I A385671 #13 Aug 04 2025 08:03:36
%S A385671 1,22,774,30458,1260886,53731512,2333065354,102643195068,
%T A385671 4559878830006,204091261040552,9189096061165784,415734554486178378,
%U A385671 18884084064916032026,860673634902720476392,39339618388269633525564,1802605962076744803396888,82777622289467318635747446
%N A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
%F A385671 a(n) = [x^n] (1+2*x)^(4*n+1)/(1-3*x)^(3*n+1).
%F A385671 a(n) = [x^n] 1/((1-2*x) * (1-5*x)^(3*n+1)).
%F A385671 a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n+1,k).
%F A385671 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k,k).
%o A385671 (PARI) a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));
%Y A385671 Cf. A016127, A385669, A385670.
%Y A385671 Cf. A384366, A386768.
%K A385671 nonn
%O A385671 0,2
%A A385671 _Seiichi Manyama_, Aug 04 2025