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A386938 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k-1,n-k).

%I A386938 #11 Aug 10 2025 10:34:16
%S A386938 1,6,57,608,6835,79170,934892,11189568,135263799,1647649850,
%T A386938 20191754297,248664799344,3074813151956,38151145101048,
%U A386938 474747568376520,5922579575399680,74047774139941503,927579860291591226,11639480787978105179,146278009406326705600,1840856649159814801515
%N A386938 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k-1,n-k).
%F A386938 a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^n.
%F A386938 a(n) = [x^n] 1/((1-x)^(2*n+2) * (1-2*x)^n).
%F A386938 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k+1,n-k).
%F A386938 a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(3*n-k+1,n-k).
%o A386938 (PARI) a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k-1, n-k));
%Y A386938 Cf. A386834, A386939.
%K A386938 nonn
%O A386938 0,2
%A A386938 _Seiichi Manyama_, Aug 10 2025