A386899 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
1, 16, 339, 7840, 189295, 4689216, 118155156, 3013479744, 77557234095, 2010176842960, 52394920516939, 1371957494204544, 36062378503314436, 950984592573500800, 25147592297769065400, 666594977732384307840, 17706778517771676847215, 471217399398861925667760
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*n+1, k)*binomial(2*n-k, n-k));
Formula
a(n) = [x^n] (1+3*x)^(3*n+1)/(1-2*x)^(n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x))^(n+1).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k,k) * binomial(2*n-k,n-k).
a(n) = 2^n*binomial(2*n, n)*hypergeom([-1-3*n, -n], [-2*n], -3/2). - Stefano Spezia, Aug 07 2025