A385320 a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(3*n-k-1,n-k).
1, 8, 118, 1970, 34714, 630548, 11678284, 219240008, 4157096266, 79429466456, 1526869550638, 29495424821354, 572100064904872, 11134578632483600, 217341014671302976, 4253067310380772400, 83409477100625759050, 1638952453699219007072, 32259670449587082804466
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Keywords
Programs
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Mathematica
Table[Sum[3^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *) -
PARI
a(n) = sum(k=0, n, 2^k*binomial(3*n, k)*binomial(3*n-k-1, n-k));
Formula
a(n) = [x^n] ( (1+2*x)^3/(1-x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^2 / (1+2*x)^3 ). See A386722.
a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(3*n,k).
a(n) ~ 3^(4*n + 3/2) / (7*sqrt(Pi*n)*2^(2*n)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n+k-1,k). - Seiichi Manyama, Aug 01 2025