A370097 a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(3*n-k-1,n-k).
1, 5, 49, 545, 6401, 77505, 956929, 11976193, 151388161, 1928363009, 24712450049, 318255628289, 4115300220929, 53396370030593, 694845537386497, 9064787191660545, 118516719269445633, 1552528215946035201, 20372392543502991361, 267736366910401413121
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[2^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(3*n, k)*binomial(3*n-k-1, n-k));
Formula
a(n) = [x^n] ( (1+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+x)^3 ). See A365842.
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 31 2025
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k-1,k). - Seiichi Manyama, Aug 01 2025
a(n) = [x^n] 1/((1-x) * (1-2*x)^(2*n)). - Seiichi Manyama, Aug 09 2025