A385438 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k).
1, 10, 208, 4888, 121132, 3092950, 80506684, 2123780536, 56581885468, 1518936682888, 41021505946468, 1113273696074968, 30335161535834212, 829405495046080612, 22742967214283811976, 625193974445825554408, 17223870801864911429404, 475423918887141016417144
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..689
Programs
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PARI
a(n) = sum(k=0, n, 3^k*binomial(3*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^4/(1-2*x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^3 / (1+x)^4 ). See A385475.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k).
a(n) = (-2)^(-3*n) - 3^n*binomial(4*n-1, n)*(hypergeom([1, 4*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(8*n + 1/2) / (11 * 3^(2*n - 3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025