A384950 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k).
1, 7, 103, 1720, 30319, 550867, 10204660, 191606380, 3633593071, 69434167357, 1334845289023, 25787841299392, 500217562201348, 9736067678711524, 190051513661403112, 3719197868485767940, 72942019051301120239, 1433317465944902210161, 28212929859612197439829
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..766
Programs
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PARI
a(n) = sum(k=0, n, 3^k*binomial(2*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^3/(1-2*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 * (1-2*x)^2 / (1+x)^3 ). See A385474.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = 4^(-n) - 3^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 3^(4*n + 3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025