A383888 a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k).
1, 4, 34, 334, 3478, 37384, 409960, 4558306, 51199558, 579554056, 6600532684, 75546800476, 868224027916, 10012494936136, 115804853315332, 1342795688895754, 15604522381828678, 181690692393744376, 2119144763079629452, 24754486729805925124, 289563977079418497748
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..928
Programs
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PARI
a(n) = sum(k=0, n, 3^k*binomial(n+k-1,k));
Formula
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^2/(1-2*x) )^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) / (1+x)^2 ).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-2)^(-n)*(1 - (-6)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 3) - 1)). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(2*n) * 3^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025