A386764 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(2*n+k-1,k).
1, 13, 319, 8872, 260511, 7885793, 243404884, 7615561092, 240662849871, 7663737420223, 245529092332599, 7904950462600512, 255541233005365956, 8289112264436610828, 269663237466343607464, 8794852773491081069132, 287467221911677590185391, 9414259968096351504747483
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, 5^k*3^(n-k)*binomial(2*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+3*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+3*x)^3 ). See A386770.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = (27/4)^n - 5^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 5/3) - 1). - Stefano Spezia, Aug 02 2025