A386763 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k-1,k).
1, 8, 114, 1862, 32246, 576768, 10529544, 194960802, 3647285766, 68772760928, 1304858513324, 24882531221292, 476462691535436, 9155397868559288, 176447193966483204, 3409285356643013082, 66020593061854488006, 1280989373915746600848, 24897996624141835608684
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, 5^k*3^(n-k)*binomial(n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+3*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+3*x)^2 ). See A386769.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-9/2)^n*(1 - (-10/9)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 5/3) - 1)). - Stefano Spezia, Aug 02 2025