A386617 a(n) = Sum_{k=0..n-1} binomial(3*k+1,k) * binomial(3*n-3*k,n-k-1).

0, 1, 10, 81, 610, 4436, 31626, 222681, 1554772, 10790721, 74560728, 513452604, 3526463304, 24168921568, 165357919850, 1129724254953, 7709039995368, 52551835079699, 357930487932282, 2436038623348521, 16568626556643738, 112626521811112464, 765201654587796312, 5196570956399432796

Offset: 0

Author

Seiichi Manyama, Jul 27 2025

Keywords

• nonn

Crossrefs

Cf. A006256, A036829, A062236, A075045.

Cf. A006419, A386612, A386614, A386616.

Programs

PARI

a(n) = sum(k=0, n-1, binomial(3*k+1, k)*binomial(3*n-3*k, n-k-1));

Formula

G.f.: g^3 * (g-1)/(3-2*g)^2 where g=1+x*g^3.

G.f.: g/((1-g)^2 * (1-3*g)^2) where g*(1-g)^2 = x.

a(n) = Sum_{k=0..n-1} binomial(3*k+1+l,k) * binomial(3*n-3*k-l,n-k-1) for every real number l.

a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(3*n+2,k).

a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(2*n+k+2,k).