A386616 a(n) = Sum_{k=0..n-1} binomial(6*k+1,k) * binomial(6*n-6*k,n-k-1).

0, 1, 19, 315, 5000, 77785, 1196667, 18282742, 278031900, 4214278350, 63723788295, 961789682008, 14495501585664, 218216042892175, 3281961694927950, 49322417450239980, 740753733463215604, 11118981305235476010, 166821561372208253850, 2501861335268901337425, 37507747177968865536840

Offset: 0

Author

Seiichi Manyama, Jul 27 2025

Keywords

• nonn

Crossrefs

Cf. A079679, A386368, A386567, A386615.

Cf. A006419, A386612, A386614, A386617.

Programs

PARI

a(n) = sum(k=0, n-1, binomial(6*k+1, k)*binomial(6*n-6*k, n-k-1));

Formula

G.f.: g^3 * (g-1)/(6-5*g)^2 where g=1+x*g^6.

G.f.: g/((1-g)^2 * (1-6*g)^2) where g*(1-g)^5 = x.

a(n) = Sum_{k=0..n-1} binomial(6*k+1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.

a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n+2,k).

a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k+2,k).