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

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1200

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).

From Seiichi Manyama, May 06 2026: (Start)

G.f.: x*(B(x)/x)^2 where B(x) is the g.f. of A025174.

a(n+1) = (1/n) * Sum_{k=0..n-1} (k+1) * (3*k+10) * 2^k * binomial(3*n+5,n-1-k) for n > 0.

a(n+1) = (1/n) * Sum_{k=0..n-1} (k+1) * (2*k+10) * 3^k * binomial(3*n+3-k,n-1-k) for n > 0. (End)

D-finite with recurrence: 243*(3*n + 5)*(3*n + 4)*a(n) - 9*(153*n^2 + 756*n + 944)*a(n + 1) + 6*(44*n^2 + 279*n + 444)*a(n + 2) - 8*(n + 4)*(2*n + 7)*a(n + 3) = 0. - Robert Israel, May 06 2026

Maple

f:= gfun:-rectoproc({243*(3*n + 5)*(3*n + 4)*a(n) - 9*(153*n^2 + 756*n + 944)*a(n + 1) + 6*(44*n^2 + 279*n + 444)*a(n + 2) - 8*(n + 4)*(2*n + 7)*a(n + 3), a(0) = 0, a(1) = 1, a(2) = 10}, a(n), remember):
map(f, [0..30]); # Robert Israel, May 06 2026

Prog

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

Crossrefs

Cf. A006256, A036829, A062236, A075045.

Cf. A006419, A386612, A386614, A386616.

Cf. A386368, A386566, A386611.

Cf. A025174, A395783, A395785.

Sequence in context: A136870 A293367 A018202 * A154236 A095004 A037541

Adjacent sequences: A386614 A386615 A386616 * A386618 A386619 A386620

Keyword

nonn

Author

Seiichi Manyama, Jul 27 2025

Status

approved