A130579 Convolution of A000108 (Catalan numbers) and A001764 (ternary trees): a(n) = Sum_{k=0..n} C(2k,k) * C(3(n-k),n-k) / [(k+1)(2(n-k)+1)].

1, 2, 6, 22, 92, 423, 2087, 10856, 58765, 327877, 1872490, 10890483, 64267612, 383773529, 2314271146, 14071475748, 86165249745, 530862665988, 3288219482754, 20464419717069, 127901478759153, 802421158028657

Offset: 0

Author

Paul D. Hanna, Jun 07 2007

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A001764.

Programs

Maple

f:= proc(n) local k; add(binomial(2*k, k)/(k+1)*binomial(3*(n-k), n-k)/(2*(n-k)+1),k=0..n) end proc:
map(f, [$0..25]); # Robert Israel, Nov 12 2024

PARI

a(n)=sum(k=0,n,binomial(2*k,k)/(k+1)*binomial(3*(n-k),n-k)/(2*(n-k)+1))

Formula

G.f.: A(x) = C(x)*T(x) where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

a(n) ~ 3^(3*n+2) / ((3^(3/2) + sqrt(11)) * sqrt(Pi) * n^(3/2) * 2^(2*n+1)). - Vaclav Kotesovec, Nov 12 2024