A130579 - OEIS

%I #7 Nov 12 2024 12:07:01

%S 1,2,6,22,92,423,2087,10856,58765,327877,1872490,10890483,64267612,

%T 383773529,2314271146,14071475748,86165249745,530862665988,

%U 3288219482754,20464419717069,127901478759153,802421158028657

%N 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)].

%H Robert Israel, <a href="/A130579/b130579.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%p 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:

%p map(f, [$0..25]); # _Robert Israel_, Nov 12 2024

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

%Y Cf. A000108, A001764.

%K nonn,changed

%O 0,2

%A _Paul D. Hanna_, Jun 07 2007