A355347 a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1) for n >= 0.

1, 7, 44, 275, 1729, 10948, 69768, 447051, 2877875, 18599295, 120609840, 784384692, 5114119724, 33417386200, 218786861392, 1434903854139, 9425348845815, 61997934676405, 408323057257500, 2692322893972635, 17770644483690945, 117406930477134480, 776363580147660960

Offset: 0

Author

Paul D. Hanna, Jul 25 2022

Keywords

• nonn

Comments

A355345(2*n*(n+1)) = (-1)^n * a(n) for n >= 1.

Limit_{n->oo} a(n)/a(n+1) = 4/27.

Examples

			G.f.: A(x) = 1 + 7*x + 44*x^2 + 275*x^3 + 1729*x^4 + 10948*x^5 + 69768*x^6 + 447051*x^7 + 2877875*x^8 + 18599295*x^9 + 120609840*x^10 + ...
such that
A(x) = G(x)^3 * (1 + x*G(x)^2) / (1 - 3*x*G(x)^2)
where G(x) = 1 + x*G(x)^3 begins
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + 246675*x^9 + 1430715*x^10 + ... + A001764(n)*x^n + ...
		

Crossrefs

Cf. A001764, A355345.

Programs

PARI

{a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1)}
for(n=0,22,print1(a(n),", "))

Formula

G.f.: G(x)^3 * (1 + x*G(x)^2) / (1 - 3*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

G.f.: G'(x) * (1 + x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

a(n) = [x^n] (1+x)/(1-x)^(2*n+4) for n >= 0.