A068765 Generalized Catalan numbers 3*x*A(x)^2 -A(x) +1 -2*x = 0.

1, 1, 6, 39, 270, 1962, 14796, 114831, 911574, 7368894, 60457428, 502162902, 4214515212, 35686162548, 304491863448, 2615468845311, 22598114065254, 196269877811574, 1712578870493316, 15005719955119698

Offset: 0

Author

Wolfdieter Lang, Mar 04 2002

Keywords

• nonn
• easy

Comments

a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.

Links

• Fung Lam, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A068764, A068766-72, A025227-30.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Formula

a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.

a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).

Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014