A064340 Generalized Catalan numbers C(2,2; n).

1, 1, 4, 28, 256, 2704, 31168, 380608, 4840960, 63458560, 851399680, 11635096576, 161396604928, 2266669453312, 32166082822144, 460531091685376, 6644185553305600, 96498260064403456, 1409750653282287616, 20702370737659052032, 305428492830594039808

Offset: 0

Author

Wolfdieter Lang, Oct 12 2001

Keywords

• nonn
• easy

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

• J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.

Crossrefs

Cf. A000108 (Catalan as C(1,1; n)), A064879, A067298.

Programs

PARI

my(x='x+O('x^30)); Vec((1+(13-3*sqrt(1-16*x))*x/2)/(1+2*x)^2) \\ Jinyuan Wang, Apr 20 2025

Formula

a(n) = ((4^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1), n >= 2, a(0) = a(1) = 1.

G.f.: (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 = c(4*x)*(3+c(4*x))/(1+c(4*x))^2 = (1+5*x+3*c(4*x)*(2*x)^2)/(1+2*x)^2 with c(x) = A(x) g.f. of Catalan numbers A000108.

(-n+1)*a(n) + 2*(7*n-20)*a(n-1) + 16*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Aug 09 2017