A064341 Generalized Catalan numbers C(3,3; n).

1, 1, 6, 81, 1566, 36126, 921456, 25055001, 711951606, 20891575566, 628237506276, 19259213633226, 599654171202156, 18911332670183856, 602840023457208516, 19392890824608619401, 628769286622411762086

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

Crossrefs

Cf. A064340.

Formula

a(n) = ((9^(n-1))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/3)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).

G.f.: (1-5*x*c(9*x))/(1-3*x*c(9*x))^2 = c(9*x)*(5+4*c(9*x))/(1+2*c(9*x))^2 = (5*c(9*x)*(3*x)^2+4*(1+4*x))/(2+3*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.

2*(-n+1)*a(n) +3*(23*n-60)*a(n-1) +54*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017