A064342 Generalized Catalan numbers C(4,4; n).

1, 1, 8, 176, 5888, 238848, 10770432, 518909952, 26156466176, 1362414338048, 72751723839488, 3961437637574656, 219123329636761600, 12278352550322765824, 695492547259800748032, 39759203500044029263872

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

A064341.

Formula

a(n)= ((16^(n-1))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/4)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).

G.f.:(1-7*x*c(16*x))/(1-4*x*c(16*x))^2 = c(16*x)*(7+9*c(16*x))/(1+3*c(16*x))^2 = (7*c(16*x)*(4*x)^2+3*(3+11*x))/(3+4*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.

3*(-n+1)*a(n) +4*(47*n-120)*a(n-1) +128*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017