A064344 Generalized Catalan numbers C(6,6; n).

1, 1, 12, 540, 39744, 3598992, 363776832, 39348690624, 4456429954560, 521760612125952, 62642882007530496, 7670452375558388736, 954216689151845302272, 120261048050627578368000

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

A064343.

Formula

a(n)= ((6^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/6)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).

G.f.:(1-11*x*c(36*x))/(1-6*x*c(36*x))^2 = c(36*x)*(11+25*c(36*x))/(1+5*c(36*x))^2 = (11*c(36*x)*(6*x)^2+5*(5+17*x))/(5+6*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.

5*(-n+1)*a(n) +6*(119*n-300)*a(n-1) +432*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017