A064347 Generalized Catalan numbers C(9,9; n).
1, 1, 18, 1701, 278478, 56542698, 12838905972, 3121895416077, 795077021525526, 209364566760439038, 56540432581528153788, 15573764062988183490786, 4358381303784085630372620, 1235729432868053981694246324
Offset: 0
Links
- J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.
Crossrefs
Formula
a(n)= ((9^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/9)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-17*x*c(81*x))/(1-9*x*c(81*x))^2 = c(81*x)*(17+64*c(81*x))/(1+8*c(81*x))^2 = (17*c(81*x)*(9*x)^2+16*(4+13*x))/(8+9*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
8*(-n+1)*a(n) +9*(287*n-720)*a(n-1) +1458*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017
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