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A064340 Generalized Catalan numbers C(2,2; n). 10
1, 1, 4, 28, 256, 2704, 31168, 380608, 4840960, 63458560, 851399680, 11635096576, 161396604928, 2266669453312, 32166082822144, 460531091685376, 6644185553305600, 96498260064403456, 1409750653282287616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..18.

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

FORMULA

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

CROSSREFS

A000108 (Catalan as C(1, 1, n)).

Sequence in context: A230640 A300050 A191801 * A002895 A294189 A141004

Adjacent sequences:  A064337 A064338 A064339 * A064341 A064342 A064343

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 12 2001

STATUS

approved

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Last modified November 13 21:26 EST 2019. Contains 329106 sequences. (Running on oeis4.)