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Generalized Catalan numbers C(2,2; n).
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%I #8 Aug 09 2017 12:27:42

%S 1,1,4,28,256,2704,31168,380608,4840960,63458560,851399680,

%T 11635096576,161396604928,2266669453312,32166082822144,

%U 460531091685376,6644185553305600,96498260064403456,1409750653282287616

%N Generalized Catalan numbers C(2,2; n).

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

%H J. Abate, W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.

%F 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).

%F 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.

%F (-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

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

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 12 2001