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%I #11 Aug 09 2017 12:10:03
%S 1,1,5,49,653,10201,174965,3188641,60623645,1189050025,23884139525,
%T 488869387249,10159948737581,213822249696121,4547793322315157,
%U 97600834975487809,2110916340429978173
%N Generalized Catalan numbers C(2,3;n)=C(3,2;n).
%C This sequence appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for alpha =2, beta = 3 (or alpha=3, beta=2). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1. See also the Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.
%D B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%D B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.
%D G. Schuetz and E. Domany, Phase Transitions in an Exactly Soluble one-Dimensional Exclusion Process, J. Stat. Phys. 72 (1993) 277-295, eq. (2.18), p. 283, with eqs. (2.13)-(2.15).
%D T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.
%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, example section 4.
%F G.f.: (1+5*x+12*x^2*c(6*x))/((1+3*x)*(1+x)) with the g.f. c(x) for A000108 (Catalan numbers).
%F a(n)=((-1)^(n+1))*(3^n-2)+6*sum(((-1)^k)*C(n-2-k)*6^(n-2-k)*(3^(k+1)-1),k=0..n-2), with C(n):=A000108(n) (Catalan numbers) and the sum is replaced by 0 for n=0,1. Proof from the g.f. after partial fraction decomposition. [_Wolfdieter Lang_, May 05 2006]
%F (1-n)*a(n) +4*(5*n-14)*a(n-1) +3*(31*n-79)*a(n-2) +36*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Aug 06 2013
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 24 2006
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