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%I #9 Aug 28 2019 16:36:09
%S 1,1,1,0,1,1,0,2,1,1,0,5,4,1,1,0,14,28,6,1,1,0,42,256,81,8,1,1,0,132,
%T 2704,1566,176,10,1,1,0,429,31168,36126,5888,325,12,1,1,0,1430,380608,
%U 921456,238848,16750,540,14,1
%N Triangle of numbers composed of certain generalized Catalan numbers.
%C The column sequences (without leading zeros) for m=0..10 give: A019590, A000108, A064340-7, A064878. Row sums give A064880.
%C The sequence for column m (m >= 1) (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for alpha = beta = m. 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 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 T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.
%H W. Lang: <a href="/A064879/a064879.txt">First 10 rows.</a>
%F a(n, m) = C(m, m; n-m) if n >= m, else 0, with C(m, m; n) := ((m^(2*(n-1)))/(n-1))*sum((k+1)*(k+2)*binomial(2*(n-2)-k, n-2-k)*((1/m)^(k+1)), k=0..n-2), n >= 2; C(m, m; 0) := 1=:C(m, m; 1).
%F G.f.: (x^m)*(1+(1-2*m)*x*c(x*m^2))/(1-m*x*c(x*m^2))^2 = (x^m)*((2*m-1)*c(x*m^2)*(m*x)^2 +(1-m)*(1-m+(1-3*m)*x))/(1-m-m*x)^2, m >= 0. For m >= 1 also: (x^m)*c(x*m^2)*(2*m-1+c(x*m^2)*(m-1)^2)/(1+(m-1)*c(x*m^2))^2.
%F In the G.f. the g.f. c(x) of A000108 (Catalan) appears.
%e {1}; {1,1}; {0,1,1}; {0,2,1,1}; {0,5,4,1,1}; ...
%K nonn,easy,tabl
%O 0,8
%A _Wolfdieter Lang_, Oct 12 2001
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