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A064879
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Triangle of numbers composed of certain generalized Catalan numbers.
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11
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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, 2704, 1566, 176, 10, 1, 1, 0, 429, 31168, 36126, 5888, 325, 12, 1, 1, 0, 1430, 380608, 921456, 238848, 16750, 540, 14, 1
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OFFSET
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0,8
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COMMENTS
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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.
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REFERENCES
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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.
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.
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.
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LINKS
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FORMULA
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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).
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.
In the G.f. the g.f. c(x) of A000108 (Catalan) appears.
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EXAMPLE
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{1}; {1,1}; {0,1,1}; {0,2,1,1}; {0,5,4,1,1}; ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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