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 A064879 Triangle of numbers composed of certain generalized Catalan numbers. 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The column sequences (without leading zeros) for m=0..10 give: A019590, A000108, A064340-7, A064878. Row sums give A064880. 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. REFERENCES 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. LINKS W. Lang: First 10 rows. FORMULA 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. EXAMPLE {1}; {1,1}; {0,1,1}; {0,2,1,1}; {0,5,4,1,1}; ... CROSSREFS Sequence in context: A122049 A238802 A229892 * A173591 A343320 A156603 Adjacent sequences:  A064876 A064877 A064878 * A064880 A064881 A064882 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Oct 12 2001 STATUS approved

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Last modified May 17 21:11 EDT 2021. Contains 343990 sequences. (Running on oeis4.)