A064879 Triangle of numbers composed of certain generalized Catalan numbers.

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

Offset: 0

Author

Wolfdieter Lang, Oct 12 2001

Keywords

• nonn
• easy
• tabl

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.

Examples

			{1}; {1,1}; {0,1,1}; {0,2,1,1}; {0,5,4,1,1}; ...
		

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.