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

0,8

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

Table of n, a(n) for n=0..53.

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