OFFSET
1,2
COMMENTS
First (k=0) column removed from Catalan triangle A009766(n,k).
In the Derrida et al. 1992 reference this triangle, called here X(alpha=1,beta=1;k=1,n,m), n >= m >= 1, is called there X_{N=n}(K=1,p=m) with alpha=1 and beta=1.
The column sequences give A000027 (natural numbers), A000096, A005586, A005587, A005557, A064059, A064061 for m=1..7. The numerator polynomials for the o.g.f. of column m is found in A062991 and the denominator is (1-x)^(m+1).
The diagonal sequences are convolutions of the Catalan numbers A000108, starting with the main diagonal.
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.
LINKS
FORMULA
a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n<m.
EXAMPLE
[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jan 13 2006
STATUS
approved