A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

1, 2, 2, 3, 5, 5, 4, 9, 14, 14, 5, 14, 28, 42, 42, 6, 20, 48, 90, 132, 132, 7, 27, 75, 165, 297, 429, 429, 8, 35, 110, 275, 572, 1001, 1430, 1430, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 16796, 11, 65, 273, 910

Offset: 1

Author

Wolfdieter Lang, Jan 13 2006

Keywords

• nonn
• easy
• tabl

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.

Examples

			[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.
		

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

• W. Lang: First 10 rows.

Crossrefs

Row sums give A001453(n+1)=A000108(n+1)-1 (Catalan -1).

Formula

a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n

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