%I #8 Aug 30 2019 03:37:50
%S 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,
%T 297,429,429,8,35,110,275,572,1001,1430,1430,9,44,154,429,1001,2002,
%U 3432,4862,4862,10,54,208,637,1638,3640,7072,11934,16796,16796,11,65,273,910
%N First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).
%C First (k=0) column removed from Catalan triangle A009766(n,k).
%C 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.
%C 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).
%C The diagonal sequences are convolutions of the Catalan numbers A000108, starting with the main diagonal.
%D 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.
%D 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.
%H W. Lang: <a href="/A115126/a115126.txt">First 10 rows.</a>
%F a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n<m.
%e [1];[2,2];[3,5,5];[4,9,14,14];...
%e a(4,2) = 9 = binomial(6,2)*3/5.
%Y Row sums give A001453(n+1)=A000108(n+1)-1 (Catalan -1).
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Jan 13 2006