OFFSET
0,3
COMMENTS
This triangle, called Y(3,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=3, beta=1. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(3,n+1):=A064063(n+1).
LINKS
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.
Wolfdieter Lang, First 10 rows.
FORMULA
a(n,n+1)=A064063(n+1) (main diagonal with M=1); a(n,n-M+2)= a(n,n-M+1) + 3*a(n-1,n-M+2), M>=2; a(n,1)=1; n>=0.
G.f. for diagonal sequence M=1: GY(1,x):=(3*c(3*x)-1)/(2+x) with c(x) the o.g.f. of A000108 (Catalan); for M=2: GY(2,x)=(1-3*x)*GY(1,x)-1; for M>=3: GY(M,x)= GY(M-1,x) - 3*x*GY(M-2,x) + 2*x^(M-2).
EXAMPLE
Triangle begins:
1;
1, 4;
1, 13, 25;
1, 40, 115, 190;
1, 121, 466, 1036, 1606;
...
466 = a(4,3) = a(4,2) + 3*a(3,3) = 121 + 3*115.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Feb 23 2006
STATUS
approved