OFFSET
0,3
COMMENTS
The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N >= 1, M=1..N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.
Consider a Dyck path as a path with steps N=(0,1) and E=(1,0) from (0,0) to (n,n) that stays weakly above y=x. T(n,m) is the number of Dyck paths of semilength n+1 where the (m+1)st north step is followed by an east step. - Lara Pudwell, Apr 12 2023
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
FORMULA
Sum_{k>=0} T(n,k) = A001700(n). T(n,k) = A067323(n,n-k), n >= k >= 0, otherwise 0. - Philippe Deléham, May 26 2005
G.f. for column sequences m >= 0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) with C(n):=A000108(n). - Wolfdieter Lang, Mar 24 2006
G.f. for column sequences m >= 0 (without leading zeros): c(x)*Sum_{k=0..m} C(m,k)*c(x)^k with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). - Wolfdieter Lang, Mar 24 2006
G.f. for triangle: Sum_{n>=0, m>=0} T(n, m)*x^n*y^m = (c(x)-c(xy))/(x(1-y)c(x)) with the g.f. c(x) of A000108 (Catalan). - Lara Pudwell, Apr 12 2023
EXAMPLE
Triangle begins
1;
1, 2;
2, 3, 5;
5, 7, 9, 14;
14, 19, 23, 28, 42;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, add(
expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10); # Alois P. Heinz, Nov 28 2015
MATHEMATICA
t[n_, k_] = Sum[CatalanNumber[n-j]*CatalanNumber[j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}]] (* Jean-François Alcover, Jul 22 2011 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved