A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

1, 1, 2, 2, 3, 5, 5, 7, 9, 14, 14, 19, 23, 28, 42, 42, 56, 66, 76, 90, 132, 132, 174, 202, 227, 255, 297, 429, 429, 561, 645, 715, 785, 869, 1001, 1430, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 4862, 6292, 7150, 7810, 8398, 8986, 9646, 10504, 11934, 16796

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn
• tabl

Comments

There are several versions of a Catalan triangle: see A009766, A008315, A028364.

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

Examples

			Triangle begins
   1;
   1,  2;
   2,  3,  5;
   5,  7,  9, 14;
  14, 19, 23, 28, 42;
		

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.

Crossrefs

Cf. A000108 (column 0 and main diagonal), A001700 (row sums), A065097 (T(2*n-1, n-1)), A201205 (central terms).

Cf. A067323, A009766, A039598, A039599, A028377, A028378, A028376.

Programs

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 *)

Formula

T(n,k) = Sum_{j>=0} A039598(k,j)*A039599(n-k,j). - Philippe Deléham, Feb 18 2004

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

T(n,n) = T(n,k) + T(n,n-1-k) = A000108(n+1), n > 0, k = 0..floor((n+1)/2). - Yuchun Ji, Jan 09 2019

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