0,3

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.

A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.

(n, m)-th entry in triangle is Sum Catalan(n-k)*Catalan(k), k=0..m.

T(n, k) = Sum_{j>=0} A039598(k, j)*A039599(n-k, j). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004

Sum_{k>=0} T(n, k) = A001700(n). T(n, k) = A067323(n, n-k), n>=k>=0, else 0 . - Philippe DELEHAM, May 26 2005

Sum_{k>=0} T(n, k) = A001700(n).

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). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006.

G.f. for column sequences m>=0 (without leading zeros): c(x)*sum(C(m,k)*c(x)^k,k=0..m) with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 24 2006.

{1}, {1, 2}, {2, 3, 5}, {5, 7, 9, 14}, {14, 19, 23, 28, 42}, etc

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

Sequence in context: A113827 A033189 A008507 this_sequence A011971 A060048 A110699

Adjacent sequences: A028361 A028362 A028363 this_sequence A028365 A028366 A028367

tabl,nonn

Wouter L. J. MEEUSSEN (wouter.meeussen(AT)pandora.be)