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A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k). 25
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

G. Chatel, 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

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, else 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(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). - Wolfdieter Lang, Mar 24 2006

EXAMPLE

   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

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

Row sums give A001700.

T(2n,n) gives A201205.

Sequence in context: A113827 A033189 A008507 * A239482 A280470 A011971

Adjacent sequences:  A028361 A028362 A028363 * A028365 A028366 A028367

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen

STATUS

approved

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Last modified February 21 14:40 EST 2018. Contains 299414 sequences. (Running on oeis4.)