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

%I A028364 #89 Jun 22 2024 18:02:50
%S A028364 1,1,2,2,3,5,5,7,9,14,14,19,23,28,42,42,56,66,76,90,132,132,174,202,
%T A028364 227,255,297,429,429,561,645,715,785,869,1001,1430,1430,1859,2123,
%U A028364 2333,2529,2739,3003,3432,4862,4862,6292,7150,7810,8398,8986,9646,10504,11934,16796
%N A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).
%C A028364 There are several versions of a Catalan triangle: see A009766, A008315, A028364.
%C 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.
%C A028364 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
%H A028364 Alois P. Heinz, <a href="/A028364/b028364.txt">Rows n = 0..140, flattened</a>
%H A028364 Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%H A028364 G. Chatel and V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">Cambrian Hopf Algebras</a>, arXiv:1411.3704 [math.CO], 2014-2015.
%H A028364 A. Sapounakis et al., <a href="http://dx.doi.org/10.1016/j.disc.2006.03.044">Ordered trees and the inorder transversal</a>, Disc. Math., 306 (2006), 1732-1741.
%F A028364 T(n,k) = Sum_{j>=0} A039598(k,j)*A039599(n-k,j). - _Philippe Deléham_, Feb 18 2004
%F A028364 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
%F A028364 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
%F A028364 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
%F A028364 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
%F A028364 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
%e A028364 Triangle begins
%e A028364    1;
%e A028364    1,  2;
%e A028364    2,  3,  5;
%e A028364    5,  7,  9, 14;
%e A028364   14, 19, 23, 28, 42;
%p A028364 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A028364       expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
%p A028364     end:
%p A028364 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b((n+1)$2)):
%p A028364 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 28 2015
%t A028364 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 *)
%Y A028364 Cf. A000108 (column 0 and main diagonal), A001700 (row sums), A065097 (T(2*n-1, n-1)), A201205 (central terms).
%Y A028364 Cf. A067323, A009766, A039598, A039599, A028377, A028378, A028376.
%K A028364 nonn,tabl
%O A028364 0,3
%A A028364 _Wouter Meeussen_