%I #19 Aug 06 2024 09:01:58

%S 1,2,3,4,10,13,8,28,54,67,16,72,180,314,381,32,176,536,1164,1926,2307,

%T 64,416,1488,3816,7668,12282,14589,128,960,3936,11568,26904,51468,

%U 80646,95235,256,2176,10048,33184,86992,189928,351220,541690,636925,512,4864

%N Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

%C This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.

%C The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).

%C The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

%H B. Derrida, E. Domany and D. Mukamel, <a href="https://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

%H Wolfdieter Lang, <a href="/A115195/a115195.txt">First 10 rows</a>.

%F G.f. m-th diagonal, m>=1: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^m)/(2*x*(1+x)) with c(x) the o.g.f. of A000108 (Catalan).

%e Triangle begins:

%e 1;

%e 2, 3;

%e 4, 10, 13;

%e 8, 28, 54, 67;

%e 16, 72, 180, 314, 381;

%e ...

%Y Row sums give A084076.

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Feb 23 2006