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A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

%I A115195 #19 Aug 06 2024 09:01:58
%S A115195 1,2,3,4,10,13,8,28,54,67,16,72,180,314,381,32,176,536,1164,1926,2307,
%T A115195 64,416,1488,3816,7668,12282,14589,128,960,3936,11568,26904,51468,
%U A115195 80646,95235,256,2176,10048,33184,86992,189928,351220,541690,636925,512,4864
%N A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).
%C A115195 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 A115195 The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).
%C A115195 The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.
%H A115195 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 A115195 Wolfdieter Lang, <a href="/A115195/a115195.txt">First 10 rows</a>.
%F A115195 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 A115195 Triangle begins:
%e A115195    1;
%e A115195    2,  3;
%e A115195    4, 10,  13;
%e A115195    8, 28,  54,  67;
%e A115195   16, 72, 180, 314, 381;
%e A115195   ...
%Y A115195 Row sums give A084076.
%K A115195 nonn,easy,tabl
%O A115195 0,2
%A A115195 _Wolfdieter Lang_, Feb 23 2006