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%I #12 Jul 25 2022 09:08:50
%S 1,3,4,9,21,25,27,90,165,190,81,351,846,1416,1606,243,1296,3834,8082,
%T 12900,14506,729,4617,16119,40365,79065,122583,137089,2187,16038,
%U 64395,185490,422685,790434,1201701,1338790
%N Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).
%C This triangle Y(1,3) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=3. See the Derrida et al. reference 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(3;n+1):= A064063(n+1).
%H B. Derrida, E. Domany and D. Mukamel, <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1090.4158">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="/A116868/a116868.txt">First 10 rows</a>.
%F G.f. m-th diagonal, m>=1:((3*x*c(3*x))^m)*(2 + 3*x*c(3*x))/(3*x*(2+x))) with c(x) the o.g.f. of A000108 (Catalan).
%e Triangle begins:
%e 1;
%e 3, 4;
%e 9, 21, 25;
%e 27, 90, 165, 190;
%e 81, 351, 846, 1416, 1606;
%e ...
%Y Row sums give A116862.
%K nonn,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Mar 24 2006
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