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A116868 Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).

%I A116868 #19 Jan 23 2025 08:32:01
%S A116868 1,3,4,9,21,25,27,90,165,190,81,351,846,1416,1606,243,1296,3834,8082,
%T A116868 12900,14506,729,4617,16119,40365,79065,122583,137089,2187,16038,
%U A116868 64395,185490,422685,790434,1201701,1338790
%N A116868 Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).
%C A116868 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 A116868 The main diagonal (M=1) gives the generalized Catalan sequence C(3;n+1):= A064063(n+1).
%H A116868 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 A116868 Wolfdieter Lang, <a href="/A116868/a116868.txt">First 10 rows</a>.
%F A116868 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 A116868 Triangle begins:
%e A116868    1;
%e A116868    3,   4;
%e A116868    9,  21,  25;
%e A116868   27,  90, 165,  190;
%e A116868   81, 351, 846, 1416, 1606;
%e A116868   ...
%Y A116868 Cf. A064063.
%Y A116868 Row sums give A116862.
%K A116868 nonn,easy,tabl
%O A116868 0,2
%A A116868 _Wolfdieter Lang_, Mar 24 2006