%I A064094
%S A064094 1,1,1,1,1,1,1,2,1,1,1,5,3,1,1,1,14,13,4,1,1,1,42,67,25,5,1,1,1,132,
%T A064094 381,190,41,6,1,1,1,429,2307,1606,413,61,7,1,1,1,1430,14589,14506,4641,
%U A064094 766,85,8,1,1,1,4862,95235
%N A064094 Triangle composed of generalized Catalan numbers.
%C A064094 The column m sequence (without leading zeros and the first 1) appears 
               in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, 
               for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida 
               et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/
               (alpha*beta)^N for N>=1.
%C A064094 The column sequences (without leading zeros) are: A000012, A000108, A064062-3, 
               A064087-93 for m=0..10, respectively. Row sums give A064095.
%D A064094 B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional 
               asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 
               1992, 667-687; eqs. (20), (21), p. 672.
%D A064094 B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of 
               a 1D asymmetric exclusion model using a matrix formulation, J. Phys. 
               A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) 
               p. 1513.
%F A064094 G.f. for column m: (x^m)/(1-x*c(m*x))= (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) 
               with the g.f. c(x) of Catalan numbers A000108.
%F A064094 a(n, m)= sum((n-m-k)*binomial(n-m-1+k, k)*(m^k)/(n-m), k=0..n-m-1) = 
               ((1/(1-m))^(n-m)*(1-m*sum(C(k)*(m*(1-m))^k, k=0..n-m-1)), n-m >= 
               1; a(n, n)=1; a(n, m)=0 if n<m; with C(k)=A000108(k) (Catalan).
%Y A064094 Diagonals : A000012, A000012, A000027, A001844, A064096, A064302, A064303, 
               A064304, A064305.
%Y A064094 Sequence in context: A069739 A066060 A008550 this_sequence A090182 A111673 
               A121391
%Y A064094 Adjacent sequences: A064091 A064092 A064093 this_sequence A064095 A064096 
               A064097
%K A064094 nonn,easy,tabl
%O A064094 0,8
%A A064094 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 
               2001