0,8

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.

The column sequences (without leading zeros) are: A000012, A000108, A064062-3, A064087-93 for m=0..10, respectively. Row sums give A064095.

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.

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.

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.

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).

Diagonals : A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.

Sequence in context: A069739 A066060 A008550 this_sequence A090182 A111673 A121391

Adjacent sequences: A064091 A064092 A064093 this_sequence A064095 A064096 A064097

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001