0,17

The sequence for column m (m >= 1) (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 (unphysical) 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. See also Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.

The unsigned column sequences (without leading zeros) are: A000012, A064310-11, A064325-33 for m=0..11, respectively. Row sums (signed) give A064338. Row sums (unsigned) give A064339.

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.

T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.

G.f. for column m: (x^m)/(1-x*c(-m*x))= (x^m)*((m+1)+m*x*c(-m*x))/((m+1)-x), m>0, 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*(m+1))^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). For m=0: a(n, 0)=1.

{1}; {1,1}; {1,1,1}; {1,0,1,1}; {1,1,-1,1,1}; {1,-2,5,-2,1,1}; ...

Sequence in context: A120294 A047921 A102786 this_sequence A061176 A124780 A108437

Adjacent sequences: A064331 A064332 A064333 this_sequence A064335 A064336 A064337

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 21 2001