OFFSET
0,17
COMMENTS
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.
REFERENCES
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.
LINKS
G. C. Greubel, Rows n =0..100 of triangle, flattened
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.
FORMULA
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.
T(n, m) = Sum_{k=0..n-m-1} (n-m-k)*binomial(n-m-1+k, k)*(-m)^k/(n-m), with T(n,0) = T(n,n)=1.
T(n,m) = (1/(1+m))^(n-m)*(1 + m*Sum_{k=0..n-m-1} C(k)*(-m*(m+1))^k ), n-m >= 1, T(n, n) = T(n,0) =1, T(n, m)=0 if n<m, with C(k)=A000108(k) (Catalan).
T(n, k) = hypergeometric([1-n+k, n-k], [-n+k], -k) if k<n else 1. - Peter Luschny, Nov 30 2014
EXAMPLE
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 0, 1, 1;
1, 1, -1, 1, 1;
1, -2, 5, -2, 1, 1; ...
MATHEMATICA
Table[If[k==0, 1, If[k==n, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]*(-k)^j/(n -k), {j, 0, n-k-1}]]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 04 2019 *)
PROG
(Sage)
def T(n, k):
return hypergeometric([1-n, n], [-n], -k) if n>0 else 1
for n in (0..10):
print([simplify(T(n-k, k)) for k in (0..n)]) # Peter Luschny, Nov 30 2014
(PARI) {T(n, k) = if(k==0, 1, if(k==n, 1, sum(j=0, n-k-1, (n-k-j)* binomial(n-k-1+j, j)*(-k)^j/(n-k))))}; \\ G. C. Greubel, May 04 2019
(Magma) [[k eq 0 select 1 else k eq n select 1 else (&+[(n-k-j)* Binomial(n-k-1+j, j)*(-k)^j/(n-k): j in [0..n-k-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 04 2019
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 21 2001
STATUS
approved