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A064334 Triangle composed of generalized Catalan numbers. 14
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, 1, 1, 1, -2, 5, -2, 1, 1, 1, 6, -25, 13, -3, 1, 1, 1, -18, 141, -98, 25, -4, 1, 1, 1, 57, -849, 826, -251, 41, -5, 1, 1, 1, -186, 5349, -7448, 2817, -514, 61, -6, 1, 1, 1, 622, -34825, 70309, -33843, 7206 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..60.

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.

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.

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; ...

PROG

(Sage)

def a(n, k):

    return hypergeometric([1-n, n], [-n], -k) if n>0 else 1

for n in (0..10):

print [simplify(a(n-k, k)) for k in (0..n)] # Peter Luschny, Nov 30 2014

CROSSREFS

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.

Cf. A064062.

Sequence in context: A146103 A245172 A196527 * A320032 A270061 A061176

Adjacent sequences:  A064331 A064332 A064333 * A064335 A064336 A064337

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved

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Last modified February 21 10:49 EST 2019. Contains 320372 sequences. (Running on oeis4.)