A064334 Triangle composed of generalized Catalan numbers.

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, -917, 85, -7, 1, 1

Offset: 0

Author

Wolfdieter Lang, Sep 21 2001

Keywords

• sign
• easy
• tabl

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.

Examples

			Triangle starts:
  1;
  1,  1;
  1,  1,  1;
  1,  0,  1,  1;
  1,  1, -1,  1, 1;
  1, -2,  5, -2, 1, 1; ...
		

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.

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.

Programs

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

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 *)

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

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

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 nA000108(k) (Catalan).

T(n, k) = hypergeometric([1-n+k, n-k], [-n+k], -k) if kPeter Luschny, Nov 30 2014