A116868 Triangle of numbers, called Y(1,3), related to generalized Catalan numbers A064063(n) = C(3;n).

1, 3, 4, 9, 21, 25, 27, 90, 165, 190, 81, 351, 846, 1416, 1606, 243, 1296, 3834, 8082, 12900, 14506, 729, 4617, 16119, 40365, 79065, 122583, 137089, 2187, 16038, 64395, 185490, 422685, 790434, 1201701, 1338790

Offset

0,2

Comments

This triangle Y(1,3) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=3. See the Derrida et al. reference given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.

The main diagonal (M=1) gives the generalized Catalan sequence C(3;n+1):= A064063(n+1).

Links

Table of n, a(n) for n=0..35.

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.

Wolfdieter Lang, First 10 rows.

Formula

G.f. m-th diagonal, m>=1: ((3*x*c(3*x))^m)*(2 + 3*x*c(3*x))/(3*x*(2+x)) with c(x) the o.g.f. of A000108 (Catalan).

Example

Triangle begins:
   1;
   3,   4;
   9,  21,  25;
  27,  90, 165,  190;
  81, 351, 846, 1416, 1606;
  ...

Crossrefs

Cf. A064063.

Row sums give A116862.

Sequence in context: A324764 A092763 A232955 * A049976 A032789 A299123

Adjacent sequences: A116865 A116866 A116867 * A116869 A116870 A116871

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 24 2006

Status

approved