A115154 Triangle of numbers related to the generalized Catalan sequence C(3;n+1) = A064063(n+1), n>=0.

1, 1, 4, 1, 13, 25, 1, 40, 115, 190, 1, 121, 466, 1036, 1606, 1, 364, 1762, 4870, 9688, 14506, 1, 1093, 6379, 20989, 50053, 93571, 137089, 1, 3280, 22417, 85384, 235543, 516256, 927523, 1338790, 1, 9841, 77092, 333244, 1039873, 2588641, 5371210

Offset

0,3

Comments

This triangle, called Y(3,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=3, beta=1. See the Derrida et al. refs. 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).

The diagonal sequences give A064063(n+1), A115188-A115192 for n+1>= M=1,..,6.

Links

Table of n, a(n) for n=0..42.

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

a(n,n+1)=A064063(n+1) (main diagonal with M=1); a(n,n-M+2)= a(n,n-M+1) + 3*a(n-1,n-M+2), M>=2; a(n,1)=1; n>=0.

G.f. for diagonal sequence M=1: GY(1,x):=(3*c(3*x)-1)/(2+x) with c(x) the o.g.f. of A000108 (Catalan); for M=2: GY(2,x)=(1-3*x)*GY(1,x)-1; for M>=3: GY(M,x)= GY(M-1,x) - 3*x*GY(M-2,x) + 2*x^(M-2).

G.f. for diagonal sequence M (solution to the above given recurrence): GY(M,x)= (x^(M-1)/(1+x))*( 3^(M+1)*x*(p(M,3*x)-(3*x)*p(M+1,3*x)*c(3*x))+1), with c(x) g.f. of A000108 (Catalan) and p(n,x):= -((1/sqrt(x))^(n+1))*S(n-1,1/sqrt(x)) with Chebyshev's S(n,x) polynomials given in A049310.

Example

Triangle begins:
  1;
  1,   4;
  1,  13,  25;
  1,  40, 115,  190;
  1, 121, 466, 1036, 1606;
  ...
466 = a(4,3) = a(4,2) + 3*a(3,3) = 121 + 3*115.

Crossrefs

Row sums give A115187.

Sequence in context: A116414 A215502 A144698 * A292270 A051928 A347486

Adjacent sequences: A115151 A115152 A115153 * A115155 A115156 A115157

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 23 2006

Status

approved