A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

1, 2, 3, 4, 10, 13, 8, 28, 54, 67, 16, 72, 180, 314, 381, 32, 176, 536, 1164, 1926, 2307, 64, 416, 1488, 3816, 7668, 12282, 14589, 128, 960, 3936, 11568, 26904, 51468, 80646, 95235, 256, 2176, 10048, 33184, 86992, 189928, 351220, 541690, 636925, 512, 4864

Offset: 0

Author

Wolfdieter Lang, Feb 23 2006

Keywords

• nonn
• easy
• tabl

Comments

This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. 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(2,n+1):=A064062(n+1).

The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

Examples

			Triangle begins:
   1;
   2,  3;
   4, 10,  13;
   8, 28,  54,  67;
  16, 72, 180, 314, 381;
  ...
		

Links

• 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.

Crossrefs

Row sums give A084076.

Formula

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