A114299 First row of Modified Schroeder numbers for q=9 (A114295).

1, 1, 1, 1, 1, 2, 5, 13, 34, 89, 288, 1029, 3794, 14113, 52624, 210428, 883881, 3805858, 16570925, 72497060, 325602364, 1498899060, 7017126473, 33185818242, 157858754637, 759960988368, 3706528583080, 18273586377144, 90805138443560, 453695642109973

Offset: 0

Author

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005

Keywords

• nonn

Comments

a(i) is the number of paths from (0,0) to (i,i) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=4x/5.

Examples

			The number of paths from (0,0) to (6,6) staying between the lines y=x and y=4x/5 using steps of length (0,1), (1,0) and (1,1) is a(6)=5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

Crossrefs

See also A112833-A112844 and A114292-A114298.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x or y<4*x/5, 0,
       `if`(x=0, 1, b(x, y-1)+b(x-1, y)+b(x-1, y-1)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Apr 25 2013

Mathematica

b[x_, y_] := b[x, y] = If[y > x || y < 4*x/5, 0, If[x == 0, 1, b[x, y-1] + b[x-1, y] + b[x-1, y-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)