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

0,6

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.

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.

Example

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.

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 *)

Crossrefs

See also A112833-A112844 and A114292-A114298.

Sequence in context: A099496 A122367 A367658 * A112842 A097417 A367657

Adjacent sequences: A114296 A114297 A114298 * A114300 A114301 A114302

Keyword

nonn

Author

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

Status

approved