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A114292
Modified Schroeder numbers for q=3.
8
1, 1, 1, 2, 2, 1, 5, 5, 2, 1, 16, 16, 6, 2, 1, 57, 57, 21, 6, 2, 1, 224, 224, 82, 22, 6, 2, 1, 934, 934, 341, 89, 22, 6, 2, 1, 4092, 4092, 1492, 384, 90, 22, 6, 2, 1, 18581, 18581, 6770, 1729, 393, 90, 22, 6, 2, 1, 86888, 86888, 31644, 8044, 1794, 394, 90, 22, 6, 2, 1
OFFSET
0,4
COMMENTS
a(i,j) is the number of paths from (i,i) to (j,j) using steps of length (0,1), (1,0) and (1,1), not passing above the line y=x nor below the line y=x/2. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 3-pillow (A112833). To calculate this quantity, let P_n = the principal n X n submatrix of this array. If J_n = the back-diagonal matrix of order n, then A112833(n)=det(P_n+J_nP_n^(-1)J_n).
REFERENCES
C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
LINKS
EXAMPLE
The number of paths from (0,0) to (3,3) staying between the lines y=x and y=x/2 using steps of length (0,1), (1,0) and (1,1) is a(0,3)=5.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 2, 1;
16, 16, 6, 2, 1;
57, 57, 21, 6, 2, 1;
224, 224, 82, 22, 6, 2, 1;
934, 934, 341, 89, 22, 6, 2, 1;
4092, 4092, 1492, 384, 90, 22, 6, 2, 1;
MAPLE
b:= proc(x, y, k) option remember;
`if`(y>x or y<x/2, 0, `if`(x=k, `if`(y=k, 1, 0),
b(x, y-1, k)+b(x-1, y, k)+b(x-1, y-1, k)))
end:
a:= (n, k)-> b(n, n, k):
seq(seq(a(n, k), k=0..n), n=0..12); # Alois P. Heinz, Apr 26 2013
MATHEMATICA
b[x_, y_, k_] := b[x, y, k] = If[y>x || y<x/2, 0, If[x == k, If[y == k, 1, 0], b[x, y-1, k] + b[x-1, y, k] + b[x-1, y-1, k]]]; a[n_, k_] := b[n, n, k]; Table[ Table[ a[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A187307 A280785 A204851 * A333724 A178518 A299499
KEYWORD
nonn,tabl
AUTHOR
Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
STATUS
approved