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A114292
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Modified Schroeder numbers for q=3.
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8
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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
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OFFSET
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0,4
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COMMENTS
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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).
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REFERENCES
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C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
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LINKS
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EXAMPLE
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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;
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
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STATUS
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approved
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