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A114295
Modified Schroeder numbers for q=9.
1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 5, 5, 5, 5, 5, 2, 1, 13, 13, 13, 13, 13, 5, 2, 1, 34, 34, 34, 34, 34, 13, 5, 2, 1, 89, 89, 89, 89, 89, 34, 13, 5, 2, 1, 288, 288, 288, 288, 288, 110, 42, 16, 6, 2, 1, 1029, 1029, 1029, 1029, 1029, 393, 150, 57, 21, 6, 2, 1
OFFSET
0,16
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=4x/5. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 9-pillow (A112842). 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 A112842(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.
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(0,6)=5.
CROSSREFS
Sequence in context: A104518 A329030 A027388 * A004216 A076489 A211666
KEYWORD
nonn,tabl
AUTHOR
Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
STATUS
approved