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%I #3 Nov 10 2007 03:00:00
%S 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,
%T 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,
%U 288,288,288,110,42,16,6,2,1,1029,1029,1029,1029,1029,393,150,57,21,6,2,1
%N Modified Schroeder numbers for q=9.
%C 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).
%D C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
%e 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.
%Y See also A112833-A112844 and A114292-A114299.
%K nonn,tabl
%O 0,16
%A Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
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