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%I #10 Mar 06 2015 06:28:44
%S 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,
%T 934,934,341,89,22,6,2,1,4092,4092,1492,384,90,22,6,2,1,18581,18581,
%U 6770,1729,393,90,22,6,2,1,86888,86888,31644,8044,1794,394,90,22,6,2,1
%N Modified Schroeder numbers for q=3.
%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=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).
%D C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
%H Alois P. Heinz, <a href="/A114292/b114292.txt">Rows n = 0..140, flattened</a>
%e 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 5, 5, 2, 1;
%e 16, 16, 6, 2, 1;
%e 57, 57, 21, 6, 2, 1;
%e 224, 224, 82, 22, 6, 2, 1;
%e 934, 934, 341, 89, 22, 6, 2, 1;
%e 4092, 4092, 1492, 384, 90, 22, 6, 2, 1;
%p b:= proc(x, y, k) option remember;
%p `if`(y>x or y<x/2, 0, `if`(x=k, `if`(y=k, 1, 0),
%p b(x, y-1, k)+b(x-1, y, k)+b(x-1, y-1, k)))
%p end:
%p a:= (n, k)-> b(n, n, k):
%p seq(seq(a(n,k), k=0..n), n=0..12); # _Alois P. Heinz_, Apr 26 2013
%t 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_ *)
%Y See also A112833-A112844 and A114293-A114299.
%K nonn,tabl
%O 0,4
%A Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
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