A114294 Modified Schroeder numbers for q=7.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 5, 5, 5, 5, 2, 1, 13, 13, 13, 13, 5, 2, 1, 34, 34, 34, 34, 13, 5, 2, 1, 110, 110, 110, 110, 42, 16, 6, 2, 1, 393, 393, 393, 393, 150, 57, 21, 6, 2, 1, 1449, 1449, 1449, 1449, 553, 210, 77, 21, 6, 2, 1, 5390, 5390, 5390, 5390, 2057, 781

Offset: 0

Author

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005

Keywords

• nonn
• tabl

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=3x/4. The Hamburger Theorem implies that we can use this table to calculate the number of domino tilings of an Aztec 7-pillow (A112839). 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 A112839(n)=det(P_n+J_nP_n^(-1)J_n).

Examples

			The number of paths from (0,0) to (5,5) staying between the lines y=x and y=3x/4 using steps of length (0,1), (1,0) and (1,1) is a(0,5)=5.
		

References

• C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

Crossrefs

See also A112833-A112844 and A114292-A114299.