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A214457
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Table read by antidiagonals in which entry T(n,k) in row n and column k gives the number of possible rhombus tilings of an octagon with interior angles of 135 degrees and sequences of side lengths {n, k, 1, 1, n, k, 1, 1} (as the octagon is traversed), n,k in {1,2,3,...}.
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1
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8, 20, 20, 40, 75, 40, 70, 210, 210, 70, 112, 490, 784, 490, 112, 168, 1008, 2352, 2352, 1008, 168, 240, 1890, 6048, 8820, 6048, 1890, 240, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 440, 5445, 29040, 76230, 104544, 76230, 29040, 5445, 440
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OFFSET
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1,1
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COMMENTS
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Proof of the formula for T(n,k) is given in [Elnitsky].
So-called "generalized Narayana numbers" (see A145596), linking rhombus tilings of polygons to certain walks or paths through the square lattice.
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LINKS
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Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.
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FORMULA
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T(n,k) = 2*(n+k+1)!*(n+k+2)!/[n!*k!*(n+2)!*(k+2)!].
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EXAMPLE
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See [Jeffery]. T(1,1) = 8 because there are eight ways to tile the proposed octagon with rhombuses.
Table begins as
8 20 40 70 112 ...
20 75 210 490 1008 ...
40 210 784 2352 6048 ...
70 490 2352 8820 27720 ...
112 1008 6048 27720 76230 ...
...
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MATHEMATICA
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Table[2*(# + k + 1)!*(# + k + 2)!/(#!*k!*(# + 2)!*(k + 2)!) &[n - k + 1], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Feb 26 2024 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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