[seqfan] guess the relation

[D. S. McNeil]

While trying to find a formula to count certain binary matrices for a
problem that came up on SO, I found plausible empirical generating
functions for the rectangular generalization.  [See
http://stackoverflow.com/questions/21149014/finding-a-better-way-to-count-matricesif
you're interested.]

One of the reasons they're so believable is how nicely they factor:

4  (x - 6) * (x - 4) * (x - 2)
5  (x - 12) * (x - 8) * (x - 6) * (x - 4) * (x - 2)
6  (x - 24) * (x - 16) * (x - 12) * (x - 10) * (x - 8) * (x - 6) * (x - 4)
* (x - 2)
7  (x - 48) * (x - 32) * (x - 24) * (x - 20) * (x - 18) * (x - 16) * (x -
12) * (x - 10) * (x - 8) * (x - 6) * (x - 4) * (x - 2)

Should be able to get 8 as well, but beyond that seems unlikely.

Dividing out the factors of 2, we get

4  [1, 2, 3]
5  [1, 2, 3, 4, 6]
6  [1, 2, 3, 4, 5, 6, 8, 12]
7  [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 24]

as the numbers to be reproduced, and I feel like there's some
divisibility-related pattern there I'm missing.  Any numerologists have a
guess?


Doug

PS: If anyone can just look at the problem and write down a combinatorial
formula at once, I'm sure the OP would appreciate that too. :^)