I have a proof technique in search of examples. I'm looking for combinatorially meaningful sequences $\{a_n\}$ so that $a_{n+1}/a_n$ is known or conjectured to be an integer, such that there is a relation between the $n$th case and $n+1$st, but **not** an obvious $a_{n+1}/a_n\to 1$ map. This means $a_n$ is the $n$th partial product of an infinite sequence of integers, but there isn't an obvious product structure.

The prototype was an enumeration of domino tilings of an Aztec diamond of order $n$, $a_n = 2^{n(n+1)/2}$, so $a_{n+1}/a_n = 2^{n+1}$. (There is a nice $2^{n+1}$ to 1 map unrelated to my technique, but it isn't obvious.)

Another application was a proof that $\det \{B_{i+j}\}_{i,j=0}^n = \prod_{i=1}^n i! $ where $B_n$ is the $n$th Bell number, equation 25 in the linked page.

The counts of alternating sign matrices 1, 2, 7, 42, ... are

**not**an example, since $ASM(n+1)/ASM(n) = \frac{ (3n+1)!n!}{2n! (2n+1)!}$ which is not always an integer, e.g, 7/2 is not.

What are some other interesting combinatorial families whose ratios $a_{n+1}/a_n$ are known or (preferably) conjectured to be integers?

Thanks.