In proving Gould's "Star of David" conjecture, Hillman and Hoggat generalized the binomial coefficient. First, they demand that $a_n$ be a sequence with the two properties that
$$\gcd(a_m, a_n) \mid a_{m+n}$$
$$\gcd(a_m, a_{m+n}) \mid a_{n}$$
The then say $b(n, 0) = 1$, and $$b(n, k) = \frac{a_n}{a_k} b(n-1, k-1)$$
Using the constraints on $a_n$, they are able to show that SoD holds for $b(n,k)$ and therefore holds for both the standard binomial as well as variants of it (including the Fibonomial coefficient).
They use the claim that $b(n,k)$ will be an integer. This is what confuses me. If $a_n$ is the $n$th prime, then the constraints holds vacuously, but $b(n,k)$ is often times not an integer. I think I am missing something in my understanding of Hillman and Hoggat's constraint on $a_n$
The paper I am referencing is called "A Proof of Gould's Pascal Hexagon Conjecture".
If $a_n$ is the $n$th prime, it is not the case that $\gcd(a_n,a_n)\vert a_{n+n}$.