Divisibility Properties for Integer Sequences

For a sequence $f = (f_1, f_2, \dots)$ of nonzero integers, define $\Delta(f)$ to be the numerical triangle that lists all the generalized binomial coefficients \[ \abin{n}{k}_{f} \;=\; \frac{f_nf_{n-1}\cdots f_{n-k+1}}{\!\!\!\!\!f_k\,f_{k-1}\phantom{i}\cdots\phantom{i} f_1}. \] Sequence $f$ is called \emph{\bin} if all entries of $\Delta(f)$ are integers. For $I = (1, 2, 3, \dots)$, $\Delta(I)$ is Pascal's Triangle and $I$ is \bin. Surprisingly, every row and column of Pascal's Triangle is also \bin. For any $f$, the rows and columns of $\Delta(f)$ generate their own triangles and all those triangles fit together to form the ``\Bim\ Pyramid'' $\bp(f)$. Sequence $f$ is \emph{\bin\ at every level} if all entries of $\bp(f)$ are integers. We prove that several familiar sequences are \bin\ at every level. For instance, every sequence $L$ satisfying a linear recurrence of order 2 has that property provided $L(0) = 0$. The sequences $I$, the Fibonacci numbers, and $(2^n - 1)_{n \ge 1}$ provide examples