Divisibility Properties of Integer Sequences

Daniel B. Shapiro

Published 2023-02-04Version 1

A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[ \left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f \ = \ \frac{ f_nf_{n-1}\cdots f_{n-k+1} }{ f_kf_{k-1}\cdots f_1 }. \] Let $\Delta(f)$ be the infinite triangle with those numbers as entries. When $I = (1, 2, 3, \dots)$ then $\Delta(I)$ is Pascal's Triangle so that $I$ is binomid. Surprisingly, every row and column of Pascal's Triangle is also binomid. For any $f$, each row and column of $\Delta(f)$ generates its own triangle and all those triangles fit together to form the ``Binomid Pyramid'' $\mathbb{BP}(f)$. Sequence $f$ is ``binomid at every level'' if all entries of $\mathbb{BP}(f)$ are integers. We prove that several familiar sequences have that property, including the Lucas sequences. In particular, $I = (1, 2, 3, \dots )$, the sequence of Fibonacci numbers, and $(2^n - 1)_{n \ge 1}$ are binomid at every level.