My question concerns *holonomic* sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form
$$
a(n+k) = \sum_{i=0}^{k-1} p_i(n)a(n+i), \quad n = 1, 2, \ldots,
$$
where $p_i(x) \in \mathbb{Q}(x)$ are rational functions. In other words, the generating function $f(X) := \sum_{n} a(n)X^n \in \mathbb{Q}[[X]]$ and fulfills a linear ODE with polynomial coefficients. 

Define $D_n$ to be the least common denominator of the rational numbers $a(1), a(2), \ldots, a(n)$. That is the minimum positive integer such that all $a(i) \in \frac{1}{D_n}\mathbb{Z}$ for $i \leq n$. 

**Question.** *If $D_n$ is unbounded, does it grow at least exponentially in $n$?*