Question
Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? i.e. when $$ a_n= \sum_{i=0}^{n-1} f_i(n) a_{i} + g(n) $$ where $$ f_i(n)=\frac{P_i(n)}{Q_i(n)} r^n $$ for $P_i, Q_i$ polynomials and $r \in \mathbb{C}$ and also $g(n)$ is the product of a rational function for some geometric sequence.
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
Motivation
There is a complete characterization of the case in which the coefficients are constant (see comments below). In many circumstances using the recursive formula simplify the calculation, so I'm wondering if it is possible to expand this characterization to the case in which the coefficients are not constants
The conjecture
Let $$ F(X):= \sum_{n=0}^\infty \frac{a_n}{n!} X^n $$ and assume that the radius of convergence of the series is not zero.
Conjecture: the sequence can be expressed as a recurrence relation iif $F(X)$ satisfies a linear ODE $$ \sum_{i=0}^k c_i(X) \frac{d^i}{dX^i}F(X) - g(X)=0 $$ for some $c_i(X), g(X)$ analytic near the origin such that $$ c^{(k)}_i(0)=\frac{\tilde{P}_i(k)}{\tilde{Q}_i(k)} r^k $$ with $\tilde{P}_i,\tilde{Q}_i$ polynomials and $r \in \mathbb{C}$ and same for the Taylor coefficients of $g(X)$
The condition is sufficient as $a_n$ satisfies the recurrece relation $$ \sum_{i=0}^k \left(\sum_{r=0}^n \frac{c_i^{(n-r)}(0)}{(n-r)!r!} a_{r+i}\right) - \frac{g^{(n)}(0)}{n!}=0 $$ but i don't know if it is necessary.
Is this condition really necessary? Is there any simpler characterization of such sequences?