Let $f(n), g(n,m), h(n)$ be an arbitrary functions which equal to the non-negative integers.
Let $$ R(n,q) = \sum\limits_{j=0}^{f(q)}g(q,j)R(n-1,j),\\ R(0,q) = h(q) $$
In the comment to the one of my previous questions Will Sawin noted that
$R(n,0)$ is the number of sequences $q_1,\cdots,q_n$ of nonnegative integers with $q_1=0$ and $q_{i+1}\leqslant q_i+q_i\operatorname{mod}3+1$.
In the question mentioned above, we have $f(n)=n+n\operatorname{mod}3+1$, $g(n,m)=h(n)=1$. I guess the same works for any $f(n)$ (I mean $q_{i+1}\leqslant f(q_i)$).
I asked Will if there is a similar combinatorial interpretation for the more general case (i.e. for $R(n,0)$ from this question). He gave a positive answer:
Yes, as long as the $g$ and $h$ are nonnegative.
But when I asked to explain this case in more detail, I did not receive an answer.
So my question is the following: what is a similar combinatorial interpretation for the more general case of $R(n,0)$?
