For a combinatorial enumeration problem that requires finding $f(n)$, I managed to come up with a recursive formula that involves around $n$ parameters. For example, while the Fibonacci recursion $F(n) = F(n-1) + F(n-2)$ involves only $1$ parameter, the formula I obtained looks like $F(s_1,s_2,\dots,s_n)$ equaling to an expression of $F$'s with smaller inputs. I tried my best to come up with a recursion that involves a constant number of parameters but could not do so.
Before this problem, I have never encountered recurrence relations where the number of parameters depends on the size of the input, namely $n$. So, how "acceptable" are these recursions as solutions to enumeration problems, as inelegant as they seem? I ask this because I have not seen them in any combinatorics literature I have read and don't know how people deal with them.
Sorry for not being able to provide a specific example. The problem I am working on is for research that I cannot disclose.