# Coin Change Problem Recurrence Relation with one parameter

I have been looking through the recursive formulation for the coin change problem here and am wondering if it is possible to define the function $$C(N, m)$$ in one parameter as $$C(N)$$, therefore not depending on the set $$S_{m}$$? What would this function look like in general? If a concrete example is needed what would it look like for the set of coins {3 cents coin, 4 cents coin, 5 cent coin}?

One possible definition is $$C(N) = \sum_{i \in S_m} C(N-i),$$ with base cases $$C(0) = 1$$ and $$C(x) = 0$$ for negative $$x$$. This counts the number of ways to write $$N$$ as a sum of elements of $$S_m$$ where order matters, that is, if $$a \neq b$$ then $$a + b$$ is different from $$b + a$$. As an example, if $$S_m = \{1,2\}$$ then $$C(3) = 3$$ since $$3 = 1 + 1 + 1 = 1 + 2 = 2 + 1$$.
It is, however, possible to obtain a one-parameter recurrence relation, using inclusion exclusion: $$C(N) = \sum_{\emptyset \neq A \subseteq S_m} (-1)^{|A|+1} C(N-\Sigma A),$$ where $$\Sigma A$$ is the sum of elements in $$A$$. For example, if $$S_m = \{3,4,5\}$$, then the recurrence is $$C(N) = C(N-3)+C(N-4)+C(N-5)-C(N-7)-C(N-8)-C(N-9)+C(N-12).$$ The initial values are $$C(0) = 1, C(1) = C(2) = 0, C(3) = \cdots = C(7) = 1, C(8) = \cdots = C(11) = 2,$$ or in table format: $$\begin{array}{c|ccccccccccc} N & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\\hline C(N) & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 \end{array}$$