Coin Change Problem Recurrence Relation with one parameter

Question

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}?