E.g there is a formula for a function that gives a number of idempotent functions for a set with finite size. The solution is $\sum_{k=1}^n{n\choose k}k^{n-k}$ but you need to sum intermediate results.
So the number of operations is dependent on the size of the set. But is there any solution for this function that does require only a constant amount of operations? How can I prove it? Is there a general way to conclude that function can have a formula with a constant amount of operations given the fact that we already have a formula with the number of operations dependent on the input or have proven that such must exist?