Let $A$ be a finite set of integers. The generating function for the number of ways of writing a given integer $n$ as the sum of $k$ elements from $A$ not necessarily distinct is given by:
$$\left(\sum_{a \in A}{x^a}\right)^k=\sum_n{r(n,k)x^n}$$
Is there a generating function for the number of ways of writing an integer $n$ as a sum of $k$ distinct elements of $A$?