I have a question on the number of unique ways to exchange cash, for instance we have a target cash n and k kinds of cash from 1 to k(i.e., if k = 3, we can use unlimited $1, $2 or $3) to exchange.
Now I already know $m(n,k) = m(n-k, k) + m(n, k-1)$, is their any combinatorial way to quickly calculate the final result? like generating functions?