Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set {1,2,...,n}, that is consider the set

{1^{a_1} * 2^{a_2} * ... * n^{a_n}: a_i>=0, a_1+a_2+...+a_n=m }.

How many results can be obtained?

Denote this number with P(m,n) (the number of elements of the set defined above). Then for example, if m=1, then P(1,n)=n. Similarly, we have: P(m,1)=1 P(m,2)=m+1

Is there a general method that would give precise value of P(m,n) for any m and n? Or at least an approximation of P(m,n)?