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

$\{1^{a_1} \cdot 2^{a_2} \cdot ... \cdot n^{a_n}\mid a_i\geq0, a_1+a_2+...+a_n=m \}$.

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 $P(m,n)$?