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

$\{1^{a_1} \cdot 2^{a_2} \cdot \ldots \cdot n^{a_n}\mid a_i\geq 0, a_1+a_2+\ldots+a_n=m \}$.

How many results can be obtained?

Denote this number with $P(m,n)$ (the number of elements of the set defined above). 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 $P(m,n)$?