This is a follow up and motivated by this question, Number of ways to divide a stick of integer length $N$, Suppose we have a stick of integer length $N$. I'm looking for (preferably closed-form) formula that gives the numbers of different ways in which we can divide the stick into $m$ parts with integral lengths.
In other words, Given some $N$, count the number of all possible different ways to divide stick $N$ into $m$ parts such that the length of any part in any valid division is an integer between $1$ an $N$. Note a crude upper bound is $C_m^N$. I want an exact count or a very tight estimate. Naturally, $m \lt N$. Two divisions of a stick are different if they differ in the length of at least one part. Order of parts inside a division does not matter.
EDIT: The comment of Hagen von Eitzen partially answers my question. I'm more interested in a closed form formula. For instance, find closed-form formula for $F(N, N/2)$.