This formula is actually pretty simple to understand.

First, the $2^8$ is the number of possible $D4$ states. Then for each (indistinguishable) $D0$, they can be in either a fermionic or bosonic state, of which there are $8$ each.

Next, the coefficient of $q^n$ in $(1+q)^8$ is the number of ways for $n$ independent $D0$ branes to fit in $8$ fermionic states.

The coefficient of $q^n$ in $(1-q)^{-8}$ is the number of ways for $n$ independent $D0$ branes to fit in $8$ bosonic states.

We multiply these two to allow $D0$ branes to occupy either bosonic or fermionic states.

By taking the products over $q^k$, we allow $D0$ branes to first form $k$-tuply bound states which occupy a single $D0$ state.

This post imported from StackExchange Physics at 2014-03-05 14:52 (UCT), posted by SE-user Ryan Thorngren