See Definition 1 in this paper: http://arxiv.org/pdf/1111.4689v3.pdf
The left-hand-side of the second formula appears to suggest some kind of recursion, but the right hand side is not a recursive expression.
The right hand side depends on $i$ but the left hand side may possibly be written similarly for several values of $i$.
I believe this formula is missing something but I cannot figure out what.
Yes, this formula is slightly malformed. To find ${\Bbb P}({\bf Z}=\ell)$ given $\ell\ne 0$, you should take all possible $i$s so that $\ell_i>0$. Then, for each of these $i$s, substitute $i$ and ${\bf k}=\ell-{\bf e}_i$ into the right-hand side of the formula. Summing all these values of the right-hand side up then gives ${\Bbb P}({\bf Z}=\ell)$: $$ {\Bbb P}({\bf Z}=\ell)=\sum_{i\mid \ell_i>0} \phi(\ell-{\bf e}_i,i), $$ where $$\phi({\bf k},i)=\frac{h_i m^k}{(1+m)^{k+1}} \binom{k}{k_1,k_2,\dots} {\bf g}^{\bf k}, \qquad \ \ k={\bf k}{\bf 1}^t$$ is the right-hand side of the formula in the text. To understand the distribution of $\bf Z$, it may help to look at the pgf for $\bf Z$ in the next displayed equation, or the expansion at the bottom of the page.
