Alternating series of compositions of triangular numbers I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls.  Each time I add a ball to $S$, it may dislodge another ball with probability proportional to the fraction of the total balls already in $S$.
I've gotten as far as this expression for the expected number of balls $n_k$ after $k$ additions, and I'm hoping someone might show me how to find a closed form expression for this series, or at least some bounds:
$$\langle n_{k}\rangle=k-
\frac{1}{n_{A}}\sum_{a=1}^{k-1}1+
\frac{1}{{n_{A}}^2}\sum_{a=1}^{k-2}\sum_{b=1}^{a}1-
\frac{1}{{n_{A}}^3}\sum_{a=1}^{k-3}\sum_{b=1}^{a}\sum_{c=1}^{b}1+
\frac{1}{{n_{A}}^4}\sum_{a=1}^{k-4}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}1-...$$

Thanks to @hypergeometric for the insight that led to this.  Now I'm off to look for implementations of the biomial coefficient.   ;^)
$$\langle n_{k}\rangle=\sum_{r=0}^{k-1} {(-1)^r \frac 1{n_A^r}\binom {k}r}$$
