Let $(c_1, c_2, \cdots)$ be an $m$-periodic sequence of natural numbers and let $n$ and $k$ be integers with $0\leq k \leq n$. I am trying to simplify $$ \sum_{\substack{I \subseteq \{1, \cdots, n\}\\ |I|=k}} \prod_{i \in I} c_i $$ For example, if $c_i = 1$ for all $i$, this is just $n \choose k$.
I'm wondering if there is a name for this type of thing. I've tried searching for terms like "weighted combination" or "weighted permutation", but have had no luck. If there is not, does anyone have tips on simplifying this expression, or getting some asymptotics for large $n$? I suspect that any sort of asymptotics will include terms like the geometric mean $\sqrt[m]{c_1 \cdots c_m}$.
Basically, what you want is the $x^k$-coefficient in $\displaystyle\prod_{i = 1}^{n}(1+c_ix) = (1+c_1x)(1+c_2x)\cdots(1+c_nx)$.
I don't know if there is an easy way to compute this in general.
EDIT: If you want asymptotics, you need to specify the behavior of $c_n$ for large $n$.