Counting distinct set covers
Thanks to Andreas Björklund, I realized that $c_k$ is the number of all tuples with repetitions. I reread the article carefully and reformulated the problem in the following way:

Let $B = 2^{\mathcal{F}}$, $A_i = \{S \in 2^\mathcal{F} \mid S \cap \{i\} = \emptyset \}$ (i.e. set of all subsets of $\mathcal{F}$ that avoid singleton set $\{i\}$), and $C(\mathcal{F})$ is the number of distinct covers of $\mathcal{F}$. Then from the inclusion-exclusion principle we have:
$$C(\mathcal{F}) = \sum_{X \subseteq N} (-1)^{|X|} 2^{a(X)}$$
As before, $a(X)$ stands for the number of sets in $\mathcal{F}$ that avoid all of $X$.

I've tested this formula on a couple of examples, and it works like a charm. For the simple example from the question above we have:
$$2^3 - 2^1 - 2^1 + 2^0 = 5$$