Chromatic polynomial of a cycle - Interpreting its terms
There is a theorem of Whitney answering your question. Let
$$ P(G,\lambda) = \sum_{k=0}^n (-1)^k a_k \lambda^{n-k}. $$
Then $a_k$ is the number of $k$-subsets of the edges of $G$ not containing any broken cycle. A broken cycle of $G$ is obtained from a cycle of $G$ by removing its maximum edge according to some fixed ordering of the edges.

For example, if $G = C_n$ then for $k &lt; n-1$, $a_k = \binom{n}{k}$, and $a_{n-1} = \binom{n}{n-1} - 1$, since we miss one broken cycle.

For a proof and references to other proofs, see this paper.