For a smooth projective variety X and a coherent sheaf S on it, you can consider the projective cone $\pi:\mathbb{P}S \rightarrow X$. You can consider $\mathcal{O}(1)$ on $\mathbb{P}S$ and define the Segre series a la Fulton:

$$ s(S,t) = \pi_*\left(\frac {1}{t-c_1(\mathcal{O}(1))}\right) \in A^*(X) $$ I can also take a finite projective resolution of $S$ by vector bundles, and ask how to express $s(S,t)$ in terms of the Chern classes of these bundles.