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In Fulton's book Intersection Theory, Theorem 15.3 and Example 15.3.1 (p. 297 of my edition) it is proven, by using GRR, that $$c_k(\mathscr{O}_Z)=(-1)^{k-1}(k-1)![Z].$$ By the short exact sequence $$0 \to \mathscr{I}_Z \to \mathscr{O}_X \to \mathscr{O}_Z \to 0$$ one has $1=c(\mathscr{O}_X)=c(\mathscr{O}_Z)c(\mathscr{I}_Z)$ and this implies $$c_k(\mathscr{I}_Z)=(-1)^k (k-1)![Z].$$