Hoste has a combinatorial formula for the Casson invariant for integral homology $3$-spheres. Let $\Sigma$ be an integral homology $3$-sphere obtained via surgery on a framed link $L = K_1 \cup \dots \cup K_n$ in $S^3$ with framings $1/q_i$. Additionally, we assume $\mathrm{lk}(K_i, K_j) = 0$ for $i \neq j$ (this can always be arranged, see Lemma 12.2 in Saveliev's *Lectures on the Topology of $3$-Manifolds*. In fact, that lemma says that we can take $q_i = \pm 1$). We can write the Conway polynomial of $L$ in the form
$$\nabla_L(z) = z^{n-1}(a_0(L) + a_1(L) z^2 + \cdots + a_m(L) z^{2m}).$$
With the above notations, Hoste's formula is
$$\lambda(\Sigma) = \sum_{L' \subset L} \left( \prod_{i : K_i \subset L'} q_i\right) a_1(L').$$
The sum is taken over all sublinks $L'$ of $L$.

You can see the details of Hoste's formula in his original paper:

Hoste, Jim. *A formula for Casson's invariant*. Trans. Amer. Math. Soc. 297 (1986), 547-562

Hoste's original paper does not show that $\lambda$ as defined by his formula is actually an invariant of integral homology $3$-spheres, however. The missing ingredient is a Kirby move that preserves the unlinking of all components of $L$ as required for Hoste's formula to be stated. Such a move is constructed by Habiro in the following paper:

Habiro, Kazuo. *Refined Kirby calculus for integral homology spheres*. Geometry & Topology 10 (2006), 1285–1317.

Combining the results of the above two papers gives a nice combinatorial construction of the Casson invariant as an invariant of integral homology $3$-spheres.