Galois theory in combinatorics

Question

When trying to find an explicit formula, how often shall we admit such a formula may not exist? To be more precise, suppose we are trying to find an explicit formula of a function $f(n)$ that returns the number of isomorphism classes of $n$-vertex graphs. Before going on, shouldn't we ask ourselves whether this formula exists at all? The question is rhetorical, but there are not so many precedents when mathematicians do this pre-work. The good examples are Galois, Lobachevsky and P. Cohen. The first gave a criterion when a polynomial $f$ is not soluble by radicals (when its Galois group is not soluble). Two others used models (of geometry and set theory, respectively) where some assertion doesn't hold (so it was useless to prove this assertion). Basically, my question is whether we can extend Galois' (or Lobachevsky's) approach to show the non-existence of some general class of explicit combinatorial formulas.