I don't know that it is of the type you wish, but there is a formula of sorts.

Consider the more general case of a finite group G acting diagonally by conjugation on the set of k-tuples of elements of G.  For a fixed g in G, the number of fixed points of g is |C_G(g)|^k.  Let g_1,...,g_c be a set of representatives for the conjugacy classes of G.  Applying Burnside's Lemma and grouping together elements in the same conjugacy class, we see that the number of orbits of G in the given action is the sum over all g_i of |C_G(g_i)|^{k-1}.

So, in your case, we have the sum of |C_G(g_i)| over a set of representatives for the conjugacy classes of S_n.  As you noted, these classes are parameterized by partitions of n, and if such a partition p has a_j parts of size j for each j in [n], the order of the corresponding centralizer is the product over all such j of (a_j)!j^{a_j}.  Thus we get the sum over all partitions of n of such products.

Maybe it is worth remarking that, for any G, when k=2 the number in question is the sum of the square norms of the entries of the character table of G.