Kervaire and Milnor found a formula for the number of smooth structures on the $4n - 1$ sphere (see, e.g. the last part of this MO answer). It is relatively easy to compute the number of smooth structures on the $k$ sphere for other values of $k$ (aside from $k = 4$).

Has there been work finding formulas for the number of smooth structures on elements of other infinite classes of manifolds?