  We compute the number of (weak) equivalence classes of branched covers from a
surface of genus g to the sphere, with 3 branching points, degree 2k, and local
degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2),
(d_1,...,d_m), for several values of g and h. We obtain explicit formulae of
arithmetic nature in terms of the d_i's. Our proofs employ a combinatorial
method based on Grothendieck's dessins d'enfant.
