  This paper presents a new methodology to count the number of numerical
semigroups of given genus or Frobenius number. We apply generating function
tools to the bounded polyhedron that classifies the semigroups with given genus
(or Frobenius number) and multiplicity. First, we give theoretical results
about the polynomial-time complexity of counting the number of these
semigroups. We also illustrate the methodology analyzing the cases of
multiplicity 3 and 4 where some formulas for the number of numerical semigroups
for any genus and Frobenius number are obtained.
