  From the perturbative expansion of the exact Green function, an exact
counting formula is derived to determine the number of different types of
connected Feynman diagrams. This formula coincides with the Arqu\`es-Walsh
sequence formula in the rooted map theory, supporting the topological
connection between Feynman diagrams and rooted maps. A classificatory
summing-terms approach is used, in connection to discrete mathematical theory.
