  We completely generalize previous results related to the counting of
connected Feynman diagrams. We use a generating function approach, which
encodes the Wick contraction combinatorics of the respective connected
diagrams. Exact solutions are found for an arbitrary number of external legs,
and a general algorithm is implemented for this calculus. From these solutions,
we calculate many asymptotics expansion terms for a simple analytical tool
(Taylor expansion theorem). Our approach offers new perspectives in the realm
of Feynman diagrams enumeration (or zero-dimensional quantum field theory).
