#### Abstract

We treat the problem of normally ordering expressions involving the standard boson operators a, a where [a, a] = 1. We show that a simple product formula for formal power series — essentially an extension of the Taylor expansion — leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions — in essence, a combinatorial field theory. We apply these techniques to some examples related to specific physical Hamiltonians.