Hypergraphs and a functional equation of Bouwkamp and de Bruijn

Let Φ(u, v) = ∑ ∞ m=0 ∑ ∞ n=0 cmnu v. Bouwkamp and de Bruijn found that there exists a power series Ψ(u, v) satisfying the equation tΨ(tz, z) = log (∑ ∞ k=0 t k k! exp(kΦ(kz, z)) ) . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u, 0) and Ψ(u, 0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversion to count hypertrees by number of vertices and number of edges of a specified size.