Hypergraphs and a functional equation of Bouwkamp and de Bruijn

Let Φ ( u , v ) = ∑ m = 0 ∞ ∑ n = 0 ∞ c mn u m v n . Bouwkamp and de Bruijn found that there exists a power series Ψ ( u , v ) satisfying the equation t Ψ ( tz , z ) = log ∑ k = 0 ∞ t k k ! exp ( k Φ ( 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.