Logarithms of Catalan Generating Functions: A Combinatorial Approach

We analyze the combinatorics behind the operation of taking the logarithm of the generating function Gₖ for kᵗh generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of tree graphs. Using explicit bijections, we are able to recover known closed expressions for the coefficients of log Gₖ by purely combinatorial means of enumeration. The non-algebraic proof easily generalizes to higher powers logᵃ Gₖ, a≥ 2.