We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra . Given two non-negative integer sequences and , with , it is well-known that if the identity holds for the generating functions and , then is the number of rooted planar trees with vertices such that each vertex having children may be colored with any one of distinct colors. Kuperberg proved a specific case when this identity holds, namely when , where is the 7-dimensional fundamental representation of , and is the number of triangulations of a regular -gon such that each internal vertex has degree at least . He also observed that and conjectured that this estimate is sharp, or, in terms of power series, that the radius of convergence of is exactly . We prove this conjecture by introducing a new criterion for sharpness in the analogous estimate for general power series and satisfying . Moreover, by way of singularity analysis performed on a recently discovered generating function for , we significantly refine the conjecture by deriving an asymptotic formula for the sequence .
Mathematics Subject Classifications: 05A16, 05E10
Keywords: Analytic combinatorics