On the Asymptotic Analysis of a Class of Linear Recurrences

Many problems in enumerative combinatorics can be expressed via linear recurrences whose generating functions satisfy a functional equation. I present the asymptotic analysis for a class of recurrences which lead to a functional equation involving a transformation with a parabolic fixed point. The method used relies heavily on analytic iteration theory. Examples given are Bell numbers, partition lattice enumerations, and the Takeuchi recursion.