Analytic Combinatorics – A Worked Example

This article is technically way beyond me, but can anybody explain to me what on earth a "worked example" is? That expression is completely incoherent.

It's an idiom in math meaning "a problem, together with its solution, that illustrates a general principle or technique".

A worked example of the quadratic formula would be solving a particular equation by plugging in values of a, b, c into the formula.

A worked example of analytic combinatorics is solving a combinatorics problem with techniques from real/complex analysis.