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Analytic Combinatorics – A Worked Example
Chris Grossack's math blog and professional website.
I’m a sucker for these kinds of combinatorial problems, and after finding a generating function solution I wanted to push myself to get an asymptotic approximation using Flajolet–Sedgewick style analytic combinatorics! Here the obvious marked point is our singularity $\omega$, and we’ve chosen a branch cut for $T$ (shown in blue) so that $T$ is holomorphic in the region where the pink curve lives. As a fun exercise, you might modify this code (using s.add_term()) to compute a longer puiseux series and get asymptotics valid up to a multiplicative error of $O \left ( \frac{1}{n^2} \right )$ for the number of rooted ternary trees.
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