TITLE: What is Lagrange Inversion good for?
QUESTION [71 upvotes]: I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of very few applications for it -- basically just enumeration of trees and some slight variants on this. I checked van Lint and Wilson, Enumerative Combinatorics II (but not the exercises) and Concrete Mathematics, and they all only present this application.
So, besides counting trees, where can we use Lagrange inversion?

REPLY [4 votes]: An elementary proof of Rodrigues' formula for Legendre polynomials (which is usually done via the orthogonality properties of the polynomials). If we define the polynomials by the classical generating function,
$$ \frac{1}{\sqrt{1-2z_{0}w+w^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(z_{0}\right)w^{n} $$
and consider the solution $z(w)$ of $$w=f(z)=2\frac{z-z_{0}}{z^{2}-1}$$ fixed by $z(0)=z_0$, then the direct application of the Lagrange's formula (formulated for the solution $z(w)$ of the equation $w=f(z)$ with $w_0=f(z_0)$)
$$\frac{\Phi\left(z\left(w\right)\right)}{1-\left(w-w_{0}\right)\left.\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{z-z_{0}}{f(z)-w_{0}}\right)\right|_{z=z(w)}}=\sum_{n=0}^{\infty}c_{n}\left(w-w_{0}\right)^{n},\quad c_{n}=\frac{1}{n!}\left.\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\left[\Phi\left(z\right)\left(\frac{z-z_{0}}{f(z)-w_{0}}\right)^{n}\right]\right|_{z=z_{0}}$$
to $f(z)$, $w_0=f(z_0)=0$ and $\Phi(z)\equiv 1$ gives
$$P_{n}(z)=\frac{1}{2^{n}n!}\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\left[\left(z^{2}-1\right)^{n}\right].$$