*The reps of the Lagrange inversion formula (LIF) in different “coordinate systems” are intrinsically interesting*.

Consider a compositional inverse pair of functions, $h$ and $h^{-1}$, analytic at the origin with $h(0)=0=h^{-1}(0)$.

Then with $\omega=h(z)$ and $g(z)=1/[dh(z)/dz]$,

$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]f(z) = \exp \left[ {t\frac{d}{{d\omega }}} \right]f[{h^{ - 1}}(\omega )] = f[{h^{ - 1}}[t + \omega]] = f[{h^{ - 1}}[t + h(z)]],$$ so
$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]z |_{z=0}=h^{-1}(t)$$

(see OEIS A145271 and A139605 for more relations).

With the power series rep $h(z)= c_1z + c_2z^2 + c_3z^3 + ... ,$

$$\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{c_1^{9}} [14 c_2^{4} - 21 c_1 c_2^2 c_3 + c_1^2[6 c_2 c_4+ 3 c_3^2] - 1 c_1^3 c_5],$$

which is the coefficient of the fifth order term of the power series for $h^{-1}(t)$. This is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or 3-D associahedron, with **14** vertices (0-D faces), **21** edges (1-D faces), **6** pentagons (2-D faces), **3** rectangles (2-D faces), **1** 3-D polytope (3-D faces).

This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2) term of the compositional inverse holds in general, (see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373). These refined partition polynomials are also a refined presentation of the number of diagonal dissections of a convex n-gon (A033282) or, equivalently, the refined numbers for a set of Schroeder lattice paths (A126216), which sum to the little Schroeder numbers (A001003).

If $h(z)$ is presented as a Taylor series, the LIF A134685 is obtained, which is related to the tropical Grassmannian G(2,n) and phylogenetic trees (A134991) and to the partitioning of 2n elements into n groups.

When the invertible function $h(t)$ is represented as a power series of its own reciprocal, $t/h(t)$, the refined Narayana numbers are obtained (A134264), which are the refined h-polynomials of the simplicial complexes (A001263) dual to the Stasheff associahedra and also a refined enumeration of a set of Dyck lattice paths A125181 and noncrossing partitions, which sum to the Catalan numbers A000108.

Also, the "infinitesimal generators" A145271 for these reps have very interesting associations (e.g., to permutahedra, surjections, and multiplicative reciprocals A019538/A049019, for the LIF A134685) and allow reps of the partition polynomials for A133437 as colored umbral binary trees related to refined Lah polynomials.

To illustrate an important application, you might look at OEIS-A074060 "Graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations)," as well as links in the LIF entries, to relate Lagrange inversion (or, equivalently, the Legendre transform) of series to the cohomology of moduli spaces.

For a less fancy application, the LIFs can sometimes be used, just as other transforms, such as the Fourier transform, to jump between "reciprocal" domains to simplify expressions to solve a problem, e.g., in conjunction with the OEIS to suggest generating functions for integer arrays by looking at their compositional inverses numerically.