Hi everyone!
There is a concept that is sometimes mentioned here and there. The Lagrange inversion theorem.
Let $$$x = f(y)$$$. We want to solve this equation for $$$y$$$. In other words, we want to find a function $$$g$$$ such that $$$y = g(x)$$$.
The Lagrange inversion theorem gives an explicit formula for the coefficients of $$$g(x)$$$ as a formal power series over $$$\mathbb K$$$:
In a special case $$$y = x \phi(y)$$$, that is $$$f(y) = \frac{y}{\phi(y)}$$$, which is common in combinatorics, it can also be formulated as
Finally, to avoid division by $$$k$$$ one may use (see the comment by Elegia) the following formula:
which may be formulated for $$$y = x \phi(y)$$$ as
Prerequisites
Familiarity with the following:
It is required to have knowledge in the following:
- Polynomials, formal power series and generating functions;
- Basic notion of analytic functions (e.g. Taylor series);
- Basic concepts of graph theory (graphs, trees, etc);
- Basic concepts of set theory (describing graphs, trees, etc as sets, tuples, etc).
I mention the concept of fields, but you're not required to know them to understand the article. If you're not familiar with the notion of fields, assume that we're working in real numbers, that is, $$$\mathbb K = \mathbb R$$$.
It is recommended (but not required) to be familiar with combinatorial species, as they provide lots of intuition to how the equations on generating functions below are constructed.