Series expansions of inverse polynomials

Question

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$

So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the inverse value directly using a power series, i.e.:

$$x = \sum_{n=0}^\infty b_n y^n$$

Is there a method of calculating the coefficients for the power series for arbitrary $N$? how about for $N<5$? $$$$ _{*I'm aware of course that for practical applications iterative methods are the way to go, but this seems like more fun.}

Answer 1

This is known as Series Reversion.

The analytical expression for the inverse coefficients can be written out, but it gets quite complex for higher $n$.