Given a function $f$ and a point $x_0$, one should be able to find the Taylor Series of the inverse of $f$ about $f(x_0)$, using the derivatives of $f$. $$ f^{-1}(y) = \sum_{n=0}^\infty a_n (y-f(x_0))^n $$ So far, through some tedious (and kind of sketchy) working out, I have found the following: $$ \begin{align} a_0 &= x_0\\ a_1 &= \frac1{f'(x_0)}\\ a_2 &= \frac{f''(x_0)}{2f'(x_0)^3}\\ a_3 &= \frac{3f''\left(x_{0}\right)^{2}-f'\left(x_{0}\right)f'''\left(x_{0}\right)}{6f'\left(x_{0}\right)^{5}}\\ a_4 &= \frac{10f'\left(x_{0}\right)f''\left(x_{0}\right)f'''\left(x_{0}\right)-15f''\left(x_{0}\right)^{3}-f'\left(x_{0}\right)^{2}f''''\left(x_{0}\right)}{24f'\left(x_{0}\right)^{7}} \end{align} $$ Obviously, the complexity of each constant seems to grow exponentially, and I am afraid to even try to find $a_5$ because it might take all day. I don't see any obvious patterns, other than that the power in the denominator increases by $2$ each time, and that the constant in the denominator is given by $n!$.
Is there a formula for these constants?
The constants that I have am trying to find can be found by using the Lagrange–Bürmann formula, as people have helpfully brought up in the comments. $$ a_n = \lim_{w\to x_0} \frac{d^{n-1}}{dw^{n-1}}\left[\left( \frac{w-x_0}{f(w)-f(x_0)} \right)^n\right]. $$ It looks like this is the most compact general form these constants can take.