Say you want to find the inverse of a function, and it's not expressible in terms of known functions, so you want to find the taylor series for that inverse. But, you at least know the taylor series for your original function f(x), defined as $$f(x)=\sum\_{n=0}^{\infty} \frac{g(x,n)h(n)}{n!}$$ and it converges for a sufficient radius.
Since Taylor series stem from derivatives, and there is a theorem called the inverse function theorem that relatives derivatives to a functions' inverse 
$$(f^{-1})'(f(x))=\frac{1}{f'(x)},$$

is there any consequential theorem where you can derive a new Taylor series for the inverse of a function by using the original Taylor series for that original function? 
In other words, $$f^{-1}(x)=\sum\_{n=0}^{\infty} \frac{???}{n!}$$