\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{align} One obvious case of the identities above is this: $$ y = \frac x {1- x}, \qquad x = \frac y {1+y}. $$ What is known about the nature of the mapping $a = (a_1,a_2,a_3,\ldots) \mapsto b = (b_1,b_2,b_3,\ldots) \text{?}$
Clearly $a\mapsto b$ is an involution.
Each $b_n$ is a polynomial in $a_1,\ldots,a_n$. The coefficients are integers. In the $n$th term of the polynomial, the sum of the indices (i.e. the subscripts) is $n$. Thus each term corresponds to a partition of $n$. One must wonder what the combinatorial interpretation of the coefficients is.