The Nottingham group is defined as the group of formal power series over a unital ring $R$ under the operation of substitution, giving elements of the form $t(1+\sum_{i=1}^{\infty}a_{i}t^{i})$ with the coefficients $a_{i} \in R$.
The fact that this indeed forms a group was proven by S Jennings in his paper "Substitution groups of formal power series". I'm particularly interested how one calculates the inverse of a given element in this group, as the proof in Jennings shows that the coefficients of the inverse are defined inductively in terms of the element you want the inverse of. The reason this confuses me is the fact that you must also subtract the sum of some polynomials, labelled $\phi_{s}$ in the paper, and there doesn't seem to be any indication of how to calculate what these are.
For example, even taking some "nice" element such as $e_{1} := t(1+t)$ doesn't seem to have an easy solution for the inverse. We can define $a_{1} = -1$ for $e_{1}^{-1}$, but $t(1-t)$ on it's own doesn't work and requires further terms in its formal power series.
Here is a link to the paper I'm talking about: https://cms.math.ca/10.4153/CJM-1954-031-9
The result is labelled (1.1.3).