Multiplicative Inverse of a Power Series

Write $F = 1 + F'$ where $F'$ has constant coefficient $0$. The inverse $F^{-1}$ satisfies $$(1 + F')F^{-1} = 1$$ Therefore $$F^{-1} = (1 + F')^{-1}$$ Now expand using the generalized binomial theorem, \begin{align*} F^{-1} & = \displaystyle\sum_{n \geq 0} \binom{-1}{n}(F')^n\\ &= \displaystyle\sum_{ n \geq 0} (-1)^n (F')^n \end{align*} Letting $F = \displaystyle\sum_{n \geq 0} a_nx^n$, $F' = \displaystyle\sum_{n \geq 1} a_{n}x^n$,

\begin{align*} F^{-1} &= \displaystyle\sum_{n \geq 0} (-1)^n \Big(\sum_{i \geq 1} a_{i}x^i\Big)^n\\ &= \displaystyle\sum_{n \geq 0} (-1)^n \Big(\sum_{\substack{ \beta_1, \beta_2,\dots,\\\sum_{i} \beta_i = n}} \binom{n}{\beta_1, \beta_2,\dots} \prod_{i \geq 1} (a_{i}x^i)^{\beta_i} \Big)\\ &= \displaystyle\sum_{n \geq 0}\sum_{\substack{\beta_1, \beta_2,\dots,\\\sum_{i} \beta_i = n}} \Big((-1)^n\binom{n}{\beta_1,\beta_2,\dots}\prod_{i \geq 1} a_{i}^{\beta_i}\Big) x^{\sum_i i\beta_i} \end{align*}

where the inner sum is over all natural number sequences $\langle \beta_i\rangle$, and $\binom{n}{\beta_1,\beta_2, \dots}$ is a multinomial coefficient. Grouping terms by exponent on $x$, we have the somewhat-closed form $$F^{-1} = \displaystyle\sum_{n \geq 0} \Bigg(\sum_{\substack{\beta_1, \beta_2, \dots\\\sum_{i}i\beta_i= n}} (-1)^{\sum_i \beta_i}\binom{\sum_i \beta_i}{\beta_1, \beta_2, \dots} \prod_{i \geq 1} a_i^{\beta_i}\Bigg) x^n $$

For the multiplicative inverse, applying Faà di Bruno's formula (Wikipedia: Bell polynomials - Faà di Bruno's formula) yields the following formula.

$$q_{i}=\frac{1}{i!}\sum_{k=0}^{i}(-1)^{k}k!p_{0}^{-(k+1)}B_{i,k}(1!p_{1},2!p_{2},...,(i-k+1)!p_{i-k+1})$$

$B_{i,k}$: Partial exponential Bell polynomial (Wikipedia: Bell polynomials - Exponential Bell polynomials)

This is written e.g. in Singh, M.: nth-order derivatives of certain inverses and the Bell polynomials. J. Phys. A 23 (1990) (12) 2307-2313.

$$q_{i}=\sum_{k=0}^{i}(-1)^{k}p_{0}^{-(k+1)}\hat{B}_{i,k}(p_{1},p_{2},...,p_{i-k+1})$$

$\hat{B}_{i,k}$: Partial ordinary Bell polynomial (Wikipedia: Bell polynomials - Ordinary Bell polynomials)

For certain formal power series, there is a general formula for $B_{i,k}$ and $\hat{B}_{i,k}$.

I wrote an article "On partial Bell polynomials for the higher derivatives of composed functions".