Multiplicative Inverse of a Power Series For a formal power series
$$F(x) = \sum p_i x^i$$
a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion
$$q_0 =\frac{1}{p_0}\\
q_{n} = -\frac{1}{p_0}\sum_{0 \leq i < n}p_{n-i}q_{i}$$
What's the closed form of this recurrence? Writing out a bunch of terms hasn't yet revealed to me a pattern. It's helpful to assume $p_0 = 1$.
 A: 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
$$F^{-1} = \displaystyle\sum_{n \geq 0} \Bigg(\sum_{\substack{\beta_1, \beta_2, \dots\\\sum_{i}i\beta_i= n}} (-1)^{\sum_i \beta_i}\binom{\sum_i \beta_i}{\beta_1, \beta_2, \dots} \prod_{i \geq 1} a_i^{\beta_i}\Bigg) x^n $$
A: 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".
A: The formula given by Chris Jones seems to be the best available, and I just want to make a couple of remarks.
1) With $F = 1 + \displaystyle\sum_{n \geq 1} a_nx^n $, the formula 
\begin{align*}
F^{-1} &= \displaystyle\sum_{n \geq 0} (-1)^n \Big(\sum_{i \geq 1} a_{i}x^i\Big)^n
\end{align*}
is just a consequence of the geometric series expansion:
$$
\frac1{1+w} = \sum_{n\geq 0} (-1)^n w^n,
$$
with $w=\sum_{i \geq 1} a_{i}x^i$, ie, the "generalized binomial theorem" is not really needed.
2) The formula for $F^{-1}$ can be rewritten in terms of partitions of the natural number $n$. 
Indeed, partitions of $n$ (into equal or different non-zero parts) are in bijection with sequences of non-negative integers $(\beta_1,\cdots,\beta_n)$, satisfying $$\sum_{i}i\beta_i= n.$$
This gives the interpretation of each $\beta_i$ as the number of parts of size equal to $i$ (it can, of course, be zero). Moreover $\sum_{i}\beta_i$ becomes now the total number of parts of the partition (usually called the length of the partition).
Then, the formula admits this shorter version:
$$F^{-1} = \displaystyle\sum_{n \geq 0} \Bigg(\sum_{\beta=(\beta_i)\in P_n} (-1)^{|\beta|} |\beta| !  \prod_{i=1}^n \frac{a_i^{\beta_i}}{\beta_i !} \Bigg) x^n $$
where $P_n$ denotes the finite set of all partitions of $n$, and $|\beta|$ is the length of the partition $\beta = (\beta_1,\cdots,\beta_n)\in P_n$
A: I think that the solution can be derived from the formula
\begin{equation}\label{Sitnik-Bourbaki}\tag{1}
\frac{\textrm{d}^k}{\textrm{d}z^k}\biggl(\frac{u}{v}\biggr)
=\frac{(-1)^k}{v^{k+1}}
\begin{vmatrix}
u & v & 0 & \dotsm & 0\\
u' & v' & v & \dotsm & 0\\
u'' & v'' & 2v' & \dotsm & 0\\
\dotsm & \dotsm & \dotsm & \ddots & \dotsm\\
u^{(k-1)} & v^{(k-1)} & \binom{k-1}1v^{(k-2)} &  \dots & v\\
u^{(k)} & v^{(k)} & \binom{k}1v^{(k-1)} & \dots & \binom{k}{k-1}v'
\end{vmatrix}
\end{equation}
where $u=u(z)$ and $v=v(z)\ne0$ are differentiable functions.
For more information on the formula \eqref{Sitnik-Bourbaki}, please refer to another answer of mine at the site https://math.stackexchange.com/a/4261705/945479.
A: The Wronski's formula \eqref{f(t)g(t)=1=determ} below is a best answer to this question.
If $a_0\ne0$ and
$$
P(t)=a_0+a_1t+a_2t^2+\dotsm
$$
is a formal series, then the coefficients of the reciprocal series
$$
\frac{1}{P(t)}=b_0+b_1t+b_2t^2+\dotsm
$$
are given by
\begin{equation}\label{f(t)g(t)=1=determ}\tag{1}
b_r=\frac{(-1)^r}{a_0^{r+1}}
\begin{vmatrix}
a_1 & a_0 & 0 & 0 & \dotsm & 0& 0 & 0\\
a_2 & a_1 & a_0 & 0 & \dotsm & 0 & 0& 0\\
a_3 & a_2 & a_1 & a_0 & \dotsm & 0 & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots &\vdots & \vdots\\
a_{r-2} & a_{r-3} & a_{r-4} & a_{r-5} & \dotsm & a_1 & a_0 & 0\\
a_{r-1} & a_{r-2} & a_{r-3} & a_{r-4} & \dotsm & a_2 & a_1 & a_0\\
a_{r} & a_{r-1} & a_{r-2} & a_{r-3} & \dotsm & a_3 & a_2 & a_1
\end{vmatrix}, \quad r=1,2,\dotsc.
\end{equation}
Wronski's formula \eqref{f(t)g(t)=1=determ} can be found on page 17 Theorem 1.3 in the book [1], on page 347 in the paper [2], in Lemma 2.4 of the paper [3], in Section 2 of the paper [4], and the paper [5].
References

*

*P. Henrici, Applied and Computational Complex Analysis, Volume 1, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

*A. Inselberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl. 63 (1978), no. 2, 347--353; available online at https://doi.org/10.1016/0022-247X(78)90080-X.

*Feng Qi and Robin J. Chapman, Two closed forms for the Bernoulli polynomials, Journal of Number Theory 159 (2016), 89--100; available online at https://doi.org/10.1016/j.jnt.2015.07.021.

*H. Rutishauser, Eine Formel von Wronski und ihre Bedeutung fur den Quotienten-Differenzen-Algorithmus, Z. Angew. Math. Phys. 7 (1956), 164--169; available online at https://doi.org/10.1007/BF01600787. (German)

*M. H. Wronski, Introduction a la Philosophie des Mathematiques: Et Technie de l'Algorithmie, Chez COURCIER, Imprimeur-Libraire pour les Matheooatiqtte, quai des Augustins, n°57, Paris, 1811; available online at https://gallica.bnf.fr/ark:/12148/bpt6k6225961k. (French)

