Multivariate Lagrange inversion with zero powers (also asked in MO)
The multivariate Lagrange inversion formula, which I found in a couple of papers (such as this and this), is as follows. If $f_i=t_ig_i(f)$, $1\le i\le k$, then
$$[\boldsymbol{t^n}]h(\boldsymbol{f(t)})=\frac{1}{n_1n_2\cdots n_k}[\boldsymbol{x^{n-1}}]\sum_T \frac{\partial (h,g_1^{n_1},...,g_k^{n_k})}{\partial T},$$
where $\boldsymbol{t^n}=t_1^{n_1}\cdots t_k^{n_k}$ and the derivative is taken with respect to some trees (as discussed in those papers).
Not one of the papers in question has addressed the question of how this formula is to be used when some of the powers are zero, $n_j=0$, something that does not happen in the one-variable case (due to the assumption that $g(0)=0$) but can happen in the multivariable one.
 A: Let's see how the $n_i$ come as reciprocal factors into the formula. They come from the factorial denominators of the Taylor series terms.
Look at the one-variable Lagrange-Bürmann formula:
$$[t^n]h(f(t))=\frac{1}{n}[x^{n-1}](h'(x)g(x)^n).$$
It can be proved that
$$(h(f(t)))^{(n)}|_0=(h'(x)g(x)^n)^{(n-1)}|_0$$
Now we translate this from the derivatives to the Taylor series terms:
$$n![t^n](h(f(t)))^{(n)}|_0=(n-1)![x^{n-1}](h'(x)g(x)^n)^{(n-1)}|_0$$
$$n[t^n](h(f(t)))^{(n)}|_0=[x^{n-1}](h'(x)g(x)^n)^{(n-1)}|_0$$
$$[t^n](h(f(t)))^{(n)}|_0=\frac{1}{n}[x^{n-1}](h'(x)g(x)^n)^{(n-1)}|_0$$
This is your multivariate Lagrange-Bürmann formula for $k=1$.
In the multivariate case, $n!$ becomes $n_1!...n_k!$, and $(n-1)!$ becomes $(n_1-1)!...(n_k-1)!$. So we get the factor $\frac{1}{n_1...n_k}$.
That means, the formula cannot be applied for multinomial terms where an $n_i$ is $0$.
For $n=0$, Lagrange inversion formula and Lagrange-Bürmann formula have particular formulas.
$\ $
See Rosenkranz, M: Lagrange Inversion. Diploma thesis, RISC Linz 1997:
page 40: "It turns out that the inversion formulas in their second form can be generalized to the multivariate case in a very natural way. (For the first form of the inversion formulas, the multivariate generalizations are very complicated.)"
See theorem 42 at page 38, corollary 43 at page 39, and theorem 47 at page 41.
Corollary 43 (univariate case) and theorem 47 (multivariate case) present formulas for the general series coefficients without the factor $\frac{1}{\boldsymbol{n}}$.
