(Also asked on MSE)

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 $$ [t^n]h(f(t))=\frac{1}{n_1n_2\cdots n_k}[x^{n-1}]\sum_T \frac{\partial (h,g_1^{n_1},...,g_k^{n_k})}{\partial T},$$ where $t^n=t_1^{n_1}\cdots t_k^{n_k}$ 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 variabe case (due to the assumption that $g(0)=0$) but can happen in the multivariable one.

Enumerative Combinatorics, vol. 1, second ed. I give three proofs and some references. This result is also stated in my solution to mathoverflow.net/questions/350882. $\endgroup$