I was wondering about this. Consider a formal power series
$\sum_{n=1}^{\infty} a_n x^n$.
We can find its formal exponential, given by
$\exp\left(\sum_{n=1}^{\infty} a_n x^n\right) = \sum_{n=0}^{\infty} \frac{B_n(1! a_1, 2! a_2, \cdots, n! a_n)}{n!} x^n$
where $B_n$ is a complete Bell polynomial.
So what I wonder is, is there some kind of generalization, some kind of "super Bell polynomials" if you will, for a multivariate series like
$\sum_{\substack{(n,k) \in \mathbb{N}^2 \\ (n,k) \ne (0,0)}} a_{n,k} x^n y^k$
?
EDIT 1: Actually, I'm interested more in the case where the indices are like
$\sum_{n=1}^{\infty} \sum_{k=0}^{\infty} a_{n,k} x^n y^k$.
EDIT 2: I'm looking for a way to isolate the $a_{n,k}$ (that is, have a formula for them like we can get via the Bell polynomials in the univariate case.).