Is it possible to find the (distinct) coefficients of monomials such as
$$(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})^4\cdot(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\cdot(x_{1}+x_{2}+x_{3}+x_{4})^{2}$$
using the multinomial theorem?
Motivation
Below is a table of all of the different combinations of outer and inner exponents totalling $4$:
where the top row are the product inputs, and the subsequent rows show the coefficients of the right column. (More information on the application of this type of table is given in this article (PDF link), $§\ 4.3$, eg $12$.)
I am searching for a fast way to determine distinct coefficients of expansions of this type, given exponent sets, since the latter can be generated easily.
If this is not possible, is there an upper bound on the number of variables needed to generate the full set as shown above? I have gone with a safe $4$, but the article linked shows that only $2$ are needed for the fourth column, for example.
NB This is a companion question to this one on Mathematica SE.