Suppose $S$ is a infinite set and $R\subset S$ is also infinite. Now, we want to find the number of multisets $(M,\nu)$, with $M\subset S, |(M,\nu)|=n$, and having an additional property that for every $x\in M$ with $2\nmid \nu(x)$, we must have, $x\in R$. How can we find the number of all such multisets $(M,\nu)$ of cardinality $n$. I have no clue how to tackle this. I am looking for a generating function for $f(n)$, where $f(n)$ denotes the required number for every $n\in \mathbb{N}$.

Suppose, if we omit the additional property, that is, if we try to find all multisets $(M,\nu)$ with cardinality $n$, then it is simply the coefficient of $u^n$ in

$$\prod_{u\in S} (1+u+u^2+\ldots)=\prod_{u\in S}(1-u)^{-1}$$ I don't understand how to get my result, by using the above, or otherwise.

Note: A multiset is a pair $(M,\nu)$, where $M$ is a set, and $\nu:M\to \mathbb{N}$. By cardinality of $(M,\nu)$, we mean $\sum_{x\in M}\nu(x)$.

Thank you for any kind of help!