Interpretation of generating function infinite product

Question

Let $P$ denote the set of primes and let $s\in\{-1,1\}$. How can you interpret the coefficient of $x^n$ in the power series expansion of $$\prod_{p\in P} (1+sx^p)^s$$

for either choice of $s$? I found how to give an interpretation to $s=1$ but have no idea with $-1$. Thank you for helping

EDIT: for s=1 i said that i can interpret the coeficient as the number of ways to pay the amount k cents with only coins of prime cents (with only one coin for each prime).

Answer 1

Community wiki answer based on the comments so that the question can be marked as answered:

For $s=1$, the coefficient counts the number of ways of writing $n$ as a sum of distinct primes.

For $s=-1$, the coefficient counts the number of ways of writing $n$ as a sum of primes.