Can we turn this specific product of sums into a sum of products?

Question

I am interested in turning this product into a sum (especially in the case N=4 and $A_j$ is a scalar): $$\prod_{k=1}^N\sum_{j=1}^k A_j$$ Do you see a nice way to do the combinatorics?

Answer 1

To add to Axel Kemper's response, we could move the product all the way to the right, but this would necessitate many sums.

$$\prod_{k=1}^N\sum_{j=1}^kA_j=\sum_{k_1=1}^1\sum_{k_2=1}^2...\sum_{k_N=1}^N\prod_{j=1}^NA_{k_j}$$