I have checked that the following expression 

\begin{align}
\sum_{i=1}^N\sum_{j=1\\
j\ne i}^N\frac{A_iA_j(A_i+A_j)}{(A_i-A_j)^3}\prod_{k=1\\
k\ne i\\
k\ne j}^N\frac{A_i A_k}{(A_i-A_k)^2}
\end{align}
is zero for $N$ up to 7. I would like a general proof that it is zero for $N\in \mathbb Z_{+}$. I tried induction but I didn't find any inductive structure that I could make use of. 

This was previously asked on math stack exchange ( https://math.stackexchange.com/questions/4770113/prove-that-the-expression-is-identically-zero ) and since there was no answer, it was suggested that I post the question here.