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Does this expression always vanish?
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 …