  In their study of water waves, Massimiliano Berti, Livia Corsi, Alberto
Maspero, and Paulo Ventura, came up with two intriguing conjectured identities
involving certain weighted sums over the Boolean lattice. They were able to
prove the first one, while the second is still open. In this methodological
note, we will describe how to generate many terms of these types of weighted
sums, and if in luck, evaluate them in closed-form. We were able to use this
approach to give a new proof of their first conjecture, and while we failed to
prove the second conjecture, we give overwhelming evidence for its veracity. In
this second version, we are happy to announce that Mark van Hoeij was able to
complete the proof of the second conjecture, by explicitly solving the
second-order recurrence mentioned at the end.
