This is a translation of a previous Portuguese post of mine.
Show that for all integers the following identity holds
In order to have a short notation and because these numbers are the Apéry Numbers, let´s denote them ,
because for every term equals zero and also .
Multiplying both sides of the identity proved in my post “Uma proposição da análise combinatória” (here) [see “caderno” in the PDFs page] by
Now summing in , we get successively
which proves the proposition.