I noticed that
In the difference on the RHS, it is apparent that terms without or will vanish. Thus, all the negative terms which are not cancelled out have a and all such positive terms have a . Combinatorially, all terms of degree with can be generated by multiplying on all terms of degree . Analogous holds for the positive terms. The terms with only and are cancelled out with the exception of the that remains.
This recurrence appears in calculation of the determinant of the Vandermonde matrix.