Consider the expression $ a_{22}-2a_{23}+a_{24}-2a_{32}+4a_{33}-2a_{34}+a_{42}-2a_{43}+a_{44}$. It can be directly verified that (1) This expression remains invariant when an arithmetic progression is added to any row or column. (2) If each row of numbers in the table is an arithmetic progression, the expression is 0. (3) The expression is 4 for the initial table.

The polynomial in question has an $ x^2 y^2$ term, so has a nonzero fourth partial derivative $ f_{xxyy}$. But arithmetic progressions are linear functions, so their second derivative vanishes. Taking a $ f_{xxyy}$ derivative cancels both the row progressions and the column progressions, so the fourth derivative never changes. The strange looking invariant above is just the discrete analogue of this derivative. Alternatively, you can arrive at the expression by realizing that if you want to do an invariant argument, you want something that adds to 0 for arithmetic progressions. For one-dimensional progressions, you have $ a_{i-1}+a_{i+1}=2a_i$, and the question is just what the proper 2-dimensional analogue of this is.