From Simplest Recursion to the Recursion of Generalizations of Cross Polytope Numbers

My research project involves investigations in the mathematical field of combinatorics. The research study will be based on the results of Professors Steven Edwards and William Griffiths, who recently found a new formula for the cross-polytope numbers. My topic will be focused on ”Generalizations of cross-polytope numbers”. It will include the proofs of the combinatorics results in Dr. Edwards and Dr. Griffiths’ recently published paper. E(n,m) and O(n,m), the even terms and odd terms for Dr. Edward’s original combinatorial expression, are two distinct combinatorial expressions that are in fact equal. But there is no obvious algebraic evidence to show that they are equal. There are induction proofs in the paper. But I wondered if there is a better way to explain that at the undergraduate level, so I proved it algebraically with combinatorial identities. Ek(n,m) and Ok(n,m), which are the generalized forms for E(n,m) and O(n,m), are in fact equal and share the same recurrence formula with E(n,m) and O(n,m). We can call those numbers from the table of Ek and Ok the generalizations of the cross-polytope numbers.