Parameterized Finite Binomial Sums.

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑ k = 0 , k ≠ n m + n (− 1) k n − k m + n k = (− 1) n m + n n (H m − H n) , where n , m are non-negative integers and H n is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. [ABSTRACT FROM AUTHOR]