A Combinatorial Identity and Its Related Conjecture

Abstract

In recent researches on a discriminant for polynomials, I faced a recursive (combinatorial) sequence \(\lambda_{n,m}\) whose first four terms and identities are \(\lambda_{0,m} := \binom{m}{0}\), \(\lambda_{1,m} := \binom{m}{1}=\binom{m}{m-1}\), \(\lambda_{2,m} := {\binom{m}{2}}^2 – \binom{m}{2}=\binom{m+1}{m-1}\), and \(\lambda_{3,m} = {\binom{m}{1}}^3 – 2\binom{m}{1}\binom{m}{2} + \binom{m}{3}=\binom{m+2}{m-1}\). In this paper, I introduce this sequence, prove an identity concerning it, and leave a problem and a conjecture regarding its properties.