Repeated sums and binomial coefficients

Binomial coefficients have been used for centuries in a variety of fields and accumulated numerous definitions. In this paper, we introduce new way defining binomial as repeated sums ones. A multitude coefficient identities will be shown order to prove definition. Using definition, simplify some particular such the Harmonic sum Binomial-Harmonic sum. We derive formulae simplifying general <i> sums</i> well variant containing coefficients. Additionally, study \(m\)-th difference sequence show how sequences whose is constant can related