Sums of Powers of Natural Numbers via Combinatorial Sequences

We have developed a combinatorial approach based on Pascal’s Triangle to address the classical problem of summing powers of natural numbers. This method relies on three key sequences derived from the triangle: the binomial coefficients, the Narayana numbers, and a variant of the binomial coefficients. Through these sequences, we have formulated three systematic strategies: the Direct Method (D.M), the Coefficient Method (C.M), and the Coefficient Analysis (C.A). These strategies enhance both arithmetic reasoning and the recognition of numerical patterns, yielding results equivalent to those obtained through purely algebraic approaches, such as telescoping sums.