Conjectures on the Sums of Powers of Natural Numbers Via Combinatorial Sequences

<p dir="ltr">We introduce a novel combinatorial framework for expressing the sums of powers of natural numbers, σₚ(n) = ∑ kᵖ, using binomial coefficients, their consecutive sums, and Narayana numbers. New identities and conjectures are presented, verified computationally up to p = 13, revealing unexpected patterns in the coefficients and a surprising connection to the denominators of Bernoulli numbers via the von Staudt-Clausen theorem. This approach offers an elementary yet deep perspective on a classical problem, bridging combinatorial insight with arithmetic computation.</p>