Generalized Nested Summation of Powers of Natural Numbers

TL;DR

This paper introduces a simple, unified framework using a special triangular array and binomial coefficients to derive formulas for nested summations of powers of natural numbers, enhancing understanding and computation.

Contribution

It presents a novel, straightforward method for calculating nested sums of powers of natural numbers using a triangular array and binomial coefficients.

Findings

Provides a general formula for nested summations of powers

Introduces a special triangular array for computation

Simplifies the process of deriving nested sums

Abstract

In this paper, we provide a general framework for obtaining the formula for nested summation of powers of natural numbers. We define a special triangular array of numbers from which we can obtain the formula for nested summation of natural numbers at any particular power. Binomial coefficients play a key role in nesting. Our framework is very simple to understand.