A Fast Algorithm to Calculate Power Sum of Natural Numbers

TL;DR

This paper introduces a fast algorithm for calculating the power sum of natural numbers by deriving the coefficient matrix and its inverse, utilizing permutation properties and generating functions.

Contribution

It presents a novel method for computing power sums efficiently through matrix derivations and permutation-based inverse matrix generation.

Findings

Coefficient matrix is lower triangular.

Inverse matrix is upper triangular.

Derived generating function for power sums.

Abstract

Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix while the inverse matrix is upper triangular. Permutations of n-th order are used to generate the inverse matrix. The generation function of natural numbers' power sum is derived to calculate the power sum.