A Fast Algorithm to Calculate Power Sum of Natural Numbers

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.