There have been derivations for the Sums of Powers published since the sixteenth century. All techniques have used recursive processes, producing the following formula in the series. I present a new method that calculates the Sums of Powers and Harmonic Numbers. Starting with a novel relationship between Pascal’s Numbers and Stirling’s Numbers of the First Kind, the Sums of Powers is developed. This formula, published previously using a different methodology, is in terms of Pascal Numbers multiplied by constant coefficients. However, a further step is introduced. A recursive relationship is discovered among the coefficients of these formulae. A double sigma master formula is developed, allowing one to calculate all formulae for Sums of Powers without needing Bernoulli Numbers. Finally, from the Sums of Powers master formula, I derive a formula to calculate the Bernoulli Numbers. I further develop a summation formula for the Harmonic Numbers using the same relationships.

Publication Date


Content Type


Publisher's Site:

View at Publisher Website


This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent.

Open Access

Available to all.