Sums of Powers of Integers and Bernoulli Numbers Clarified

Abstract

We exposes a very simple method for calculating at the same time the sums of powers of the first integers  and the Bernoulli numbers  .  This is possible thank to integrations of the equation  which lead to a formula saying that the vector   is the transform of the vector  by a matrix built from the Pascal triangle and obtainable by a simple algorithm. Very useful relations between the sums  , the Bernoulli numbers  are deduced, leading straightforwardly to known and new properties of them.  The proof of the Faulhaber formulae on powers sums are outlined briefly at the end.

We exposes a very simple method for calculating at the same time the sums of powers of the first integers  and the Bernoulli  numbers  .  This is possible thank to integrations of the equation  which lead to a formula saying that the vector   is the transform of the vector  by a matrix built  from the Pascal triangle and obtainable by a simple algorithm. Very useful relations between the sums  , the Bernoulli numbers  are deduced, leading straightforwardly to known and new properties of them.  The proof of the Faulhaber formulae on powers sums are outlined briefly at the end.