Faulhaber's Formula - Faulhaber Polynomials

The term Faulhaber polynomials is used by some authors to refer to something other than the polynomial sequence given above. Faulhaber observed that if p is odd, then

is a polynomial function of

In particular:

 A000537


 A000539


 A000541


 A007487


 A123095

More generally,

\begin{align} 1^{2p+1} + 2^{2p+1} &+ 3^{2p+1} + \cdots + n^{2p+1}\\ &= \frac{1}{2^{2p+2}(2p+2)} \sum_{q=0}^p \binom{2p+2}{2q} (2-2^{2q})~ B_{2q} ~\left. \end{align}

The first of these identities, for the case p = 3, is known as Nicomachus's theorem. Some authors call the polynomials on the right hand sides of these identities "Faulhaber polynomials in a". The polynomials in the right-hand sides are divisible by a 2 because for j > 1 odd the Bernoulli number Bj is 0.

Faulhaber also knew that if a sum for an odd power is given by

then the sum for the even power just below is given by

Since a = n(n + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in n having factors n2 and (n + 1)2, while for an even power the polynomial has factors n, n + ½ and n + 1. As an application, the atomic numbers of every other alkaline earth metal (Be, Ca, Ba) are given by (4/3)n(n + 1/2)(n + 1).

Read more about this topic:  Faulhaber's Formula