  About four centuries ago, Johann Faulhaber developed formulas for the power
sum $1^n + 2^n + \cdots + m^n$ in terms of $m(m+1)/2$. The resulting
polynomials are called the Faulhaber polynomials. We first give a short survey
of Faulhaber's work and discuss the results of Jacobi (1834) and the less known
ones of Schr\"oder (1867), which already imply some results published
afterwards. We then show, for suitable odd integers $n$, the following
properties of the Faulhaber polynomials $F_n$. The recurrences between $F_n$,
$F_{n-1}$, and $F_{n-2}$ can be described by a certain differential operator.
Furthermore, we derive a recurrence formula for the coefficients of $F_n$ that
is the complement of a formula of Gessel and Viennot (1989). As a main result,
we show that these coefficients can be expressed and computed in different ways
by derivatives of generalized reciprocal Bernoulli polynomials, whose values
can also be interpreted as central coefficients. This new approach finally
leads to a simplified representation of the Faulhaber polynomials. As an
application, we obtain some recurrences of the Bernoulli numbers, which are
induced by symmetry properties.
