  Early 17th-century mathematical publications of Johann Faulhaber contain some
remarkable theorems, such as the fact that the $r$-fold summation of
$1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd
number. The present paper explores a computation-based approach by which
Faulhaber may well have discovered such results, and solves a 360-year-old
riddle that Faulhaber presented to his readers. It also shows that similar
results hold when we express the sums in terms of central factorial powers
instead of ordinary powers. Faulhaber's coefficients can moreover be
generalized to factorial powers of noninteger exponents, obtaining asymptotic
series for $1^{\alpha}+2^{\alpha}+...+n^{\alpha}$ in powers of
$n^{-1}(n+1)^{-1}$.
