  For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold
summation (or hyper-sum) over the first $n$ positive integers to the $m$th
powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we
derive a new determinantal formula for $S_m^{(r)}(n)$. Specifically, we show
that, for all integers $r\geq 0$ and $m \geq 1$, $S_m^{(r)}(n)$ is proportional
to $S_1^{(r)}(n)$ times the determinant of a lower Hessenberg matrix of order
$m-1$ involving the Bernoulli numbers and the variable $N_r = n + \frac{r}{2}$.
Furthermore, whenever $r\geq 1$, evaluating this determinant gives us
$S_m^{(r)}(n)$ as $S_1^{(r)}(n)$ times an even or odd polynomial in $N_r$ of
degree $m-1$, depending on whether $m$ is odd or even.
