  By applying the Newton-Gregory expansion to the polynomial associated with
the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a
couple of infinite families of explicit formulas for $S_k(n)$. One of the
families involves the $r$-Stirling numbers of the second kind
$\genfrac{\{}{\}}{0pt}{}{k}{j}_r$, $j=0,1,\ldots,k$, while the other involves
their duals $\genfrac{\{}{\}}{0pt}{}{k}{j}_{-r}$, with both families of
formulas being indexed by the non-negative integer $r$. As a by-product, we
obtain three additional formulas for $S_k(n)$ involving the numbers
$\genfrac{\{}{\}}{0pt}{}{k}{j}_{n+m}$, $\genfrac{\{}{\}}{0pt}{}{k}{j}_{n-m}$
(where $m$ is any given non-negative integer), and
$\genfrac{\{}{\}}{0pt}{}{k}{j}_{k-j}$, respectively. Moreover, we provide a
formula for the Bernoulli polynomials $B_k(x-1)$ in terms of
$\genfrac{\{}{\}}{0pt}{}{k}{j}_{x}$ and the harmonic numbers.
