Sums of powers of integers and generalized Stirling numbers of the second kind

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 ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_r$, $j=0,1,\dots ,k$, while the other involves their duals ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_{-r}$, 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 ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_{n+m}$, ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_{n-m}$ (where $m$ is any given non-negative integer), and ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_{k-j}$, respectively. Moreover, we provide a formula for the Bernoulli polynomials $B_k(x-1)$ in terms of ${\left\{\genfrac{}{}{0px}{}{k}{j}\right\}}_x$ and the harmonic numbers.