summation - A formula for the power sums: $1^n+2^n+dotsc +k^n=,$?

Is there explicit formula for the expression $1^n + 2^n + \dotsc + k^n\,$?

I know that for $n=1$ the explicit formula becomes $S=k(k+1)/2$ and for $n=3$ the formula becomes $S^2$. But what about general $n$?

I know there is a way using the Taylor expansion of $f(x)=1/(1-x)=1+x+x^2+\dotsc\;$, by differentiating it and then multiplying by $x$ and then differentiating again. Repeating this $n$ times, we get

$$\frac{d}{dx}(x\frac{d}{dx}(\dots x\frac{d}{dx}f(x))\dots )=1+2^nx^n+3^nx^n\dots.$$

Now do the same process but with the function $g(x)=x^{k+1}f(x)$. Then subtract them and we get $1+2^nx^n+\dots k^nx^n$. Because we have the explicit formulas $f(x)$ and $g(x)$ we can find the explicit formula by this process for arbitrary $n$. A big problem is that as $n$ grows, it is going take a lot of time finding the explicit formula. My question is therefore: are there other ways?