\[\sum_{j=0}^{n}j^{k}x^{j}=\sum_{j=0}^{k}S\left(k,j\right)x^{j}\frac{{\mathrm{d}%
}^{j}}{{\mathrm{d}x}^{j}}\left(\frac{1-x^{n+1}}{1-x}\right),\]