Calculating the regularized sum
$$\sum_{n=1}^{\infty}n^{k}$$
in two different ways gives the following relation:
$$\zeta(-k)=(-1)^{k-1}\sum_{\ell=1}^{k}\ell!S_{2}(k,\ell)G_{\ell+1}$$
where $S_{2}(k,\ell)$ are Stirling numbers of second kind and $G_{\ell}$ are the Gregory coefficients,
$$\frac{z}{\text{log}(1+z)}=\sum_{\ell=0}^{\infty}G_{\ell}z^{\ell}$$
Is there a proof of this relation?