We know the Dirichlet $\eta$-function is defined as the analytic continuation of $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} \quad \Re(s)>0$$ I find an identity for the values of this function at negative integers: $$\eta(-n)+ \sum_{k=0}^n {n\choose k}(-1)^k \eta(k-n) =0.$$ where $1<n\in\mathbb{Z}$.
I found this using some "computation" on divergent series. But this wrong computation cannot serve as a proof.
I think this identity should be true, though I haven't found a simple proof.
Do you know anything about this identity? If it's true, how to prove it? Is there any generalization (for arbitrary $s$ rather than $-n$) of this identity? Could you provide me some reference?