Prove or disprove:

$$m(n,k,s)=\sum_{a_1=1}^n \sum_{a_2=1}^n \cdots \sum_{a_k=1}^n \min(a_1, a_2,\cdots, a_k)^s =$$

$$ \sum _{i=0}^{k-1} \frac{(-1)^i}{i!} F(n,i+s) \sum _{j=0}^{k-1} \frac{\partial ^i \left(n^j \binom{k}{j}\right)}{\partial n^i}$$

Where $$F(n,s)=\sum_{t=1}^{n}{t^s}$$ is Faulhaber's formula