Solve this recurrence relation via a first order partial differential equation?

Question

Find a general formula for $a_{n,k}$ , for $n,k\geq1$.

We have initial values $a_{1,1}=1$, and $a_{1,k}=0$ for $k>1$.

The recurrence relation is:

$a_{n+1,1}=-a_{n,1}$ , for $n\geq1$

and

$a_{n+1,k}=-ka_{n,k}+ka_{n,k-1}$ , for $n\geq1$ and $k\geq2$ .