I have the following system of $m+n$ linear equations for $m+n+1$ variables $x_0,\dots, x_{n+m}$:
For $k<n$ ($n$ equations):
$$x_k=0$$
and for $k<m$ ($m$ equations):
$$\sum_{l=0}^{m+n-k}{k+l \choose l}x_{k+l}=0$$
I know that (up to rescaling by a constant) the solution to this system of equations is given by
$$x_k=(-1)^k{m \choose k-n}$$
But I'm not sure how I would go about deriving that (without knowing the answer in advance). Could I have some assistance?
EDIT: In fact, I can't even prove that the equations are satisfied by these $x_k$!
$$\sum_{l=0}^{m+n-k}(-1)^{k+l}{k+l \choose l}{m \choose k+l-n}=0$$
I "know" that it is true (for $k<m$) from experiments in Mathematica, but can't seem to make progress beyond that.