The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 1$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.