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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
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Showing an alternating integer series is never $0$
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 neg …