Abstract
We study the following family of integral-valued alternating sums, where and are integers: We first consider for and non-negative integers and show that it is of the form , where may be represented as a polynomial of degree in , or expressed as a non-polynomial closed form given by a sum of binomial numbers. We then consider for a negative integer and for a non-negative integer. This reveals, in particular, that for , that for . We also show that is a polynomial of degree in , for fixed , with , and we give expressions for the coefficients.
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