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You might find of some interest my paper at arxiv.org/abs/1807.09290 which gives an application of products of cyclotomic polynomials with all coefficients 0 or 1.
The identity is a special case of Saalschütz's theorem. It can be generalized to $$\sum_{t\geq 0} \frac{a}{a+t} \binom{-k}{t} \binom{k-1}{n-t} = (-1)^n\frac{\binom{a}{k}}{\binom{n+a}k}.$$
