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Let $a_n$ is a binomial sum, for example $$ a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k} \quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...