On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $\sum_{k=-n}^n\binom{2n}{n-k}k^{2r}$ is divisible by $2^{2n-\min\{α(n),α(r)\}}$, where $α(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.

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