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On 2-adic orders of some binomial sums
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On 2-adic orders of some binomial sums
classification
math.CO
math.NT
keywords
alphabinomialadicbinarybinomconfirmsconjecturedenotes
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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 $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ 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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