On 2-adic orders of some binomial sums

Details On 2-adic orders of some binomial sums On 2-adic orders of some binomial sums

2009-09-27

arxiv.org/abs/0909.4945v3

J. Number Theory 130(2010), no.12, 2701-2706

Mathematics

Combinatorics

6 pages

Scientific paper

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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