On 2-adic orders of some binomial sums

We prove that for any nonnegative integers n and r the binomial sum∑k=−nn(2nn−k)k2r is divisible by 22n−min{α(n),α(r)}22n−min{α(n),α(r)}, where α(n)α(n) denotes the number of 1s in the binary expansion of n. This confirms a recent conjecture of Guo and Zeng [J. Number Theory 130 (2010) 172–186].