On 2-adic Orders of Some Binomial Sums

We prove that for any nonnegative integers n and r the binomial sum n ∑ k=−n ( 2n n− k ) k is divisible by 22n−min{α(n),α(r)}, where 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]. In 1976 Shapiro [3] introduced the Catalan triangle ( k n ( 2n n−k ) )n>k>1 and determined the sum of entries in the nth row; namely, he showed that