A refinement of Guo's theorem concerning divisibility properties of binomial coefficients

Let s(n, k) = [GRAPHICS] /((2n - 1) [GRAPHICS] . Recently, Guo confirmed a conjecture of Z.-W. Sun by showing that s(n, k) is an integer for k = 0, 1, ..., n. Let d = (3n + 2)/ gcd(3n + 2, 2n - 1). In this paper, we prove that s(n, k) is a multiple of the odd part of d for k = 0, 1, ..., n. Furthermore, if gcd(k, n) = 1, then s(n, k) is also a multiple of n. We also show that the 2-adic order of s(n, k) is at least the sum of the digits in the binary expansion of 3n.