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

作者：Yang, Quan-Hui^*

来源：Ars Combinatoria, 2017, 135: 213-221.

• binomial coefficients
• p-adic order
• divisibility properties

Let s(n, k) =
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/((2n - 1)
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.
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.

• 出版日期2017-10
• 单位南京信息工程大学