Abstract
For all positive integers $n$, we prove the following divisibility properties:\[ (2n+3){2n\choose n} \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers $m$ and $n$, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by $m+n$. In fact, the latter result can be further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.
Highlights
In [18, 19], Z.-W
For all positive integers n, we prove the following divisibility properties: 2n 6n 3n
We show that, for all positive integers m and n, the product am am+bm−1 am an+bn an is divisible by m + n
Summary
Sun proved some divisibility properties of binomial coefficients, such as. In this paper we first prove the following two results conjectured by Z.-W. Theorem 1.3 Let n be a positive integer. Let Z denote the set of integers. Another result in this paper is the following. Letting a = b = 1 in (1.10), we get the following result, of which a combinatorial interpretation was given by Gessel [9, Section 7]. Corollary 1.5 Let m, n be positive integers. The proofs of Theorems 1.1–1.3 will be given in Sections 3–5 respectively.
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