2013/12/29 by Victor J. W. Guo, Guo, Victor J. W.
Mathematics · #05A10 #05A30 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:05A30 #msc:11B65
paper · pdf · doi:10.48550/arxiv.1312.7548
13 pages, add many new results
arxiv created 2014/01/02 · arxiv updated 2014/01/03
For all positive integers n, we prove the following divisibility properties: (2n+3)2n\choose n | 36n\choose 3n3n\choose n, and (10n+3)3n\choose n | 2115n\choose 5n 5n\choose n. 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 amam+bm-1\choose aman+bn\choose an is divisible by m+n. In fact, the latter result can be generalized to the q-binomial coefficients and q-integers case, which generalizes the positivity of q-Catalan numbers. We also propose several related conjectures.