2010/05/06 by Zhi-Wei Sun, Sun, Zhi-Wei
Mathematics · #05A10 #11A07 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65
paper · pdf · doi:10.48550/arxiv.1005.1054
18 pages. The current (1.12) is new
arxiv created 2010/05/31 · arxiv updated 2010/06/01
Let k and n be positive integers. We mainly show that (ln+1) | k\binomkn+lnkn, 2\binomknn | \binom 2nnC2n(k-1), \binomknn | (2k-1)Cn\binom2kn2n, \binom2nn | (k+1)Cn(k-1)\binom2knkn, 2k-1\binom2nn | \binom2(2k-1)n(2k-1)nCn(2k-2), (6n+1)\binom5nn | \binom3n-1n-1C3n(4), and \binom3nn | \binom5n-1n-1C5n(2), where Cn denotes the Catalan number \binom2nn/(n+1), and Cm(h) refers to the Catalan number \binom(h+1)mm/(hm+1) of order h.