On divisibility concerning binomial coefficients

Abstract

Let k and n be positive integers. We mainly show that $$(ln+1) | k\binom{kn+ln}{kn},$$ $$2\binom{kn}n | \binom {2n}{n}C_{2n}^{(k-1)}$$, $$\binom{kn}n | (2k-1)C_n\binom{2kn}{2n},$$ $$\binom{2n}n | (k+1)C_n^{(k-1)}\binom{2kn}{kn},$$ $$2^{k-1}\binom{2n}{n} | \binom{2(2^k-1)n}{(2^k-1)n}C_n^{(2^k-2)},$$ $$(6n+1)\binom{5n}{n} | \binom{3n-1}{n-1}C_{3n}^{(4)},$$ and $$\binom{3n}{n} | \binom{5n-1}{n-1}C_{5n}^{(2)},$$ where C_n denotes the Catalan number $\binom{2n}{n}/(n+1)$, and C_m^{(h)} refers to the Catalan number $\binom{(h+1)m}{m}/(hm+1)$ of order h.