%0 Journal Article
%T Some congruences for the second-order Catalan numbers
%A Li-Lu Zhao
%A Hao Pan
%A Zhi-Wei Sun
%J Mathematics
%D 2009
%I arXiv
%X Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.
%U http://arxiv.org/abs/0909.3733v2