Some congruences for the second-order Catalan numbers

Li-Lu Zhao, Hao Pan, Zhi-Wei Sun

Let p be any odd prime. We mainly show that $$\sum _{k=1}^{p-1}binomial(3k,k)\ast 2^k/k=0(modp)$$ and $$\sum _{k=1}^{p-1}{2}^{k-1}{C}_k^{(2)}=(-1{)}^{(p-1)/2}-1(modp),$$ where $C_k^{(2)}=b…