Proof of Some Conjectural Congruences Involving Binomial Coefficients and Apery-like Numbers

In this paper, we mainly establish some congruences involving binomial coefficients and Apery-like numbers, for example, we prove the following result which was conjectured by Z.-H. Sun: Let p > 3 be a prime. Then & sum;(p-1)(k=0)(2kk) Wk/ (-12)(k)equivalent to{L-2-2p(mod p(2)) if p equivalent to 1(mod3) & 4p=L-2+27M(2), 0(modp(2)) if p equivalent to 2(mod3), where L, M are integers and W-n=& sum;& LeftFloor;n3 & RightFloor;k=0(2kk)(3kk)(n3k)(-3)(n-3k) are the second kind Apery-like numbers.