SOME CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that Tp-1 equivalent to(p/3)3(p-1) (mod p(2)), where the central trinomial coefficient T-n is the constant term in the expansion of (1 + x + x(-1))(n). We also prove three congruences modulo p(3) conjectured by Sun, one of which is Sigma(p-1)(k=0) (p-1 k) (2k k) ((-1)(k) - (-3)(-k)) equivalent to (p/3) (3(p-1) - 1) (mod p(3)). In addition, we get some new combinatorial identities.