NEW CONGRUENCES FOR SUMS INVOLVING APERY NUMBERS OR CENTRAL DELANNOY NUMBERS

The Apery numbers A(n) and central Delannoy numbers D-n are defined by A(n) = Sigma(n)(k=0) (n+k 2k)(2) (25 k)(2), D-n = Sigma(n)(k=0) (n+k 2k)(2) (25 k). Motivated by some recent work of Z.-W. Sun, we prove the following congruences: Sigma(n-1)(k=0) (2k+1)(2r+1) A(k) equivalent to Sigma(n-1)(k=0) epsilon(k) (2k+1)(2r+1) D-k equivalent to 0 (mod n), where n >= 1, r >= 0, and epsilon = +/- 1. For r = 1, we further show that Sigma(n-1)(k=0) (2k+1)(3) A(k) equivalent to 0 (mod n(3)), Sigma(p-1)(k=0) (2k+1)(3) A(k) equivalent to p(3) (mod 2p(6)), where p > 3 is a prime. The following congruence Sigma(n-1)(k=0) (n+k k)(2) (n-1 k)(2) equivalent to 0 (mod n) plays an important role in our proof.