Some conjectural supercongruences related to Bernoulli and Euler numbers

We prove some supercongruences involving the Apery polynomials A(n)(x) = Sigma(k=0)(n) ((k) (n))(2) ((k) (n + k))(2) x(k) (n is an element of N = {0, 1,...,}), the generalized Domb numbers D-n(A, B, C) = Sigma(k=0)(n) ((k) (n))(A) ((k)2(k))(B) ((n - k) (2n - 2k))(C) (n is an element of N) and Q(n) = Sigma(k=0)(n) ((k) (n)) ((k) (n - k)) ((k) (n + k)) (n is an element of N), which were conjectured by Z.-W. Sun. For example, we show that for any prime p > 3 and positive integer r we have A(p)(r) (-1)- A(p)(r)- 1 (-1)/p(3r) equivalent to 29/6 Bp-3 (mod p) and Q(p)(r) - Q(p)(r-1) p(3r) = - 1/9 B-p (- 3) (mod p), where B-0, B-1, B-2,... are the Bernoulli numbers. The following supercongruences hold modulo p: D-p(r) ( A, 1, 1) - D-p(r-1) (A, 1, 1)/p((A+1)r) = {8 (-1/p(r)) Ep-3, if A = 1, 16/3 Bp-3, if A = 2, where (center dot/p) denotes the Legendre symbol and E-0, E-1, E-2,... are the Euler numbers.