Apéry Numbers and Central Trinomial Coefficients 3

Define the Apéry polynomial of degree n by A n (x) = n k=0 n k 2 n + k k 2. We determine p−1 k=0 (−1) k A k (1/4) and p−1 k=0 (−1) k A k (1/16) modulo a prime p > 3. Let b and c be integers and let the generalized trinomial coefficient T n (b, c) be the coefficient of x n in the expansion of (x 2 +bx+c) n. We establish the following new congruence p−1 k=0 T k (b, c) 2 (b 2 − 4c) k ≡ c(b 2 − 4c) p (mod p) provided that p ∤ b 2 − 4c, where (−) denotes the Legendre symbol. We also show that n | n−1 k=0 T k (b, c 2)(b − 2c) n−1−k for all n = 1, 2, 3,. .. ; moreover, if b ≡ 2c (mod p) then 2c p−1 k=0 T k (b, c 2) (b − 2c) k ≡ −bp + (b + 2c)p b 2 − 4c 2 p (mod p 2) and p−1 k=0 T k (b, c 2) 2 (b − 2c) 2k ≡ −c 2 p (mod p). Besides these we also get some other congruences and raise several challenging conjectures.