Some parametric congruences involving generalized central trinomial coefficients.

For n = 0 , 1 , 2 , … and b , c ∈ Z , the nth generalized central trinomial coefficient T n (b , c) is the coefficient of x n in the expansion of (x 2 + b x + c) n . In particular, T n = T n (1 , 1) (n = 0 , 1 , 2 , …) are central trinomial coefficients. Let p be an odd prime. For any b , c ∈ Z with p ∤ b c (b + 2 c) , we determine ∑ k = 0 p - 1 2 k k T k (b , c 2) 4 k (b + 2 c) k and ∑ k = 0 p - 1 2 k k T k (b , c) (4 b) k <graphic href="13398_2023_1502_Article_Equ17.gif"></graphic> modulo p 2 . As consequences, ∑ k = 0 p - 1 2 k k 12 k T k ≡ p 3 3 p - 1 + 3 4 (mod p 2) <graphic href="13398_2023_1502_Article_Equ18.gif"></graphic> provided p > 3 (where (-) denotes the Legendre symbol), and ∑ k = 0 p - 1 2 k k T k (2 , - 1) 8 k ≡ 2 x - p / (2 x) (mod p 2) if p = x 2 + 4 y 2 (x , y ∈ Z) and 4 ∣ x - 1 , 0 (mod p 2) if p ≡ 3 (mod 4). <graphic href="13398_2023_1502_Article_Equ19.gif"></graphic> [ABSTRACT FROM AUTHOR]