On a parametric congruence concerning generalized central trinomial coefficients

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{C}$, the $n$-th generalized central trinomial coefficient is defined as the constant term in the Laurent expansion of $(b+x+cx^{-1})^n$. In this paper, utilizing the constant term method and generating functions, we prove a parametric congruence concerning generalized central trinomial coefficients. As applications, we confirm several conjectures of Z.-W. Sun.