Congruences concerning generalized central trinomial coefficients

For any $n\in \mathbb{N}=\{0,1,2,\dots \}$ and $b,c\in \mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c{)}^n$. Let $p$ be an odd prime. In this paper, we determine the summation ${\sum }_{k=0}^{p-1}{T}_k(b,c{)}^2\mathrm{/}{m}^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375–1400].