For $n\in\mathbb{ N}=\{ 0,1,2,\ldots\}$ and $b,c\in\mathbb{ Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{ k=0}^{ p-1}\frac{ \binom{ 2k}{ k}}{ 12^k}T_k\equiv\left(\frac{ p}{ 3}\right)\frac{ 3^{ p-1}+3}{ 4}\pmod{ p^2} $$ and $$ \sum_{ k=0}^{ p-1}\frac{ T_kH_k}{ 3^k}\equiv\frac{ 3+\left(\frac{ p}{ 3}\right)}{ 2}-p\left(1+\left(\frac{ p}{ 3}\right)\right)\pmod{ p^2}, $$ where $(-)$ denotes the Legendre symbol and $H_k:=\sum_{ j=1}^k1/j$ denotes the $k$th harmonic number. These confirm two conjectural congruences of the second author.
Categories: math.NT math.CO