## Abstract

The generalized central trinomial coefficient ${T}_{n}(b,c)$ introduced by Noe is defined as the coefficient of ${x}^{n}$ in the expansion of ${({x}^{2}+bx+c)}^{n}$, where $n\in \mathbb{N}=\{0,1,2,\dots \}$ and $b,c\in \mathbb{Z}$. Let $p$ be an odd prime. In this paper, we determine ${\sum}_{k=0}^{p-1}k{T}_{k}{(b,c)}^{2}\u2215{m}^{k}$ modulo ${p}^{2}$ for any integer $m$ satisfying the equation ${(m-d)}^{2}=16mc$, where $d={b}^{2}-4c$. As applications, we prove that for any prime $p>3$, we have

$$\sum _{n=0}^{p-1}n{\left(\sum _{k=0}^{n}\frac{1}{{2}^{k}}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)\phantom{\rule{-0.17em}{0ex}}\right)}^{\phantom{\rule{-0.17em}{0ex}}2}\equiv \frac{3}{4}\left(\frac{3}{p}\right)p-\left(\frac{-1}{p}\right)\phantom{\rule{0.3em}{0ex}}(\text{mod}\phantom{\rule{0.3em}{0ex}}{p}^{2})$$

and

$$\sum _{n=0}^{p-1}n{\left(\sum _{k=0}^{n}\frac{1}{{(-6)}^{k}}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)\phantom{\rule{-0.17em}{0ex}}\right)}^{\phantom{\rule{-0.17em}{0ex}}2}\equiv -\frac{1}{4}\left(\frac{3}{p}\right)p\phantom{\rule{0.3em}{0ex}}(\text{mod}\phantom{\rule{0.3em}{0ex}}{p}^{2}),$$

as conjectured by Sun (*Finite Fields Appl.* **46** (2017), 179–216), where $\left(\frac{\cdot}{p}\right)$ stands for the Legendre symbol.

## Citation

Chen Wang. Jia-Yu Chen. "CONGRUENCES INVOLVING SQUARES OF GENERALIZED CENTRAL TRINOMIAL COEFFICIENTS." Rocky Mountain J. Math. 53 (3) 959 - 968, June 2023. https://doi.org/10.1216/rmj.2023.53.959

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