Supercongruences for central trinomial coefficients

Abstract

For each $n=0,1,2,\dots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1{)}^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the quotient $(T_{pn}-T_n)/(pn{)}^2$ is always a $p$-adic integer. In this paper, we confirm this conjecture, and further prove that $$\frac{T_{pn}-T_n}{(pn{)}^2}\equiv \frac{T_{n-1}}{6}\left(\frac{p}{3}\right)B_{p-2}\left(\frac{1}{3}\right)(\mathrm{mod}p),$$ where $(\frac{p}{3})$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

9 pages, final version