  For each $n=0,1,2,\ldots$ the central trinomial coefficient $T_n$ is the
coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. In 2016 the second
author conjectured that for any prime $p>3$ and positive integer $n$ the
quotient $(T_{pn}-T_n)/(pn)^2$ is a $p$-adic integer. In this paper we confirm
this conjecture and prove further that
$$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac
p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the
Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.
