  The Sun polynomials $g_n(x)$ are defined by \begin{align*}
g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that,
for any positive integer $n$, there hold \begin{align*}
&\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x) \in\mathbb{Z}[x],\quad\text{and}\\
&\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\equiv 0\pmod{n}. \end{align*} The first
one confirms a recent conjecture of Z.-W. Sun, while the second one partially
answers another conjecture of Z.-W. Sun. We give three different proofs of the
former. One of them depends on the following congruence: $$ {m+n-2\choose
m-1}{n\choose m}{2n\choose n}\equiv 0\pmod{m+n}\quad\text{for $m,n\geqslant
1$.} $$

Number Theory;Combinatorics