On the divisibility properties concerning sums of binomial coefficients

For any integer \(n> 1,\) we prove

$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$

The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.