Let $S \in \Sigma^*$ be a string over a finite alphabet. A compressibly repeated substring is a substring $t \leqslant S$ (let that mean substring), such that $t\gamma t \leqslant S$ and $|t| \geq 3$, or $t\gamma t \gamma' t \leqslant S$ and $|t| = 2$, for some strings $\gamma, \gamma'\in \Sigma^...

Hint $$\log \left(\binom{n^2}{n}\right)=\log\left(\frac{(n^2)!}{n!(n^2-n)!}\right)=\log((n^2)!)-\log(n!)-\log((n^2-n)!)$$ Now, use Stirling approximation $$\log(m!)=m (\log (m)-1)+\frac{1}{2} \left(\log (2 \pi )+\log (m)\right)+O\left(\frac{1}{m}\right)$$ Apply it to each term and simplify.