Consider $n\in\mathbb{N}.$ Find the sum of:$$S=\left(\dfrac{C_n^0}{1} \right)^2+\left(\dfrac{C_n^1}{2} \right)^2+\cdots+\left( \dfrac{C_n^n}{n+1}\right)^2$$
I don't know how to solve it, i don't have any ideas, so i am very happy to hear from you. Thank you very much.
Outline: $$ \frac{{n \choose k}}{k+1} =\frac{n!}{k! (n-k)!(k+1)}=\frac{n!}{(k+1)!(n-k)!} = \frac{1}{n+1}\cdot {n+1 \choose k+1} $$ so $$ \sum_{k=0}^n \Big(\frac{{n \choose k}}{k+1}\Big)^2 = \Big(\frac{1}{n+1}\Big)^2 \sum_{k=0}^n {n+1 \choose k+1}^2 $$
Now, it is known that $$ \sum_{l=0}^{m} {m \choose l}^2 = {2m \choose m} $$ (you can do this inductively, see here). From here... can you finish?