Prove or disprove this lemma for Catalan Numbers

Question

Prove or disprove that for all non-negative integers $n$ and $r$ with $r+1$ is less than or equal to $n$, $C(n,r+1)=C(n,r)\times\frac{n-r}{r+1}$.

Answer 1

The question does not appear to be about Catalan numbers, but about binomial coefficients. I will write $C(n,k)$ in the more common form $\binom{n}{k}$. Then $$\binom{n}{r+1}=\frac{n!}{(r+1)!(n-r-1)!}=\frac{n!}{r!(n-r-1)!}\frac{1}{r+1}=\frac{n!}{r!(n-r)!}\frac{n-r}{r+1}=\binom{n}{r}\frac{n-r}{r}.$$

Remark: There are various ways to define the $n$-th Catalan number, which is often called $C_n$, sometimes $C(n)$. One way is $C_n=\frac{1}{n+1}\binom{2n}{n}$. The Catalan numbers turn up often in combinatorial problems, but not here.