prove that $\sum_{k=0}^n \frac{1}{k+1}\binom{2k}{k}\binom{2n-2k}{n-k}=\binom{2n+1}{n}$

Question

How to prove this formula? I tried Catalan numbers but failed.
I am interested in a solution, not a clue.

Answer 1

Write $C_n=\frac1{k+1}\binom{2k}k$. Then your sum is the coefficient of $x^n$ in the power series for $C(x)D(x)$ where $C(x)=\sum C_n x^n =(1-\sqrt{1-4x})/(2x)$ and $D(x)=\sum\binom{2n}n x^n=1/\sqrt{1-4x}$. So $$C(x)D(x)=\frac{1}{2x}\left(\frac1{\sqrt{1-4x}}-1\right)$$ etc.