Assuming the result

$\displaystyle P^{'}_{n+1}(z)-P^{'}_{n-1}(z)=(2n+1)P_{n}(z)$

I need to show that

$\displaystyle \int^{1}_{-1}z^{n}P_{n}(z)dz={\frac{n}{2n+1}}\int^{1}_{-1}z^{n-1}P_{n-1}(z)dz$

and then deduce that

$\displaystyle \int^{1}_{-1}z^{n}P_{n}(z)dz={\frac{2^{n+1}(n!)^{2}}{(2n+1)!} }$

Any pointers on where to start with this?