I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values. \begin{equation} \int_{-1}^{1}\sqrt{\frac{1-x}{2}} P_n(x) \text{d}{x}=\frac{-4}{(2n-1)(2n+1)(2n+3)}. \end{equation}

This formula is motivated by research about the random walk on the sphere.