Is there any known probability distribution whose pdf is the following?
Given $y\in[0,1]$, $n\in\mathbb{N}$ and $F(x)$, a CDF of some random variable $X$ (with density $f(x)$), $h(x)$ is given by $$h(x)=(n+1)\sum^{n}_{k=0}{n\choose k}^2y^k(1-y)^{n-k}F(x)^k[1-F(x)]^{n-k}f(x)$$ $h(x)$ is a weighted average of pdf of $k$-th order statistics of $X$ with $n+1$ samples.
What I wonder is whether there's any known or famous random variable or associated probability distribution whose pdf is the same as $h(x)$.