Sequence of real numbers $a_0,a_1,\dots,a_{n}$ are called concave if for each $0<i<n$, $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.$a_{0}=0$ Find the largest $c(n)$ such that for every concave sequence of non-negative real numbers: such $$c(n)\sum_{i=1}^{n}a^2_{i}\le \left(\sum_{i=1}^{n}a_{i}\right)^2$$ I know $c(n)=\dfrac{n-1}{2}$ is well Khinchine inequality.but for best I can't it

A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$. 

Find the largest $c(n)$ such that for every concave sequence $a_0,a_1,\dots,a_n$ of non-negative real numbers, we have $$c(n)\sum_{i=1}^{n}a^2_{i}\le \left(\sum_{i=1}^{n}a_{i}\right)^2 .$$

I know $c(n)=\dfrac{n-1}{2}$ is well Khinchine inequality. But is it the best $c(n)$?