Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum _{j=1}^{\infty } a_i a_{i+j} A_j\right)$$ and $$Y=\sum _{j=1}^{\infty } A_j^2.$$

It have the inequality $X/Y\le \sqrt{2}/{2}$ but maybe this is not sharp.

Experiments with mathematica suggest that $X/Y \le 1/2$.