Double series of Harmonic Numbers

Question

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}}{(n+1) (n+s+3)^{3}} \end{align} such that \begin{align} \sum_{k=1}^{\infty} \frac{(-1)^{k-1} H_{k}}{k+1} \left(\sum_{n=1}^{\infty} \frac{H_{n}}{(n+1) (n+s+3)^{3}} \right) \end{align} has a closed form. Alternatively, one could calculate the series \begin{align} A_{n} = \sum_{s=1}^{n} \frac{(-1)^{s-1} \, H_{n-s} H_{s} }{ (s+1)(n-s+1) } \end{align} and then find a closed form of the series \begin{align} \sum_{n=1}^{\infty} \frac{A_{n}}{(n+3)^{3}} \end{align}