<s>[INST] Generalized Harmonic Number Summation $ \sum_{n=1}^{\infty} {2^{-n}}{(H_{n}^{(2)})^2}$ 
Prove That $$ \sum_{n=1}^{\infty} \dfrac{(H_{n}^{(2)})^2}{2^n} = \tfrac{1}{360}\pi^4 - \tfrac16\pi^2\ln^22 + \tfrac16\ln^42 + 2\mathrm{Li}_4(\tfrac12) + \zeta(3)\ln2 $$
Notation : $ \displaystyle H_{n}^{(2)} = \sum_{r=1}^{n} \dfrac{1}{r^2}$

We can solve the above problem using the generating function $\displaystyle \sum_{n=1}^{\infty} (H_{n}^{(2)})^2 x^n $, but it gets rather tedious especially taking into account the indefinite polylogarithm integrals involved. Can we solve it using other methods like Euler Series Transform or properties of summation?
 [/INST] second approach suggested by Cornel Ioan Valean using summation by parts and lets start with the following sum:
with ${N \in \mathbb{N}_{\ \geq\ 1}}$
\begin{align}
\sum_{n=1}^N\frac{\left(H_{n-1}^{(2)}\right)^2}{2^n}=\sum_{n=1}^N\frac{\left(H_n^{(2)}\right)^2}{2^n}-2\sum_{n=1}^N\frac{H_n^{(2)}}{n^22^n}+\sum_{n=1}^N\frac1{n^42^n}\tag{1}
\end{align}
on the other hand:
\begin{align}
\sum_{n=1}^N\frac{\left(H_{n-1}^{(2)}\right)^2}{2^n}=\sum_{n=1}^{N-1}\frac{\left(H_{n}^{(2)}\right)^2}{2^{n+1}}=\sum_{n=1}^{N}\frac{\left(H_{n}^{(2)}\right)^2}{2^{n+1}}-\frac{\left(H_{N}^{(2)}\right)^2}{2^{N+1}}\tag{2}
\end{align}
from $(1)$ and $(2)$ we reach
$$\sum_{n=1}^N\frac{\left(H_{n}^{(2)}\right)^2}{2^n}=4\sum_{n=1}^N\frac{H_n^{(2)}}{n^22^n}-2\sum_{n=1}^N\frac{1}{n^42^n}-2\frac{\left(H_{N}^{(2)}\right)^2}{2^{N+1}}$$
letting $N$ approach $\infty$ we get
$$\sum_{n=1}^\infty\frac{\left(H_{n}^{(2)}\right)^2}{2^n}=4\sum_{n=1}^\infty\frac{H_n^{(2)}}{n^22^n}-2\sum_{n=1}^\infty\frac{1}{n^42^n}-0$$
I was able here to prove $$\begin{align*}
\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n^22^n}}=\operatorname{Li_4}\left(\frac12\right)+\frac1{16}\zeta(4)+\frac14\ln2\zeta(3)-\frac14\ln^22\zeta(2)+\frac1{24}\ln^42
\end{align*}$$
which follows $$\sum_{n=1}^\infty\frac{\left(H_{n}^{(2)}\right)^2}{2^n}=2\operatorname{Li_4}\left(\frac{1}{2}\right)+\frac14\zeta(4)+\ln2\zeta(3)-\ln^22\zeta(2)+\frac16\ln^42
$$
</s>