Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

HINT: Consider $\displaystyle \sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}=\sum\limits_{n=1}^{\infty} \frac{H_{n-1}^{(2)}}{2^nn^2}+\operatorname{Li}_4\left(\frac{1}{2}\right)$ and then express the remaining sum as a double integral. After some work, you get

$$\int_0^1 \frac{\displaystyle\log(x)\operatorname{Li}_2\left(\frac{x}{2}\right)}{x-2} \ dx+\operatorname{Li}_4\left(\frac{1}{2}\right)$$

and after letting $x\mapsto 2x$ combined with the integration by parts, you arrive at some integrals
pretty easy to finish. I'm confident you can finish the rest of the job to do.

$$\frac{\pi^4}{1440}-\frac{\pi^2}{3}\log^2(2)+\frac{1}{24}\log^4(2)+\frac{7}{24}\pi^2\log^2(2)+\frac{1}{4}\log(2)\zeta(3)+\operatorname{Li}_4\left(\frac{1}{2}\right)$$

To find the above series, we need first to prove the following equality:

$\displaystyle\sum_{n=1}^{\infty}H_n^{(2)}\left(\frac{\ln2}{(n+1)2^{n+1}}+\frac{1}{(n+1)^22^{n+1}}\right)=\frac1{16}\zeta(4)+\frac14 \ln^22\zeta(2)-\frac18\ln^42$

which appeared as a problem 348 in here proposed by Cornel Ioan Valean and here is my proof:

lets start with the following integral and using the identity :$\displaystyle\sum_{n=1}^{\infty}x^nH_n^{(2)}=\frac{\operatorname{Li_2}(x)}{1-x}$

\begin{align*} I&=-\displaystyle \int_0^{1/2}\frac{\ln x\operatorname{Li_2}(x)}{1-x}\ dx=-\sum_{n=1}^{\infty}H_n^{(2)}\int_0^{1/2}x^n\ln x\ dx\\ &=\displaystyle\sum_{n=1}^{\infty}H_n^{(2)}\left(\frac{\ln2}{(n+1)2^{n+1}}+\frac{1}{(n+1)^22^{n+1}}\right)\tag{1} \end{align*} on the other hand : \begin{align*} I=-\int_0^{1/2}\frac{\ln x\operatorname{Li_2}(x)}{1-x}\ dx=-\int_0^{1}\frac{\ln x\operatorname{Li_2}(x)}{1-x}\ dx+\int_{1/2}^{1}\frac{\ln x\operatorname{Li_2}(x)}{1-x}\ dx \end{align*} using $\displaystyle\sum_{n=1}^{\infty}x^nH_n^{(2)}=\sum_{n=1}^{\infty}x^{n-1}\left(H_n^{(2)}-\frac{1}{n^2}\right)=\frac{\operatorname{Li_2}(x)}{1-x}$ for the first integral and letting $x\mapsto1-x$ for the second integral , we have

\begin{align*} I&=-\sum_{n=1}^{\infty}\left(H_n^{(2)}-\frac{1}{n^2}\right)\int_0^1 x^{n-1}\ln x\ dx+\underbrace{\int_{0}^{1/2}\frac{\ln(1-x)\operatorname{Li_2}(1-x)}{x}\ dx}_{IBP}\\ &=-\sum_{n=1}^{\infty}\left(H_n^{(2)}-\frac1{n^2}\right)\left(-\frac1{n^2}\right)+\left(-\operatorname{Li_2^2}\left(\frac12\right)+\int_0^{1/2}\frac{\ln x\operatorname{Li_2}(x)}{1-x}\right)\\ 2I&=\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n^2}-\zeta(4)-\operatorname{Li_2^2}\left(\frac12\right)\tag{2} \end{align*}

Combining (1) and (2) and using $\displaystyle \sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n^2}=\frac74\zeta(4)$ and $\displaystyle \operatorname{Li_2}\left(\frac12\right)=\frac12\zeta(2)-\frac12\ln^22$ proves our equality.

Now we ready to calculate the target sum : we proved \begin{align*} \frac1{16}\zeta(4)+\frac14 \ln^22\zeta(2)-\frac18\ln^42&=\sum_{n=1}^{\infty}H_n^{(2)}\left(\frac{\ln2}{(n+1)2^{n+1}}+\frac{1}{(n+1)^22^{n+1}}\right)\\ &=\sum_{n=1}^{\infty}H_{n-1}^{(2)}\left(\frac{\ln2}{n2^n}+\frac{1}{n^22^n}\right)\\ &=\ln2\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n2^n}}+\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n^22^n}}-\ln2\sum_{n=1}^{\infty}\frac{1}{{n^32^n}}-\sum_{n=1}^{\infty}\frac{1}{{n^42^n}} \end{align*} Rearranging the terms, we get \begin{align*} \sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n^22^n}}=\frac1{16}\zeta(4)+\frac14\ln^22\zeta(2)-\frac18\ln^42-\ln2\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n2^n}}+\ln2\operatorname{Li_3}\left(\frac12\right)+\operatorname{Li_4}\left(\frac12\right) \end{align*} Plugging $\displaystyle\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{{n2^n}}=-\frac58\zeta(3)$ and $\displaystyle \operatorname{Li_3}\left(\frac12\right)=\frac78\zeta(3)+\frac16\ln^32-\frac12\ln2\zeta(2)$

we get the closed form \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*}

Different approach:

By Cauchy product we have

$$\operatorname{Li}^2_2(x)=\sum_{n=1}^\infty x^n\left(\frac{4H_n}{n^3}+\frac{2H_n^{(2)}}{n^2}-\frac{6}{n^4}\right)$$

take $x=1/2$ and rearrange the terms to get

$$\sum_{n=1}^\infty\frac{H_n^{(2)}}{n^22^n}=\frac12\operatorname{Li}^2_2\left(\frac12\right)+3\operatorname{Li}_4\left(\frac12\right)-2\sum_{n=1}^\infty\frac{H_n}{n^32^n}$$

Plugging $ \sum_{n=1}^\infty \frac{H_n}{n^32^n}=\operatorname{Li}_4\left(\frac12\right)+\frac18\zeta(4)-\frac18\ln2\zeta(3)+\frac1{24}\ln^42$ and $\operatorname{Li_2}\left(\frac12\right)=\frac12\zeta(2)-\frac12\ln^22$

gives

\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*}