A series involves harmonic number

Here's another solution. I'll denote various versions of the sum

$$ \sum_{k=1}^\infty\sum_{j=1}^k\frac1j\frac1{k^2} $$

by an $S$ with two subscripts indicating which parities are included, the first subscript referring to the parity of $j$ and the second to the parity of $k$, with '$\mathrm e$' denoting only the even terms, '$\mathrm o$' denoting only the odd terms, '$+$' denoting the sum of the even and odd terms, i.e. the regular sum, and '$-$' denoting the difference between the even and the odd terms, i.e. the alternating sum. Then

$$ \begin{align} \sum_{n=1}^\infty\frac{H_n}{(2n+1)^2} &= 2\sum_{n=1}^\infty\sum_{i=1}^n\frac1{2i}\frac1{(2n+1)^2} \\ &= 2S_{\mathrm{eo}} \\ &= 2(S_{++}-S_{\mathrm o+}-S_{\mathrm{ee}}) \\ &= 2\left(S_{++}-S_{\mathrm o+}-\frac18S_{++}\right) \\ &= 2\left(\frac38S_{++}+\left(\frac12S_{++}-S_{\mathrm o+}\right)\right) \\ &= \frac34S_{++}+S_{-+} \\ &= \frac32\zeta(3)+\sum_{k=1}^\infty\sum_{j=1}^k\frac{(-1)^j}j\frac1{k^2}\;, \end{align} $$

where I used the result $\sum_nH_n/n^2=2\zeta(3)$ from the blog post Aeolian linked to and reduced the present problem to finding the analogue of that result with the sign alternating with $j$, which we can rewrite as

$$ \begin{align} \sum_{k=1}^\infty\sum_{j=1}^k\frac{(-1)^j}j\frac1{k^2} &= \sum_{k=1}^\infty\sum_{j=1}^\infty\frac{(-1)^j}j\frac1{k^2}-\sum_{k=1}^\infty\sum_{j=k+1}^\infty\frac{(-1)^j}j\frac1{k^2} \\ &= -\zeta(2)\log2+\sum_{j=1}^\infty\frac{(-1)^j}{j+1}\sum_{k=1}^j\frac1{k^2}\;. \end{align} $$

This last double sum can be evaluated by the method applied in the blog post, making use of the fact that summing the coefficients of a power series in $x$ corresponds to dividing it by $1-x$:

$$ \begin{align} \sum_{j=1}^\infty x^j\sum_{k=1}^j\frac1{k^2}=\def\Li{\operatorname{Li}}\frac{\Li_2(x)}{1-x}\;, \end{align} $$

where $\Li_2$ is the dilogarithm. Thus

$$ \begin{align} \sum_{j=1}^\infty\frac{(-1)^j}{j+1}\sum_{k=1}^j\frac1{k^2} &= \int_0^1\sum_{j=1}^\infty (-x)^j\sum_{k=1}^j\frac1{k^2}\mathrm dx \\ &= \int_0^1\frac{\Li_2(-x)}{1+x}\mathrm dx \\ &= \left[\Li_2(-x)\log(1+x)\right]_0^1+\int_0^1\frac{\log^2(1+x)}x\mathrm dx \\ &=-\frac{\zeta(2)}2\log2+\frac{\zeta(3)}4\;, \end{align} $$

where the boundary term is evaluated using $\Li_2(-1)=-\eta(2)=-\zeta(2)+2\zeta(2)/4=-\zeta(2)/2$ and the integral in the second term is evaluated in this separate question. Putting it all together, we have

$$ \begin{align} \sum_{n=1}^\infty\frac{H_n}{(2n+1)^2} &= \frac74\zeta(3)-\frac32\zeta(2)\log2 \\ &= \frac74\zeta(3)-\frac{\pi^2}4\log2\;. \end{align} $$

I believe all the rearrangements can be justified, despite the series being only conditionally convergent in $j$, by considering the partial sums with $j$ and $k$ both going up to $M$; then all the rearrangements can be carried out within that finite square of the grid, and the sums of the remaining terms go to zero with $M\to\infty$.

I gave an integral representation for a more general form. Here is an integral representation for your sum

$$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}= \frac{1}{4}\,\int_{0}^{1}\!{\frac {\ln \left( 1-z \right) \ln \left( z\right) }{z\sqrt {1-z}}}{dz}= \frac{1}{4}(7\,\zeta \left( 3 \right) -{\pi }^{2}\ln \left( 2 \right))\sim 0.393327464. $$

The above integral can be evaluated through beta function. Here is the technique from previous problems. Basically, you need to consider the integral

$$ \int_{0}^{1} z^s (1-z)^{w-1/2} dz. $$

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\sum_{n = 1}^{\infty}{H_{n} \over \pars{2n + 1}^{2}}:\ {\large ?}}$.

Lets consider $\ds{\fermi\pars{x}\equiv \sum_{n = 1}^{\infty}{H_{n} \over \pars{2n + 1}^{2}}\,x^{2n + 1}. \qquad\fermi\pars{1}={\large ?}\,,\quad \fermi\pars{0} = 0}$.

\begin{align} \fermi'\pars{x}&=\sum_{n = 1}^{\infty}{H_{n} \over 2n + 1}\,x^{2n} \ \imp\ \bracks{x\fermi'\pars{x}}'=\sum_{n = 1}^{\infty}H_{n}\,x^{2n} =-\,{\ln\pars{1 - x^{2}} \over 1 - x^{2}}\,,\qquad\fermi'\pars{0} = 0 \end{align} where we used the Harmonic Number Generating Function.

Then \begin{align} &x\fermi'\pars{x}=-\int_{0}^{x}{\ln\pars{1 - t^{2}} \over 1 - t^{2}}\,\dd t \\[3mm]&\imp \fermi\pars{1}=-\int_{0}^{1}{\dd x \over x}\int_{0}^{x} {\ln\pars{1 - t^{2}} \over 1 - t^{2}}\,\dd t =-\int_{0}^{1}{\ln\pars{1 - t^{2}} \over 1 - t^{2}}\int_{t}^{1}{\dd x \over x} \,\dd t \end{align}

$$\begin{array}{|c|}\hline\\ \quad\sum_{n = 1}^{\infty}{H_{n} \over \pars{2n + 1}^{2}} =\int_{0}^{1}{\ln\pars{t}\ln\pars{1 - t^{2}} \over 1 - t^{2}}\,\dd t\quad \\ \\ \hline \end{array} $$

\begin{align} &\color{#c00000}{\sum_{n = 1}^{\infty}{H_{n} \over \pars{2n + 1}^{2}}} =\int_{0}^{1}{\ln\pars{t^{1/2}}\ln\pars{1 - t} \over 1 - t}\,\half\,t^{-1/2} \,\dd t ={1 \over 4}\int_{0}^{1}{t^{-1/2}\ln\pars{t}\ln\pars{1 - t} \over 1 - t}\,\dd t \\[3mm]&={1 \over 4}\lim_{\mu\ \to\ 0 \atop{\vphantom{\LARGE A}\nu\ \to\ 0}} \partiald{}{\mu}\partiald{}{\nu}\int_{0}^{1}t^{\mu - 1/2} \pars{1 - t}^{\nu - 1}\,\dd t ={1 \over 4}\lim_{\mu\ \to\ 0 \atop{\vphantom{\LARGE A}\nu\ \to\ 0}} \partiald{}{\nu}\Gamma\pars{\nu}\partiald{}{\mu} \bracks{\Gamma\pars{\mu + 1/2} \over \Gamma\pars{\mu + \nu + 1/2}} \\[3mm]&={1 \over 4}\lim_{\nu\ \to\ 0} \partiald{}{\nu}\braces{% \Gamma\pars{\nu}\,{\Gamma\pars{1/2} \over \Gamma\pars{\nu + 1/2}} \bracks{\Psi\pars{\half} - \Psi\pars{\nu + \half}}} \\[3mm]&=-\,{1 \over 4}\,\Gamma\pars{\half}\lim_{\nu\ \to\ 0} \partiald{}{\nu}\bracks{% {\Gamma\pars{\nu + 1} \over \Gamma\pars{\nu + 1/2}}\, {\Psi\pars{1/2 + \nu} - \Psi\pars{1/2} \over \nu}} \\[3mm]&=-\,{1 \over 4}\,\Gamma\pars{\half}\lim_{\nu\ \to\ 0} \partiald{}{\nu}\braces{% {\Gamma\pars{\nu + 1} \over \Gamma\pars{\nu + 1/2}}\, \bracks{\Psi'\pars{\half} + \half\,\Psi''\pars{\half}\nu}} \\[3mm]&={\pi^{2}\gamma + \pi^{2}\Psi\pars{1/2} + 14\zeta\pars{3} \over 8} \quad\mbox{where we used}\quad\Psi\pars{1} = -\gamma\,,\quad \Psi''\pars{\half} = -14\zeta\pars{3}. \end{align}

With $\ds{\Psi\pars{\half} = -2\ln\pars{2} - \gamma}$: $$ \color{#66f}{\large\sum_{n = 1}^{\infty}{H_{n} \over \pars{2n + 1}^{2}} ={1 \over 4}\,\bracks{7\zeta\pars{3} - \pi^{2}\ln\pars{2}}} \approx {\tt 0.3933} $$