How to compute the sum $$\sum_{n\geq 1}(-1)^n\Big(H_n^2-(\gamma+\log(n))^2\Big)$$
I had computed some similar sums
$\sum_{n\geq 1}\frac{H_n-\log(n)-\gamma}{n}$ and $\sum_{n\geq 1}(-1)^n (H_n-\log(n)-\gamma)$
For the second one I considered to write it as $\sum_{n=1}^{2N}(-1)^n (H_n-\log(n)-\gamma)$
Break the sum up and Take $N$ to infinity
For the first one, the idea is to let $s\to 0$ and consider $$\sum_{n\geq 1}\frac{H_n-\log(n)-\gamma}{n^{1+s}}$$
and use Euler Macluarin formula of $\mathcal{H}(s)=\sum_{n\geq 1}\frac{H_n}{n^s}$.
But for this one, my first idea is to use Euler-Boole instead, but it didn't work. I tried to break the sum up but it didn't work too.
Any help are appreciated