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$$\sum_{n=1}^{\infty}\frac{{2n \choose n}^2}{2^{5n}}\frac{n}{n+1}H_{n+1}=\frac{\Gamma^2\left(\frac{1}{4}\right)}{4\sqrt{\pi}}\left(1-\frac{4\ln{2}}{\pi}\right)+\frac{4\sqrt{\pi}}{\Gamma^2\left(\frac{1}{4}\right)}(4\ln{2}+\pi-6)$$

$$\sum_{n=2}^{\infty}\frac{(-1)^n}{n}\left[\frac{P(n)}{(2n-1)^7(n-1)}+\frac{Q(n)}{(2n+1)^7(n+1)}\right]=\frac{4941\pi^7}{20480}$$

Where $$P(n)=280208(n^2-n)^3+210156(n^2-n)^2+50352(n^2-n)+4196$$

and $$Q(n)=280208(n^2+n)^3+210156(n^2+n)^2+51081(n^2+n)+4196$$