Non Classical Series; Bino-Skew Harmonic series.

If the series $\sum_{n=1}^{\infty}\frac{\bar{\mathcal{H}_{2n}}}{16^n}\left[{4n\choose 2n}\right]\frac{1}{n}$ can be expressed for some positive integers $A$ being prime as $\frac{A\pi^2}{12}-\frac{\log^2(2)}{2}-4\operatorname{Li}_2\left(\frac{1}{\sqrt 2}\right)+2\log^2\left(\delta_S\right)+\frac{1}{2}\log\left(\frac{1}{2}\right)\log\left(17+12\sqrt{2}\right)$ , then find the value of $A^2+9$ .

Notation: $\bar{\mathcal{H}_n}=\sum_{k= 1}^n (-1)^{k+1}\frac{1}{k}$ is nth Skew Harmonic number , $\delta_S$ is silver ratio and $\operatorname{Li}_2(x)$ is dilogarithm function .

( The best way to crack this hard nut is to utilize the generating function. This is an original problem and I believe this sort of series must be new in mathematical literature .)