Infinite Summation + Infinite Products

$\begin{aligned} \sum_{n=1}^\infty \prod_{k=n+1}^\infty \left(1 - \left(\frac{n}{k}\right)^2 \right) & = \sum_{n=1}^\infty \prod _{k=n+1} ^\infty \frac{k^2 - n^2}{k^2} \\ & = \sum_{n=1}^\infty \prod _{k=n+1} ^\infty \frac{(k+n)(k-n)}{k^2} \\ & = \sum_{n=1}^\infty \prod _{j=1} ^\infty \frac{(2n+j)j}{(n+j)^2} \\ & = \sum_{n=1}^\infty \lim_{m \to \infty} \prod _{j=1} ^m \frac{(2n+j)j}{(n+j)^2} \\ & = \sum_{n=1}^\infty \lim_{m \to \infty} \frac{m!(2n+m)!(n!)^2}{(2n)![(n+m)!]^2} \\ & = \sum_{n=1}^\infty \frac{(n!)^2}{(2n)!} \\ & = \sum_{n=1}^\infty \frac{1}{\begin{pmatrix} 2n \\ n \end{pmatrix}} = \frac{1}{3} + \frac{2\pi}{9\sqrt{3}} \quad \color{#D61F06}{[\text{See Note}]} \end{aligned}$

$\Rightarrow A+B+C+D = 1+3+2+9 = \boxed{15}$

$\color{#D61F06}{\text{Note: }}$ See (37)