It's kinda weird summation - 2

Calculus Level 4

$\large \mathcal{S} = \sum_{n=0}^{\infty} \dfrac{(-1)^n (2n+1)!}{(n+2)! \cdot n! \cdot 4^{2n+3}}$

Given that $$S$$ is equal to $$\dfrac{A - B\sqrt C}{D}$$ where $$A,B,C$$ and $$D$$ are positive integers, with $$A$$ and $$B$$ coprime, $$C$$ square-free, find the value of $$A+B+C+D$$.

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