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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