\[\large \sum _{ n=0 }^{ \infty }{ \dfrac { { 2 }^{ n+1 }n!^{ 2 } }{ (2n+1)!(n+1)^{ 2 } } } =A\pi G-\dfrac { B }{ C } \zeta (D)+\dfrac { { \pi }^{ E } }{ F } \ln { H } \]
The equation above holds true for positive integers \(A,B,C,D,E,F\) and \(H\), where \(B,C\) are coprime and \(H\) is minimized.
Find \(A+B+C+D+E+F+H\).
Notations:
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