Binomially Amazing! 3

Calculus Level 4

\[\large \sum _{ n=0 }^{ 1729 }{ \frac { { \left( -1 \right) }^{ n }{ 1729 \choose n }}{ 4n+2 } ={ 2 }^{ A }\frac { B! }{ C!! } } \]

Find \(A+B+C\).

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