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Find the first $4$ coefficients of the Taylor series expansion of the function $e^{2x}$ centered at $0$.

A

$ 1,\ 2,\ 4,\ 8$

B

$ 1,\ 1,\ \cfrac{1}{4},\ \cfrac{1}{36}$

C

$1,\ 2,\ 1,\ \cfrac{1}{2}$

D

$ 1,\ 2,\ 2,\ \cfrac{4}{3}$

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