Wonderfully Scary

Algebra Level 5

What is the coefficient of \(x^{101}\) in the expansion of \[\left(1-x \right)\left(1-2x\right)\left(1-2^{2}x\right) \cdots \left(1-2^{100}x\right)\left(1-2^{101}x\right) ? \]

Clarification: The coefficient \(x\) in \(1729x\) is \(1729\).

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