But I Can't Expand So Many Terms!

Consider the polynomial \(\displaystyle P\left( x \right) =\Large{\prod _{ k=0 }^{ 10 }{ ({ x }^{ { 2 }^{ k } }+{ 2 }^{ k })} } \)

The coefficient of \({ x }^{ 2016 }\) is equal to \({ 2 }^{ m }\). What is the value of \(m\)?

Bonus: For \(\displaystyle P(x)=\prod _{ k=0 }^{ n }{ ({ x }^{ { 2 }^{ k } }+{ 2 }^{ k }) } \), generalize the coefficient of \({ x }^{ q }\), with \(0\le q\le { 2 }^{ n+1 }-1\).

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