But I Can't Expand So Many Terms!

Probability Level 3

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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1 solution

Mateo Matijasevick
Apr 14, 2016

If $0\le q\le { 2 }^{ n+1 }-1$ and $n={ \sum { _{ j\in A }^{ }{ { 2 }^{ j } } } }$ for $A\subseteq \left\{ 0,\quad 1,\quad 2,\quad ...\quad ,\quad n \right\}$ , the coefficient of ${ x }^{ q }$ is ${ 2 }^{ m }$ , where $m=\left( \begin{matrix} n+1 \\ 2 \end{matrix} \right) -\sum { _{ j\in A }^{ }{ j } }$ .

But I Can't Expand So Many Terms!

1 solution

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