Question

what is the coefficient of x^15 in the product of (1-x)(1-2^2x)(1-2^3x)(1-2^4x).........(1-2^15x)

Answer 1

Answer:

(1−x)(1−2x)(1−2^2x)...........(1−2^15x)=2^(0+1+2+3+....+15)x16 * ((1/x)−1)

((1/2x)−1).......((1/2^15x)−1)

So ,

=2^120x^16[(1−(1/x))⋅(1−(1/2x))⋅(1−(1/2^2x))..........(1−(1/2^15x))]

So ,

=2^120*x^16[1−(1/x)(1+1/2+1/2^2+..........+1/2^15)+.....]

So Coefficient of x^15 in above expression

=−2^120(1+1/2+1/2^2+..........+1/2^15)=−2120[1−(1/2^16)]⋅1/(1/2)

So we get

=−2^121[1−(1/2^16)]=−2^121+2^105=2^105−2^121