If $P_r$ the coefficient of $x^r$ in the expansion of $(1+x)^2 \left(1+\frac{x}{2}\right)^2 \left(1+\frac{x}{2^2}\right)^2 \cdots$

Question

If $P_r$ the coefficient of $x^r$ in the expansion of $$(1+x)^2 \left(1+\frac{x}{2}\right)^2 \left(1+\frac{x}{2^2}\right)^2 \cdots$$

then prove that $$P_r=\frac{2^2}{2^r-1}(P_{r-1}+P_{r-2})$$

How to prove this result.. Any help thank you

Answer 1

Put $F(x)=\prod_{k\geq 0}(1+\frac{x}{2^k})^2$. Then $F(2x)=(1+2x)^2F(x)=(1+4x+4x^2)F(x)$. Now the coefficient of $x^r$ for $r\geq 2$ is $2^rP_r$ on one side, and $P_r+4P_{r-1}+4P_{r-2}$ on the other side.