Creating Generating Function

Question

Given the generating function, $A(x)=\sum_{n\geq 0}a_nx^n=a_0+a_1x+a_2x^2+...$, find a formula for the generating function \begin{equation*} \tilde{A}(x)=a_1+a_0x+a_3x^2+a_2x^3+a_5x^4+a_4x^5+... \end{equation*} in terms of $A(x)$.

Now i know how to get the generating function for even parts and odd even i believe is $\frac{A(x)+A(-x)}{2}$ and the difference would be odd. How do i use this method to get the coefficients of A(x) to interchange, with a generating function .

Answer 1

This generating function is $x$ times the even part of the orginal plus $x^{-1}$ times that of the odd part of the original: something like $$x\frac{A(x)+A(-x)}2+x^{-1}\frac{A(x)-A(-x)}2.$$