Question about generating functions
Hint: we have $A(x^2) = \sum_{n=0}^\infty a_nx^{2n} = a_0 + a_1x^2 + a_2x^4+ \cdots$. What can we say about the coefficients of odd powers ($x, x^3, \dots$)?
Now do the same for $xB(x^2)$, what can we say about  the its even power coefficients?
What does this mean for their sum?