Let's say we have a sequence $T(n)$ with the corresponding generating function

$$A(t) = \sum_{n = 0}^\infty T(n) t^n$$

Is there some relationship between the two functions $A(t)$ and $A(t^2)$? And for that matter is there some generalization for any integer power or $t$?

**Edit:** I'm actually trying to solve for the generating function $A(t)$ in the equation

$$A(t) + (1+t)A(t^2) = t/(1-t^2)$$

this is what inspired my question. My intuition suggested to me that I should look for some kind of relationship between $A(t^2)$ and $A(t)$, hence the vagueness of my question.