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.



Generating-functions: is there a relationship between a generating function and the corresponding squared generating function
Alright, so on the one side, you have this:


$$A(t)+(1+t)A(t^{2})=\sum\_{n=0}^{\infty}T(n)t^{n}+\sum\_{n=0}^{\infty}T(n)t^{2n}+\sum\_{n=0}^{\infty}T(n)t^{2n+1}$$


On the other side, you have:


$$\frac{t}{1-t^{2}}=\sum\_{n=0}^{\infty}t^{2n+1}$$


Equating the coefficients of $x^{2k}$, you have the relation: $T(2k)+T(k)=0$.


Equating the coefficients of $x^{2k+1}$, you have the relation: $T(2k+1)+T(k)=1$.


Now you can start computing the coefficients: $T(0)=0$, $T(1)=1$, $T(2)=-1$, $T(3)=0$, etc.


sigfpe correctly identified [the sequence](http://oeis.org/A065359 "the sequence"). You can even see these recurrences mentioned in the formula section.


