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}, $x^{2k}$, you have the relation: T(2k)+T(k)=0$T(2k)+T(k)=0$.
Equating the coefficients of x^{2k+1}, $x^{2k+1}$, you have the relation: T(2k+1)+T(k)=1$T(2k+1)+T(k)=1$.
Now you can start computing the coefficients: T(0)=0, T(1)=1, T(2)=-1, T(3)=0, $T(0)=0$, $T(1)=1$, $T(2)=-1$, $T(3)=0$, etc.
sigfpe correctly identified the sequence. You can even see these recurrences mentioned in the formula section.
