Generating Function Question - Infinite Series - Simple Question

Question

$$ \sum_0^\infty u^{2n+1} = \frac{u}{1-u^2} $$ $$ \Rightarrow \sum_0^\infty (2n+1)u^{2n} = \frac{d}{du}\left(\frac{u}{1-u^2}\right)= \frac{u^2+1}{(1-u^2)^2} $$ In our problem, we have $u= t^n(1-t^n)x^n$. But my professor has it written down that: $$ \sum_0^\infty (2n+1)u^{2n} = \frac{1-3u^2}{(1-u^2)^2} $$ We began with: $$ \sum_0^\infty (2n+1)u^{n}. $$ and changed it to: $$ \sum_0^\infty (2n+1)u^{\frac{2n}{2}} $$ I do not understand how my professor went from line 1 to line 3. The only thing I can see causing this is going from line 4 to 5 changing it. But I don't see how.