Sum of a finite arithmetic sequence

Question

Let $S_n = \sum_{j=0}^{n-1}2j(2j-1)x^{2j}$ where $|x| < 1$. What is the finite sum corresponds to?

Edit: \begin{equation*} S_n = \sum_{j=0}^{n-1}2j(2j-1)x^{2j}\\ = x^2 \frac{d^2}{dx^2}\left[\frac{1 - x^{2n}}{1-x^2}\right] \end{equation*}

Answer 1

Note that $$\sum_{j=0}^{n-1}2j(2j-1)x^{2j}=x^2\frac{d^2}{dx^2}\left(\sum_{j=0}^{n-1}(x^2)^j\right).$$ where the finite sum on the right has a closed formula that you should know (see LINK).

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