Is there a closed form for determinant of the Hankel matrix of the sequence $\{ 1, a, a^2, \dots, a^{2n+2} \}$?

Today, at my linear algebra exam, there was this question that I couldn't solve. Prove that $$\det \begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & (n+3)^{2}\\ (n+2)^{2} & (n+3)^{2} & (n+4)^{2} \end{bmatrix} = -8$$ Clearly, calculating the determinant, with the matri...

Recall that the $n$-th Catalan number $C_n=\frac{1}{n+1}{2n\choose n}$ counts the number of paths connecting $(0, 0)$ to $(n, n)$ that travel along the grid of integer lattice points of $R^2$ where each path moves up or right in one-unit steps and no path extends above the line $y = x$. In linea...