Determinant of the matrix with $a_{i,j} = (i+j)^2$

When your matrix is big enough it gets easier.

\begin{bmatrix} 2^2&3^2\cdots &(n+1)^2\\ \vdots & \ddots &\vdots\\ (n+1)^2&&(2n)^2\end{bmatrix}

Now lets do some row opperations. Subtract the row above from every row. Leave the top row as it is

\begin{bmatrix} 2^2&3^2\cdots &(n+1)^2\\ 5&7&\cdots 2n+3\\ \vdots & \ddots &\vdots\\ 2n+1&&4n-1\end{bmatrix}

and do it again

\begin{bmatrix} 2^2&3^2\cdots &(n+1)^2\\ 1&-2\cdots &2-n^2\\ 2&2\cdots& 2\\ \vdots & \ddots &\vdots\\ 2&&2\end{bmatrix}

A singular matrix when $n\ge 4$