TITLE: A binomial determinant fomula
QUESTION [11 upvotes]: Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?

REPLY [17 votes]: This is true, and in fact you can show a slightly more general fact:
$$\det_{1\le i,j\le n}\left( \binom{x_i}{2j}+ \binom{-x_i}{2j}\right)=\prod_{i=1}^n x_i^2 \prod_{i<j} (x_j^2-x_i^2) \prod_{j=1}^n \frac{2}{(2j)!}.$$
It is easy to show that this evaluates to $1$ when you set $x_i=i$. To prove this, notice that every column is an even polynomial in the $x_i$ of degree $2j$ with leading coefficient $\frac{2}{(2j)!}$. So elementary column operations will bring the matrix in a familiar Vandermonde form.