What is the determinant of the Wronskian of the functions $\{\cos\ x, \sin\ x, \cos\ 2x, \sin\ 2x,\ldots, \cos\ nx, \sin\ nx\}$? This determinant seems to be an integer, and the sequence starts with 1, 18, 86400, 548674560000... It is not in the Encyclopedia.

** Question ** What is this sequence? I guess it is enough to prove that it consists of integers (constants), because then to compute it, one can simply put $x=0$.

** Update 1. ** The fact that the determinant of the Wronskian matrix is a constant is obvious. Take the derivative of the determinant. It is a sum of determinants of matrices each of which has two proportional columns.

** Update 2. ** The determinant is equal to the square of the Vandermonde determinant of $1, 2^2,\ldots, n^2$ times $n!$ (alternatively see Felipe Voloch's answer below). It is interesting that for $n=1$ we get just the equality $\cos^2 x+\sin^2 x=1$ (the equation of the circle). So the equality for $n > 1$ can be considered as the generalization of this equation. What is the geometry behind this identity? Of course the parametrization $(\cos x,\sin x,\ldots, \cos nx, \sin nx)$ defines some curve in $\mathbb{R}^{2n}$. What is known about that curve?

** Update 3. ** Here is an easier formula for the determinant. It is equal to
$$(1! 3!\ldots (2n-1)!)^2/n!$$

** Update 4 ** I found a related paper:
Larsen, Mogens Esrom, Wronskian harmony.
Math. Mag. 63 (1990), no. 1, 33–37. He considers the Wronskian of $\sin x, \ldots, \sin nx$.

** Update 5. ** The sequence is in the Encyclopedia now.