Putnam 2004 A3

Define a sequence $\{ u_n \}_{n=0}^\infty$ by $u_0 = u_1 = u_2 = 1$, and thereafter by the condition that $$\det\begin{pmatrix} u_n & u_{n+1}\\ u_{n+2} & u_{n+3} \end{pmatrix} = n!$$ for all $n \ge 0$. Show that $u_n$ is an integer for all $n$. (By convention, $0! = 1$.)