Product of factorials of powers of two

Question

Is there any exact formula for the product of factorials of powers of 2? Namely, I want to calculate $$ (2^k)! (2^{k-1})!(2^{k-2})! \;\cdot\; ...\;\cdot\; 2 = \prod_{j=0}^{k-1}(2^{k-j})! $$ where $k$ is a positive integer.