Wallis' sequence estimated accurately using an alternating series

An asymptotic approximation of Wallis' sequence m↦Wm:=∏k=1m4k24k2−1 is presented asWm=mπ2m+1exp⁡(2σq(m))⋅exp⁡(rq(m)), whereσq(x):=∑i=1⌊q/2⌋(1−4−i)B2ii(2i−1)⋅x2i−1(Bk are the Bernoulli coefficients), and where|rq(m)|