\begin{array}{rcl} \ln (B_{n}) & = & (\frac{1}{2} + n) \ln (n) + (\frac{1}{2} - n) \ln (π) + (\frac{3}{2} - n) \ln (2) - R (n) \\ R (n) & = & n (1 - \frac{1}{12} (1 - \frac{1}{30} (1 - \frac{2}{7} n^{- 2}) n^{- 2}) n^{- 2}) \end{array}