 Solution

Under the null hypothesis, $Y\sim \mathrm{Bin}(n,1/2)$, with expected value $n/2$ and variance $n/4$. Approximating its probability at an integer by the normal distribution density over a unit-width cell givesDividing by the prior predictive distribution from the preceding part gives the uniform-alternative binomial Bayes factorThe second approximation replaces $n+1$ by $n$. Both are large-sample approximations for counts in the central range, rather than exact identities at extreme counts.