Suppose `\sigma=\sigma_1\sigma_2\sigma_3\cdots\sigma_n` is a permutation (one-line notation). If elements of `\sigma` satisfy `\sigma_1\lt\sigma_2\gt\sigma_3\lt\sigma_4\gt\sigma_5\lt\cdots\sigma_n`, we say `\sigma` is an alternating permutation of length `n`. The exponential generating function of the numbers of alternating permutations is `\sec x+\tan x`.
`a(n)` is the number of alternating permutations of length `n` such that `\sigma_1\sigma_3\sigma_5\cdots` is monotone and `\sigma_2\sigma_4\sigma_6\cdots` is also monotone.
For example, `a(1)=1`, `a(2)=1`, `a(3)=2`, `a(4)=5`.
Find `a(n)`.