Suppose `T` is a Young tableau of shape `\lambda=n^2`. If we replace the `k`-th largest element in `T` by `k` and rotate `T` 180 degrees, then we get a new tableau, say `P`. If `P=T`, we say `T` is symmetric. As examples, in the following picture, `T_1` is a symmetric Young tableau and `T_2` is not symmetric.
`a(n)` is the number of symmetric standard Young tableaux of shape `(n^2)`.
For example, `a(1)=1`, `a(2)=1`, `a(3)=3`.
Find `a(n)`.