The total number of terms in the expansion of $${(x + y)^{100}} + {(x - y)^{100}}$$ is

The coefficient of $${x^{20}}$$ in the expansion of $${(1 + 3x + 3{x^2} + {x^3})^{20}}$$ is

In the expansion of $${(1 - 3x + 3{x^2} - {x^3})^{2n}}$$, the middle term is

Using mathematical induction, the numbers $${a_n}$$'s are defined by $${a_0} = 1,{a_{n + 1}} = 3{n^2} + n + {a_n},(n \ge 0)$$. Then, $${a_n}$$ is equal to