Question 25
Let $n \geq 1$ be integer. Sequence $a_1, a_2, \dots, a_{2n}$ is called lucky if
1) $a_1, \dots, a_n$ is a permutation of $\{1, 2, \dots, n\}$;
2) $a_k = a_{n+k}$ for all $k = 1, 2, \dots, n$;
3) there exist indices $1 \le i_1 \lt i_2 \lt \dots \lt i_n \le 2n$ such that $a_{i_k} = k$ for all $k = 1, 2, \dots, n$.
For each $n \geq 1$ find the number of lucky sequences.
1) $a_1, \dots, a_n$ is a permutation of $\{1, 2, \dots, n\}$;
2) $a_k = a_{n+k}$ for all $k = 1, 2, \dots, n$;
3) there exist indices $1 \le i_1 \lt i_2 \lt \dots \lt i_n \le 2n$ such that $a_{i_k} = k$ for all $k = 1, 2, \dots, n$.
For each $n \geq 1$ find the number of lucky sequences.