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Problem Statement

Let $X = 10^{100}$. Inaba has $N$ checker pieces on the number line, where the $i$-th checker piece is at coordinate $X^{i}$ for all $1 \leq i \leq N$.

Every second, Inaba chooses two checker pieces, $A$ and $B$, and move $A$ to the symmetric point of its current position with respect to $B$. After that, $B$ is removed. (It is possible that $A$ and $B$ occupy the same position, and it is also possible for $A$ to occupy the same position as another checker piece after the move).

After $N - 1$ seconds, only one checker piece will remain. Find the number of distinct possible positions of that checker piece, modulo $10^{9} + 7$.

Constraints

• $1 \leq N \leq 50$
• $N$ is an integer.

Sample Input 1

3

Sample Output 1

12

There are $3$ checker pieces, positioned at $10^{100}, 10^{200}, 10^{300}$ respectively. Call them $A, B, C$ respectively.

Here are two (of the $12$) possible sequence of moves :

• Let $A$ jump over $B$ in the first second, and let $A$ jump over $C$ in the second second. The final position of $A$ is $2 \times 10^{300} - 2 \times 10^{200} + 10^{100}$.

• Let $C$ jump over $A$ in the first second, and let $B$ jump over $C$ in the second second. The final position of $B$ is $-2 \times 10^{300} - 10^{200} + 4 \times 10^{100}$.

There are a total of $3 \times 2 \times 2 = 12$ possible sequence of moves, and the final piece are in different positions in all of them.

Sample Input 2

4

Sample Output 2

84

Sample Input 3

22

Sample Output 3

487772376

Remember to output your answer modulo $10^{9} + 7$.