Checkers

AtCoder

IDagc022_f

Time2000ms

Memory256MB

Difficulty—

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. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

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.

Input #2

Output #2

Input #3

Output #3

487772376

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

API Response (JSON)

{
  "problem": {
    "name": "Checkers",
    "description": {
      "content": "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,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc022_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $1 \\leq i \\leq N$.\nEvery second, Inaba chooses two checker pieces,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments