Bomb Game 2

AtCoder

IDabc333_f

Time2000ms

Memory256MB

Difficulty—

There are $N$ people standing in a line, with person $i$ standing at the $i$\-th position from the front. Repeat the following operation until there is only one person left in the line: * Remove the person at the front of the line with a probability of $\frac{1}{2}$, otherwise move them to the end of the line. For each person $i=1,2,\ldots,N$, find the probability that person $i$ is the last person remaining in the line, modulo $998244353$. (All choices to remove or not are random and independent.) How to find probabilities modulo $998244353$The probabilities sought in this problem can be proven to always be a rational number. Furthermore, the constraints of this problem guarantee that if the sought probability is expressed as an irreducible fraction $\frac{y}{x}$, then $x$ is not divisible by $998244353$. Now, there is a unique integer $z$ between $0$ and $998244352$, inclusive, such that $xz \equiv y \pmod{998244353}$. Report this $z$. ## Constraints * $2\leq N\leq 3000$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

332748118 665496236

Person $1$ will be the last person remaining in the line with probability $\frac{1}{3}$.
Person $2$ will be the last person remaining in the line with probability $\frac{2}{3}$.

Input #2

Output #2

235530465 792768557 258531487 238597268 471060930

API Response (JSON)

{
  "problem": {
    "name": "Bomb Game 2",
    "description": {
      "content": "There are $N$ people standing in a line, with person $i$ standing at the $i$\\-th position from the front. Repeat the following operation until there is only one person left in the line: *   Remove th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc333_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ people standing in a line, with person $i$ standing at the $i$\\-th position from the front.\nRepeat the following operation until there is only one person left in the line:\n\n*   Remove th...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments