Dice Game

AtCoder

IDarc154_f

Time6000ms

Memory256MB

Difficulty—

We have an $N$\-sided die where all sides have the same probability to show up. Let us repeat rolling this die until every side has shown up. For integers $i$ such that $1 \le i \le M$, find the expected value, modulo $998244353$, of the $i$\-th power of the number of times we roll the die. Definition of expected value modulo $998244353$It can be proved that the sought expected values are always rational numbers. Additionally, under the constraints of this problem, when such a value is represented as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q \neq 0 \pmod{998244353}$. Thus, there is a unique integer $R$ such that $R \times Q = P \pmod{998244353}$ and $0 \le R < 998244353$. Print this $R$. ## Constraints * $1 \le N,M \le 2 \times 10^5$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

3 3

Output #1

499122182
37
748683574

For $i=1$, you should find the expected value of the number of times we roll the die, which is $\frac{11}{2}$.

Input #2

7 8

Output #2

Input #3

2023 7

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Dice Game",
    "description": {
      "content": "We have an $N$\\-sided die where all sides have the same probability to show up. Let us repeat rolling this die until every side has shown up. For integers $i$ such that $1 \\le i \\le M$, find the expec",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc154_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have an $N$\\-sided die where all sides have the same probability to show up. Let us repeat rolling this die until every side has shown up.\nFor integers $i$ such that $1 \\le i \\le M$, find the expec...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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