Sugoroku 5

AtCoder

IDabc345_g

Time12000ms

Memory256MB

Difficulty—

There is a board game with $N+1$ squares: square $0$, square $1$, $\dots$, square $N$. You have a die (dice) that rolls an integer between $1$ and $K$, inclusive, with equal probability for each outcome. You start on square $0$. You repeat the following operation until you reach square $N$: * Roll the dice. Let $x$ be the current square and $y$ be the rolled number, then move to square $\min(N, x + y)$. Let $P_i$ be the probability of reaching square $N$ after exactly $i$ operations. Calculate $P_1, P_2, \dots, P_N$ modulo $998244353$. What is probability modulo $998244353$? It can be proved that the sought probabilities will always be rational numbers. Under the constraints of this problem, it can also be proved that when expressing the values as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, there is exactly one integer $R$ satisfying $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R < 998244353$. Find this $R$. ## Constraints * $1 \leq K \leq N \leq 2 \times 10^5$ * $N$ and $K$ are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ [samples]

Samples

Input #1

3 2

Output #1

0
249561089
748683265

For example, you reach square $N$ after exactly two operations when the die rolls the following numbers:

*   A $1$ on the first operation, and a $2$ on the second operation.
*   A $2$ on the first operation, and a $1$ on the second operation.
*   A $2$ on the first operation, and a $2$ on the second operation.

Therefore, $P_2 = \left( \frac{1}{2} \times \frac{1}{2} \right) \times 3 = \frac{3}{4}$. We have $249561089 \times 4 \equiv 3 \pmod{998244353}$, so print $249561089$ for $P_2$.

Input #2

5 5

Output #2

Input #3

10 6

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Sugoroku 5",
    "description": {
      "content": "There is a board game with $N+1$ squares: square $0$, square $1$, $\\dots$, square $N$.   You have a die (dice) that rolls an integer between $1$ and $K$, inclusive, with equal probability for each out",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 12000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc345_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a board game with $N+1$ squares: square $0$, square $1$, $\\dots$, square $N$.  \nYou have a die (dice) that rolls an integer between $1$ and $K$, inclusive, with equal probability for each out...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments