Integer Sequence Fair

AtCoder

IDabc228_e

Time2000ms

Memory256MB

Difficulty—

Integer Sequence Exhibition is taking place, where integer sequences are gathered in one place and evaluated. Here, every integer sequence of length $N$ consisting of integers between $1$ and $K$ (inclusive) is evaluated and given an integer score between $1$ and $M$ (inclusive). Print the number, modulo $998244353$, of ways to give each of the evaluated sequences a score between $1$ and $M$. Here, two ways are said to be different when there is an evaluated sequence $A = (A_1, A_2, \ldots, A_N)$ that is given different scores by the two ways. ## Constraints * $1 \leq N, K, M \leq 10^{18}$ * $N$, $K$, and $M$ are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ $M$ [samples]

Samples

Input #1

2 2 2

Output #1

16

Four sequences are evaluated: $(1, 1)$, $(1, 2)$, $(2, 1)$, and $(2, 2)$. There are $16$ ways to give each of these sequences a score between $1$ and $2$, as follows.

*   Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $1$ to $(2, 1)$, and $1$ to $(2, 2)$
*   Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $1$ to $(2, 1)$, and $2$ to $(2, 2)$
*   Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $2$ to $(2, 1)$, and $1$ to $(2, 2)$
*   Give $1$ to $(1, 1)$, $1$ to $(1, 2)$, $2$ to $(2, 1)$, and $2$ to $(2, 2)$
*   $\cdots$
*   Give $2$ to $(1, 1)$, $2$ to $(1, 2)$, $2$ to $(2, 1)$, and $2$ to $(2, 2)$

Thus, we print $16$.

Input #2

3 14 15926535

Output #2

109718301

Be sure to print the count modulo $998244353$.

API Response (JSON)

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  "problem": {
    "name": "Integer Sequence Fair",
    "description": {
      "content": "Integer Sequence Exhibition is taking place, where integer sequences are gathered in one place and evaluated. Here, every integer sequence of length $N$ consisting of integers between $1$ and $K$ (inc",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc228_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Integer Sequence Exhibition is taking place, where integer sequences are gathered in one place and evaluated. Here, every integer sequence of length $N$ consisting of integers between $1$ and $K$ (inc...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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