Multiset Mean

AtCoder

IDarc104_d

Time4000ms

Memory256MB

Difficulty—

Given positive integers $N, K$ and $M$, solve the following problem for every integer $x$ between $1$ and $N$ (inclusive): * Find the number, modulo $M$, of non-empty multisets containing between $0$ and $K$ (inclusive) instances of each of the integers $1, 2, 3 \cdots, N$ such that the average of the elements is $x$. ## Constraints * $1 \leq N, K \leq 100$ * $10^8 \leq M \leq 10^9 + 9$ * $M$ is prime. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ $M$ [samples]

Samples

Input #1

3 1 998244353

Output #1

1
3
1

Consider non-empty multisets containing between $0$ and $1$ instance(s) of each of the integers between $1$ and $3$. Among them, there are:

*   one multiset such that the average of the elements is $k = 1$: ${1}$;
*   three multisets such that the average of the elements is $k = 2$: ${2}, {1, 3}, {1, 2, 3}$;
*   one multiset such that the average of the elements is $k = 3$: ${3}$.

Input #2

1 2 1000000007

Output #2

2

Consider non-empty multisets containing between $0$ and $2$ instances of each of the integers between $1$ and $1$. Among them, there are:

*   two multisets such that the average of the elements is $k = 1$: ${1}, {1, 1}$.

Input #3

10 8 861271909

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Multiset Mean",
    "description": {
      "content": "Given positive integers $N, K$ and $M$, solve the following problem for every integer $x$ between $1$ and $N$ (inclusive): *   Find the number, modulo $M$, of non-empty multisets containing between $",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc104_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given positive integers $N, K$ and $M$, solve the following problem for every integer $x$ between $1$ and $N$ (inclusive):\n\n*   Find the number, modulo $M$, of non-empty multisets containing between $...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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