Mahjong

AtCoder

IDarc160_d

Time2000ms

Memory256MB

Difficulty—

Find the number, modulo $998244353$, of sequences of $N$ non-negative integers $A=(A_1,A_2,\dots,A_N)$ totaling $M$ that satisfy the following condition. * It is possible to make all elements of $A$ equal $0$ by repeatedly choosing one of the following operations and performing it. * Choose an integer $i$ such that $1 \le i \le N$ and decrease $A_i$ by $K$. * Choose an integer $i$ such that $1 \le i \le N-K+1$ and decrease each of $A_i,A_{i+1},\dots,A_{i+K-1}$ by $1$. ## Constraints * $1 \le K \le N \le 2000$ * $1 \le M \le 10^{18}$ ## Input The input is given from Standard Input in the following format: $N$ $M$ $K$ [samples]

Samples

Input #1

3 2 2

Output #1

5

The following five sequences satisfy the requirements.

*   $(1,1,0)$
*   $(0,1,1)$
*   $(2,0,0)$
*   $(0,2,0)$
*   $(0,0,2)$

For instance, if $A=(0,1,1)$, you can do the following to make all elements of $A$ equal $0$.

*   Perform the second operation. Choose $i = 2$ to decrease each of $A_2$ and $A_3$ by $1$, making $A=(0,0,0)$.

Input #2

100 998244353 100

Output #2

0

There may be no sequence that satisfies the requirements.

Input #3

2000 545782618661124208 533

Output #3

908877889

API Response (JSON)

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  "problem": {
    "name": "Mahjong",
    "description": {
      "content": "Find the number, modulo $998244353$, of sequences of $N$ non-negative integers $A=(A_1,A_2,\\dots,A_N)$ totaling $M$ that satisfy the following condition. *   It is possible to make all elements of $A",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc160_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Find the number, modulo $998244353$, of sequences of $N$ non-negative integers $A=(A_1,A_2,\\dots,A_N)$ totaling $M$ that satisfy the following condition.\n\n*   It is possible to make all elements of $A...",
      "is_translate": false,
      "language": "English"
    }
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}

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