Remove 2K+1 Edges

AtCoder

IDagc070_e

Time12000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $K$. Consider an undirected tree $T$ with $(2K + 1)N + 1$ vertices labeled from $1$ to $(2K + 1)N + 1$. Among all such trees, find the number, modulo $998244353$, of trees that satisfy the following condition. * It is possible to remove all edges from $T$ by repeating the following operation: * Choose a path of length $2K + 1$ and remove all edges in it. Formally, choose a sequence of distinct vertices $v_0, v_1, \dots, v_{2K+1}$ such that there exists an edge $(v_i, v_{i+1})$ for all $i$ such that $0 \leq i \leq 2K$, and remove the edge $(v_i, v_{i+1})$ for all $i$ such that $0 \leq i \leq 2K$. ## Constraints * $1 \leq N \leq 1.3 \times 10^5$ * $1 \leq K \leq 50$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ [samples]

Samples

Input #1

2 1

Output #1

8820

For example, consider a tree with edges $(1, 2)$, $(2, 3)$, $(2, 4)$, $(2, 6)$, $(3, 7)$, and $(4, 5)$. By doing the following steps, one can remove all edges from the tree, so it satisfies the condition in the problem statement:

*   Choose the path $(1,2,4,5)$ and remove edges $(1,2)$, $(2,4)$, $(4,5)$.
*   Choose the path $(6,2,3,7)$ and remove edges $(6,2)$, $(2,3)$, $(3,7)$.

![image](https://img.atcoder.jp/agc070/97f1043f2e834a849e1e3d10921cfb9b.png)
There are $8820$ such trees in total.

Input #2

70 1

Output #2

273892456

Remember to print the count modulo $998244353$.

Input #3

12 29

Output #3

898069699

Input #4

100000 50

Output #4

158600909

API Response (JSON)

{
  "problem": {
    "name": "Remove 2K+1 Edges",
    "description": {
      "content": "You are given positive integers $N$ and $K$. Consider an undirected tree $T$ with $(2K + 1)N + 1$ vertices labeled from $1$ to $(2K + 1)N + 1$. Among all such trees, find the number, modulo $998244353",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 12000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc070_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $K$. Consider an undirected tree $T$ with $(2K + 1)N + 1$ vertices labeled from $1$ to $(2K + 1)N + 1$. Among all such trees, find the number, modulo $998244353...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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