Beautiful Binary Tree

AtCoder

IDabc222_h

Time3000ms

Memory256MB

Difficulty—

For a positive integer $N$, a rooted binary tree that satisfies the following conditions is said to be a **beautiful binary tree of degree $N$**. * Each vertex has $0$ or $1$ written on it. * Each vertex that is a leaf has $1$ written on it. * It is possible to do the following operation at most $N-1$ times so that the root has $N$ written on it and the other vertices have $0$ written on them. * Choose vertices $u$ and $v$, where $v$ must be a child of $u$ or a child of "a child of $u$." Let $a_u \gets a_u + a_v, a_v \gets 0$, where $a_u$ and $a_v$ are the numbers written on $u$ and $v$, respectively. Given $N$, find the number, modulo $998244353$, of beautiful binary trees of degree $N$. ## Constraints * $1 \leq N \leq 10^7$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

1

The only binary tree that satisfies the condition is a tree with one vertex whose root has $1$ written on it.

Input #2

Output #2

6

The binary trees that satisfy the condition are the six trees shown below.
![image](https://img.atcoder.jp/ghi/37c6125e227d459cd725b6ccec96e2c8.png)

Input #3

Output #3

987355927

Input #4

Output #4

675337738

API Response (JSON)

{
  "problem": {
    "name": "Beautiful Binary Tree",
    "description": {
      "content": "For a positive integer $N$, a rooted binary tree that satisfies the following conditions is said to be a **beautiful binary tree of degree $N$**. *   Each vertex has $0$ or $1$ written on it. *   Eac",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc222_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a positive integer $N$, a rooted binary tree that satisfies the following conditions is said to be a **beautiful binary tree of degree $N$**.\n\n*   Each vertex has $0$ or $1$ written on it.\n*   Eac...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments