Ex - Alchemy

AtCoder

IDabc281_h

Time4000ms

Memory256MB

Difficulty—

Takahashi has $A$ kinds of level-$1$ gems, and $10^{10^{100}}$ gems of each of those kinds. For an integer $n$ greater than or equal to $2$, he can put $n$ gems that satisfy all of the following conditions into a cauldron to generate a level-$n$ gem in return. * No two gems are of the same kind. * Every gem's level is less than $n$. * For every integer $x$ greater than or equal to $2$, there is at most one level-$x$ gem. Find the number of kinds of level-$N$ gems that Takahashi can obtain, modulo $998244353$. Here, two level-$2$ or higher gems are considered to be of the same kind if and only if they are generated from the same set of gems. * Two sets of gems are distinguished if and only if there is a gem in one of those sets such that the other set does not contain a gem of the same kind. Any level-$1$ gem and any level-$2$ or higher gem are of different kinds. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq A \leq 10^9$ * $N$ and $A$ are integers. ## Input The input is given from Standard Input in the following format: $N$ $A$ [samples]

Samples

Input #1

3 3

Output #1

10

Here are ten ways to obtain a level-$3$ gem.

*   Put three kinds of level-$1$ gems into the cauldron.
    *   Takahashi has three kinds of level-$1$ gems, so there is one way to choose three kinds of level-$1$ gems. Thus, he can obtain one kind of level-$3$ gem in this way.
*   Put one kind of level-$2$ gem and two kinds of level-$1$ gems into the cauldron.
    *   A level-$2$ gem can be obtained by putting two kinds of level-$1$ gems into the cauldron.
        *   Takahashi has three kinds of level-$1$ gems, so there are three ways to choose two kinds of level-$1$ gems. Thus, three kinds of level-$2$ gems are available here.
    *   There are three kinds of level-$2$ gems, and three ways to choose two kinds of level-$1$ gems, so he can obtain $3 \times 3 = 9$ kinds of level-$3$ gems in this way.

Input #2

1 100

Output #2

Input #3

200000 1000000000

Output #3

797585162

API Response (JSON)

{
  "problem": {
    "name": "Ex - Alchemy",
    "description": {
      "content": "Takahashi has $A$ kinds of level-$1$ gems, and $10^{10^{100}}$ gems of each of those kinds.   For an integer $n$ greater than or equal to $2$, he can put $n$ gems that satisfy all of the following con",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc281_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi has $A$ kinds of level-$1$ gems, and $10^{10^{100}}$ gems of each of those kinds.  \nFor an integer $n$ greater than or equal to $2$, he can put $n$ gems that satisfy all of the following con...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments