AND OR Equation

AtCoder

IDarc144_d

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $K$. Find the number, modulo $998244353$, of integer sequences $\bigl(f(0), f(1), \ldots, f(2^N-1)\bigr)$ that satisfy all of the following conditions: * $0\leq f(x)\leq K$ for all non-negative integers $x$ ($0\leq x \leq 2^N-1$). * $f(x) + f(y) = f(x \,\mathrm{AND}\, y) + f(x \,\mathrm{OR}\, y)$ for all non-negative integers $x$ and $y$ ($0\leq x, y \leq 2^N-1$) Here, $\mathrm{AND}$ and $\mathrm{OR}$ denote the bitwise AND and OR, respectively. ## Constraints * $1\leq N\leq 3\times 10^5$ * $1\leq K\leq 10^{18}$ ## Input Input is given from Standard Input in the following format: $N$ $K$ [samples]

Samples

Input #1

2 1

Output #1

6

The following six integer sequences satisfy the conditions:

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

Input #2

2 2

Output #2

Input #3

100 123456789123456789

Output #3

34663745

API Response (JSON)

{
  "problem": {
    "name": "AND OR Equation",
    "description": {
      "content": "You are given positive integers $N$ and $K$. Find the number, modulo $998244353$, of integer sequences $\\bigl(f(0), f(1), \\ldots, f(2^N-1)\\bigr)$ that satisfy all of the following conditions: *   $0\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc144_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $K$. Find the number, modulo $998244353$, of integer sequences $\\bigl(f(0), f(1), \\ldots, f(2^N-1)\\bigr)$ that satisfy all of the following conditions:\n\n*   $0\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments