Zero-Sum Ranges 2

AtCoder

IDarc117_e

Time5000ms

Memory256MB

Difficulty—

How many different sequences of length $2N$, $A = (A_1, A_2, \dots, A_{2N})$, satisfy both of the following conditions? * The sequence $A$ contains $N$ occurrences of $+1$ and $N$ occurrences of $-1$. * There are exactly $K$ pairs of $l$ and $r$ $(1 \leq l \leq r \leq 2N)$ such that $A_l + A_{l+1} + \cdots + A_r = 0$. ## Constraints * $1 \leq N \leq 30$ * $1 \leq K \leq N^2$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ [samples]

Samples

Input #1

1 1

Output #1

2

For $N = 1, K = 1$, two sequences below satisfy the conditions:

*   $A = (+1, -1)$
*   $A = (-1, +1)$

Input #2

2 3

Output #2

2

For $N = 2, K = 3$, two sequences below satisfy the conditions:

*   $A = (+1, -1, -1, +1)$
*   $A = (-1, +1, +1, -1)$

Input #3

3 7

Output #3

6

For $N = 3, K = 7$, six sequences below satisfy the conditions:

*   $A = (+1, -1, +1, -1, -1, +1)$
*   $A = (+1, -1, -1, +1, +1, -1)$
*   $A = (+1, -1, -1, +1, -1, +1)$
*   $A = (-1, +1, +1, -1, +1, -1)$
*   $A = (-1, +1, +1, -1, -1, +1)$
*   $A = (-1, +1, -1, +1, +1, -1)$

Input #4

8 24

Output #4

Input #5

30 230

Output #5

761128315856702

Input #6

25 455

Output #6

0

For $N = 25, K = 455$, no sequences satisfy the conditions.

API Response (JSON)

{
  "problem": {
    "name": "Zero-Sum Ranges 2",
    "description": {
      "content": "How many different sequences of length $2N$, $A = (A_1, A_2, \\dots, A_{2N})$, satisfy both of the following conditions? *   The sequence $A$ contains $N$ occurrences of $+1$ and $N$ occurrences of $-",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc117_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "How many different sequences of length $2N$, $A = (A_1, A_2, \\dots, A_{2N})$, satisfy both of the following conditions?\n\n*   The sequence $A$ contains $N$ occurrences of $+1$ and $N$ occurrences of $-...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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