Candies

AtCoder

IDdp_m

Time2000ms

Memory256MB

Difficulty—

There are $N$ children, numbered $1, 2, \ldots, N$. They have decided to share $K$ candies among themselves. Here, for each $i$ ($1 \leq i \leq N$), Child $i$ must receive between $0$ and $a_i$ candies (inclusive). Also, no candies should be left over. Find the number of ways for them to share candies, modulo $10^9 + 7$. Here, two ways are said to be different when there exists a child who receives a different number of candies. ## Constraints * All values in input are integers. * $1 \leq N \leq 100$ * $0 \leq K \leq 10^5$ * $0 \leq a_i \leq K$ ## Input Input is given from Standard Input in the following format: $N$ $K$ $a_1$ $a_2$ $\ldots$ $a_N$ [samples]

Samples

Input #1

3 4
1 2 3

Output #1

5

There are five ways for the children to share candies, as follows:

*   $(0, 1, 3)$
*   $(0, 2, 2)$
*   $(1, 0, 3)$
*   $(1, 1, 2)$
*   $(1, 2, 1)$

Here, in each sequence, the $i$\-th element represents the number of candies that Child $i$ receives.

Input #2

1 10
9

Output #2

0

There may be no ways for the children to share candies.

Input #3

2 0
0 0

Output #3

1

There is one way for the children to share candies, as follows:

*   $(0, 0)$

Input #4

4 100000
100000 100000 100000 100000

Output #4

665683269

Be sure to print the answer modulo $10^9 + 7$.

API Response (JSON)

{
  "problem": {
    "name": "Candies",
    "description": {
      "content": "There are $N$ children, numbered $1, 2, \\ldots, N$. They have decided to share $K$ candies among themselves. Here, for each $i$ ($1 \\leq i \\leq N$), Child $i$ must receive between $0$ and $a_i$ candie",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "dp_m"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ children, numbered $1, 2, \\ldots, N$.\nThey have decided to share $K$ candies among themselves. Here, for each $i$ ($1 \\leq i \\leq N$), Child $i$ must receive between $0$ and $a_i$ candie...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments