Problem:
Score : 100 points

Problem Statement

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

Output

Print the number of ways for the children to share candies, modulo 10^9 + 7.

Sample Input 1

3 4
1 2 3

Sample 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.

Sample Input 2

1 10
9

Sample Output 2

0

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

Sample Input 3

2 0
0 0

Sample Output 3

1

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

(0, 0)

Sample Input 4

4 100000
100000 100000 100000 100000

Sample Output 4

665683269

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

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

use std::error::Error;
use std::io;

fn main() {
    if let Err(e) = run() {
        panic!("failed.");
    }
}

fn run() -> Result<(), Box<Error>> {
    let mut buffer = String::new();
    io::stdin().read_line(&mut buffer)?;
    let first_line: Vec<usize> = buffer.split_whitespace()
        .map(|x| x.parse::<usize>().unwrap())
        .collect();
    let n = first_line[0];
    let k = first_line[1];

    buffer.clear();
    io::stdin().read_line(&mut buffer)?;
    let candies: Vec<i32> = buffer.split_whitespace()
        .map(|x| x.parse::<i32>().unwrap())
        .collect();

    let mut dp = vec![vec![-1; k + 1]; n];
    println!("{}", calculate(&mut dp, &candies, n - 1, k));
    Ok(())
}

fn calculate(dp: &mut Vec<Vec<i32>>, candies: &Vec<i32>, n: usize, k: usize) -> i64 {
    if dp[n][k] != -1 {
        return dp[n][k] as i64;
    }
    if n == 0 {
        if k <= candies[n] as usize {
            return 1;
        }
        return 0;
    }

    let mut result = 0 as i64;
    for i in 0..candies[n]+1 {
        if k < i as usize {
            continue;
        }
        result += calculate(dp, candies, n - 1, k - i as usize);
        result %= 1_000_000_007;
    }
    dp[n][k] = result as i32;
    result
}