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 (1.42.0) (judge verdict: Accepted)

#![allow(unused_macros)]
#![allow(dead_code)]
#![allow(unused_imports)]

use itertools::Itertools;
use text_io::*;

fn main() {
    const U_INF: usize = 1 << 60;
    const I_INF: isize = 1 << 60;
    const MOD: i128 = 1000000007;

    let n: usize = read!();
    let k: usize = read!();
    let mut a = Vec::new();
    for _ in 0..n {
        let ai: usize = read!();
        a.push(ai);
    }
    //
    // let mut dp = vec![vec![0; k + 10]; n + 10];
    // for j in 0..=k {
    //     dp[0][j] = 1;
    // }
    // for i in 1..=n {
    //     for j in 0..=k {
    //         // dp[i-1]は累積和になっていると想定する
    //         let right = dp[i - 1][j];
    //         let left = if j < (a[i - 1] + 1) {
    //             0
    //         } else {
    //             dp[i - 1][j - a[i - 1] - 1]
    //         };
    //         dp[i][j] = (right - left) % MOD;
    //     }
    //
    //     if i != n {
    //         for j in 1..=k {
    //             dp[i][j] += dp[i][j - 1];
    //             dp[i][j] %= MOD;
    //         }
    //     }
    // }
    //
    // println!("{}", dp[n][k]);

    let mut dp = vec![vec![0; k * 3 + 10]; n + 10];
    dp[0][0] = 1;
    for i in 0..n {
        for j in 0..=k {
            let dij = dp[i][j];
            let ai = a[i];
            let ai = ai.min(k - j);
            dp[i + 1][j] += dij;
            dp[i + 1][j] %= MOD;
            dp[i + 1][j + ai + 1] -= dij;
            dp[i + 1][j + ai + 1] %= MOD;
            dp[i + 1][j + ai + 1] += MOD;
            dp[i + 1][j + ai + 1] %= MOD;
        }

        for j in 0..k {
            dp[i + 1][j + 1] += dp[i + 1][j];
            dp[i + 1][j + 1] %= MOD;
        }
    }
    println!("{}", dp[n][k]);
}