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![ModUsize::new(0); k * 3 + 10]; n + 10];
    dp[0][0] = ModUsize::new(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 + ai + 1] -= dij;
        }

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

use num_integer::Integer;
use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

#[derive(Copy, Clone)]
pub struct ModUsize {
    pub mod_value: usize,
    pub value: usize,
}

impl ModUsize {
    pub fn new(n: usize) -> Self {
        Self {
            mod_value: Self::mod1(),
            value: n % Self::mod1(),
        }
    }

    pub fn pow(self, n: usize) -> Self {
        if n == 0 {
            Self::new(1)
        } else if n % 2 == 0 {
            let t = self.pow(n / 2);
            t * t
        } else {
            self * self.pow(n - 1)
        }
    }

    pub fn inv(self) -> Self {
        self.pow(self.mod_value - 2)
    }

    pub fn mod1() -> usize {
        1000000007
    }

    pub fn mod2() -> usize {
        998244353
    }
}

impl Add for ModUsize {
    type Output = ModUsize;

    fn add(self, rhs: Self) -> Self::Output {
        let mut tmp = self;
        tmp.value += rhs.value;
        if tmp.value >= self.mod_value {
            tmp.value -= self.mod_value;
        }
        tmp
    }
}

impl AddAssign for ModUsize {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl Sub for ModUsize {
    type Output = ModUsize;
    
    fn sub(self, rhs: Self) -> Self::Output {
        let mut tmp = self;
        if tmp.value < rhs.value {
            tmp.value += self.mod_value;
        }
        tmp.value -= rhs.value;
        tmp
    }
}

impl SubAssign for ModUsize {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl Mul for ModUsize {
    type Output = ModUsize;

    fn mul(self, rhs: Self) -> Self::Output {
        let mut tmp = self;
        tmp.value *= rhs.value;
        tmp.value %= self.mod_value;
        tmp
    }
}

impl MulAssign for ModUsize {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl Div for ModUsize {
    type Output = ModUsize;

    fn div(self, rhs: Self) -> Self::Output {
        let mut tmp = self;
        tmp.value /= rhs.value;
        tmp.value %= self.mod_value;
        tmp
    }
}

impl DivAssign for ModUsize {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}