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: Accepted)

#[allow(unused_imports)]
use std::cmp::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)]
use std::iter::FromIterator;

#[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        writeln!(&mut std::io::stderr(), concat!("[DEBUG] ", $(stringify!($a), "={:?} "),*), $($a),*).unwrap();
    }
}

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord)]
struct ModInt {
    value: usize,
}
impl ModInt {
    fn new(value: usize) -> Self {
        ModInt {
            value: value % Self::modulo(),
        }
    }
    fn value(&self) -> usize {
        self.value
    }
    fn modulo() -> usize {
        1_000_000_007
    }
}
impl From<usize> for ModInt {
    fn from(value: usize) -> Self {
        ModInt { value: value }
    }
}
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, other: Self) -> Self {
        let mut value = self.value + other.value;
        if value > Self::modulo() {
            value -= Self::modulo();
        }
        ModInt { value: value }
    }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, other: Self) -> Self {
        let mut value = self.value;
        if value < other.value {
            value += Self::modulo();
        }
        value -= other.value;
        ModInt { value: value }
    }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, other: Self) -> Self {
        let value = self.value * other.value % Self::modulo();
        ModInt { value: value }
    }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, other: Self) -> Self {
        let mut exp = Self::modulo() - 2;
        let mut value = self.value;
        let mut ovalue = other.value;
        while exp > 0 {
            if exp % 2 == 1 {
                value = value * ovalue % Self::modulo();
            }
            ovalue = ovalue * ovalue % Self::modulo();
            exp /= 2;
        }
        ModInt { value: value }
    }
}
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, other: Self) {
        *self = *self + other;
    }
}
impl std::fmt::Debug for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.value)
    }
}
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.value)
    }
}

fn solve() {
    let (n,k) = get!(usize, usize);
    let a = get!([usize]);

    let mut dp = vec![vec![ModInt::new(0); k+1]; n+1];
    dp[0][0] = ModInt::new(1);

    for i in 0..n {
        let mut vec = vec![ModInt::new(0); k+1];
        vec[0] = dp[i][0];
        for j in 1..k+1 {
             vec[j] = vec[j-1] + dp[i][j];
        }
        for j in 0..k+1 {
            dp[i+1][j] = if j > a[i] {
                vec[j] - vec[j-a[i]-1]
            } else {
                vec[j]
            };
        }
    }
    println!("{}", dp[n][k]);
}