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)

//---------- begin SegmentTree Range update Point query ----------
mod segment_tree {
    pub struct RUPQ<T, F> {
        n: usize,
        b: usize,
        a: Vec<T>,
        id: T,
        op: F,
    }
    impl<T: Copy, F: Fn(T, T) -> T> RUPQ<T, F> {
        pub fn new(n: usize, id: T, op: F) -> RUPQ<T, F> {
            let mut k = 0;
            while (1 << k) < n {
                k += 1;
            }
            RUPQ {
                n: 1 << k,
                b: k,
                a: vec![id; 2 << k],
                id: id,
                op: op,
            }
        }
        fn down(&mut self, x: usize) {
            let k = x + self.n;
            let a = &mut self.a;
            for i in (1..(self.b + 1)).rev() {
                let y = k >> i;
                a[2 * y] = (self.op)(a[2 * y], a[y]);
                a[2 * y + 1] = (self.op)(a[2 * y + 1], a[y]);
                a[y] = self.id;
            }
        }
        pub fn update(&mut self, mut l: usize, mut r: usize, v: T) {
            self.down(l);
            self.down(r - 1);
            l += self.n;
            r += self.n;
            let a = &mut self.a;
            while l < r {
                if (l & 1) == 1 {
                    a[l] = (self.op)(a[l], v);
                    l += 1;
                }
                if (r & 1) == 1 {
                    r -= 1;
                    a[r] = (self.op)(a[r], v);
                }
                l >>= 1;
                r >>= 1;
            }
        }
        pub fn find(&mut self, mut x: usize) -> T {
            x += self.n;
            let mut y = self.a[x];
            x >>= 1;
            while x > 0 {
                y = (self.op)(y, self.a[x]);
                x >>= 1;
            }
            y
        }
    }
}
//---------- end SegmentTree Range update Point query ----------
// ---------- begin ModInt ----------
const MOD: u32 = 1_000_000_007;

#[derive(Clone, Copy)]
struct ModInt(u32);

impl std::ops::Add for ModInt {
    type Output = ModInt;
    fn add(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = ModInt;
    fn sub(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + MOD - rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = ModInt;
    fn mul(self, rhs: ModInt) -> Self::Output {
        ModInt((self.0 as u64 * rhs.0 as u64 % MOD as u64) as u32)
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl std::ops::Neg for ModInt {
    type Output = ModInt;
    fn neg(self) -> Self::Output {
        ModInt(if self.0 == 0 {0} else {MOD - self.0})
    }
}

/* 
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
*/

#[allow(dead_code)]
impl ModInt {
    pub fn new(n: u32) -> ModInt {
        ModInt(n % MOD)
    }
    pub fn zero() -> ModInt {
        ModInt(0)
    }
    pub fn one() -> ModInt {
        ModInt(1)
    }
    pub fn pow(self, mut n: u32) -> ModInt {
        let mut t = ModInt::one();
        let mut s = self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(self) -> ModInt {
        self.pow(MOD - 2)
    }
}
// ---------- end ModInt ----------
// ---------- begin Precalc ----------
#[allow(dead_code)]
struct Precalc {
    inv: Vec<ModInt>,
    fact: Vec<ModInt>,
    ifact: Vec<ModInt>,
}

#[allow(dead_code)]
impl Precalc {
    pub fn new(n: usize) -> Precalc {
        let mut inv = vec![ModInt::one(); n + 1];
        let mut fact = vec![ModInt::one(); n + 1];
        let mut ifact = vec![ModInt::one(); n + 1];
        for i in 2..(n + 1) {
            inv[i] = -inv[MOD as usize % i] * ModInt(MOD / i as u32);
            fact[i] = fact[i - 1] * ModInt(i as u32);
            ifact[i] = ifact[i - 1] * inv[i];
        }
        Precalc {
            inv: inv,
            fact: fact,
            ifact: ifact,
        }
    }
    pub fn inv(&self, n: usize) -> ModInt {
        self.inv[n]
    }
    pub fn fact(&self, n: usize) -> ModInt {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModInt {
        self.ifact[n]
    }
    pub fn comb(&self, n: usize, k: usize) -> ModInt {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end Precalc ----------
use std::io::Read;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    it.next();
    let k: usize = it.next().unwrap().parse().unwrap();
    let mut s = segment_tree::RUPQ::new(2 * k + 1, ModInt::zero(), std::ops::Add::add);
    s.update(0, 1, ModInt::one());
    for a in it {
        let a: usize = a.parse().unwrap();
        for i in (0..(k + 1)).rev() {
            let v = s.find(i);
            s.update(i, i + 1, -v);
            s.update(i, i + a + 1, v);
        }
    }
    let ans = s.find(k);
    println!("{}", ans.0);
}

fn main() {
    run();
}