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: Time Limit Exceeded)

use std::cmp;
use proconio::input;

macro_rules! __memoize_internal {
  (@vec $FRetT:ty) => {
    Option<$FRetT>
  };
  (@vec $FRetT:ty, $memo_param_head:ident $(, $memo_param:ident)*) => {
    Vec<__memoize_internal!(@vec $FRetT $(, $memo_param)*)>
  };
  (@indices [$($t:tt)*]) => {
    [$($t)*]
  };
  (
    @indices [$($t:tt)*]
    $memo_param_head:ident
    $(, $memo_param:ident $(($memo_param_expr:expr))*)*
  ) => {
    __memoize_internal!(
      @indices [$($t)* $memo_param_head as usize,] $($memo_param $(($memo_param_expr))*),*
    )
  };
  (
    @indices [$($t:tt)*]
    $memo_param_head:ident($memo_param_expr_head:expr)
    $(, $memo_param:ident $(($memo_param_expr:expr))*)*
  ) => {
    __memoize_internal!(
      @indices [$($t)* $memo_param_expr_head(&$memo_param_head),]
      $($memo_param $(($memo_param_expr))*),*
    )
  };
  (@get $memo:expr, $indices:expr, $i:expr) => {
    $memo
  };
  (@get $memo:expr, $indices:expr, $i:expr, $memo_param_head:ident $(, $memo_param:ident)*) => {
    __memoize_internal!(
      @get $memo.and_then(|memo| memo.get($indices[$i])), $indices, $i + 1 $(, $memo_param)*
    )
  };
  (@resize $memo:expr, $indices:expr, $i:expr) => {};
  (@resize $memo:expr, $indices:expr, $i:expr, $memo_param_head:ident $(, $memo_param:ident)*) => {
    if $memo.len() < $indices[$i] + 1 {
      $memo.resize($indices[$i] + 1, <_>::default());
    }
    __memoize_internal!(
      @resize $memo[$indices[$i]], $indices, $i + 1 $(, $memo_param)*
    )
  };
  (@index $memo:expr, $indices:expr, $i:expr) => {
    $memo
  };
  (@index $memo:expr, $indices:expr, $i:expr, $memo_param_head:ident $(, $memo_param:ident)*) => {
    __memoize_internal!(
      @index $memo[$indices[$i]], $indices, $i + 1 $(, $memo_param)*
    )
  };
}

macro_rules! memoize {
  (
    #[memo_param($($memo_param:ident $(($memo_param_expr:expr))*),+)]
    $(#[$meta:meta])*
    fn $f:ident($memo:ident: _ $(, $f_param:ident: $FParamT:ty)*) -> $FRetT:ty $f_body:block
  ) => {
    $(#[$meta])*
    fn $f(
      $memo: &mut __memoize_internal!(@vec $FRetT $(, $memo_param)+)
      $(, $f_param: $FParamT)+
    ) -> $FRetT {
      let indices = __memoize_internal!(@indices [] $($memo_param $(($memo_param_expr))*),+);

      if let Some(&Some(ref ret)) =
        __memoize_internal!(@get Some(&$memo), indices, 0 $(, $memo_param)+)
      {
        return ret.clone();
      }

      let ret = $f_body;

      __memoize_internal!(@resize $memo, indices, 0 $(, $memo_param)+);
      __memoize_internal!(@index $memo, indices, 0 $(, $memo_param)+) = Some(ret.clone());

      ret
    }
  };
}

const MOD: i64 = 1000000007;

fn main() {
  input! {
    n : usize, k: i64,
    a: [i64; n]
  }
  
  memoize! {
    #[memo_param(anum, ame)]
    fn solve(memo: _, anum: usize, ame: i64, n : usize, k: i64, a: &Vec<i64>) -> i64 {
      match (anum, ame) {
        (_, i) if i == k => 1,
        (i, _) if i >= n => 0,
        _ => (ame..=cmp::min(k, ame + a[anum])).map(|i| solve(memo, anum + 1, i, n, k, a)).fold(0, |acc, x| (acc + x) % MOD)
      }
    }
  }
  println!("{}", solve(&mut vec![], 0, 0, n, k, &a));
}