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 C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <string.h>

#define MOD 1000000007

using namespace std;

int dp[101][100001];

int func(int start, int curr_k, int *arr, int n){
    if(dp[start][curr_k]!=-1)return dp[start][curr_k];
    if(curr_k==0){
        dp[start][curr_k]=1;
        return dp[start][curr_k];
    }
    if(start==n-1){
        if(curr_k>arr[start]){
            dp[start][curr_k]=0;
            return 0;
        }
        else {
            dp[start][curr_k]=1;
            return 1;
        }
    }
    //else if the end element is not <= zero
    int ways = 0;
    for(int i=0; i<=arr[start]; i++){
        if(curr_k-i>=0)ways = (ways + func(start+1, curr_k-i, arr, n)%MOD)%MOD;
        else break;
    }
    dp[start][curr_k] = ways;
    return dp[start][curr_k];
}

int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    memset(dp, -1, sizeof(dp));
    
    int n,k;
    cin>>n>>k;
    int arr[n];
    //take the input for the array
    for(int i=0; i<n; i++){
        //now take the input
        cin>>arr[i];
    }
    //now fill the dp array using recursion
    cout<<func(0, k, arr, n)<<endl;
    
}