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# (.NET Core 3.1.201) (judge verdict: Accepted)


using System;
using System.Linq;

namespace AtCoder.DP
{
    class EDPC_M_FPS
    {
        public static void Main()
        {
            var nk = Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();
            var a = Console.ReadLine().Split(' ').Select(s => int.Parse(s)).ToArray();
            var n = nk[0];
            var k = nk[1];

            const long mod = 1000000007;
            /* Fi = (1+x+x^2+x^3+x^4+...+x^(ai)) = (1-x^(ai+1))/(1-x)
             * P/(1-x)=Q ⇒ P=Q(1-x) ⇒ p[n]=q[n]-q[n-1] ⇒ q[n]=Σp[i]
             */
             
            var q = new long[k + 1];
            q[0] = 1;

            for (int i = 0; i < n; i++)
            {
                for (int j = 1; j <= k; j++)
                    q[j] = (q[j] + q[j - 1]) % mod;
                for (int j = k; j >= a[i] + 1; j--)
                    q[j] = (q[j] - q[j - a[i] - 1]) % mod;
            }
            Console.WriteLine(q[k] < 0 ? q[k] + mod : q[k]);
        }
    }
}