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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

def max(a, b)
  a > b ? a : b
end

MOD = 10 ** 9 + 7

N, K = gets.split.map(&:to_i)
As = gets.split.map(&:to_i).sort

#N = 100
#K = 10 ** 5
#As = N.times.map{rand(0..(K-1))}

dp = Array.new(K+1, 0)
rui = Array.new(K+2, 0)

dp[0] = 1
N.times do |i|
  a = As[i]
  q = 0
  (K+1).times do |j|
    q += dp[j]
    rui[j+1] = q
    
    #m = max(0, j+1-(a+1))
    m = max(0, j-a)
    dp[j] = (q - rui[m]) % MOD    
  end
end

p dp[K]