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 Kotlin (1.0.0) (judge verdict: Accepted)

import java.io.*
import java.util.*

const val MOD = 1000000007L
// K個の飴をN人で分け合う方法の数
// 各飴の数はai以下
// N~100, K~1e5

fun <T, R> Iterable<T>.scanLeft(initial: R, operation: (acc: R, T) -> R): Sequence<R> {
    var last = initial
    return this.asSequence().map { last = operation(last, it); last }
}

fun solve(N: Int, K: Int, a: IntArray) {
    // dp[i][j] i番目まででj個を分ける方法の数
    // dp[i][j] = sum(dp[i-1][j-k]){k=0~a[i]}
    val dp = Array(N + 1) { LongArray(K + 1) }
    val cumSum = Array(N + 1) { LongArray(K + 1) }
    dp[0][0] = 1L
    for (x in 0..K) cumSum[0][x] = 1L
    for (i in 1..N) {
        for (j in 0..K) {
            dp[i][j] = (cumSum[i - 1][j] - cumSum[i - 1].getOrElse(j - a[i-1] - 1) { 0L }) % MOD
            cumSum[i][j] = cumSum[i].getOrElse(j-1){ 0L } + dp[i][j]
        }
    }
    println(dp[N][K])
    return
}

fun main(args: Array<String>) {
    fun StringArray(size: Int, init: (Int) -> String = { "\u0000" }): Array<String> {
        return Array<String>(size, init)
    }
    class Scanner(stream: InputStream) {
        private val reader = BufferedInputStream(stream).bufferedReader()
        private var st: StringTokenizer? = null
        fun next(): String {
            while (!(st?.hasMoreTokens() ?: false)) st = StringTokenizer(reader.readLine())
            return st!!.nextToken()
        }
    }
    val sc = Scanner(System.`in`)
    val N = sc.next().toInt()
    val K = sc.next().toInt()
    val a = IntArray(N)
    for (i in 0 until N) {
        a[i] = sc.next().toInt()
    }
    solve(N, K, a)
}