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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using BitArray = System.Collections.BitArray;
using BigInteger = System.Numerics.BigInteger;

namespace AtCoderProject
{
    public class Program
    {
        public object Calc()
        {
            var line = consoleReader.Split.Int;
            N = line[0];
            K = line[1];
            a = consoleReader.Split.Int;
            dp = new Mod[N, K + 1];
            for (int i = 0; i < N; i++)
                dp[i, 0] = 1;
            for (int i = 0; i <= a[0]; i++)
                dp[0, i] = 1;


            var sumCount = a[0];
            for (int i = 1; i < N; i++)
            {
                sumCount += a[i];
                var max = Math.Min(sumCount, K);
                var sum = dp[i - 1, 0];
                for (int j = 1; j <= max; j++)
                {
                    if (j > a[i])
                    {
                        sum -= dp[i - 1, j - a[i] - 1];
                    }
                    sum += dp[i - 1, j];

                    dp[i, j] = sum;
                }
            }
            return dp[N - 1, K];
        }
        Mod Calc(int index, int count)
        {
            if (count < 0) return 0;
            if (count == 0) return 1;
            if (index == 0)
            {
                return dp[index, count] = count <= a[0] ? 1 : 0;
            }
            if (dp[index, count].val > 0) return dp[index, count];

            Mod sum = 0;
            for (int i = 0; i <= a[index]; i++)
            {
                sum += Calc(index - 1, count - i);
            }

            return dp[index, count] = sum;
        }
        int N;
        int K;
        int[] a;
        Mod[,] dp;
        struct Goods
        {
            public static Goods Create(int[] l) => new Goods { weight = l[0], value = l[1] };
            public int weight;
            public int value;
        }

        #region いつもの
        private readonly ConsoleReader consoleReader;
        public Program(ConsoleReader consoleReader) { this.consoleReader = consoleReader; }
        static void Main() => Console.WriteLine(new Program(new ConsoleReader()).Calc()); static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source);
        #endregion
    }
    struct Mod : IEquatable<Mod>
    {
        public const long mod = 1000000007;
        public readonly long val;
        public Mod(long val) { this.val = val; }
        public override bool Equals(object obj) => (obj is Mod) ? this == ((Mod)obj) : false;
        public bool Equals(Mod obj) => this == obj;
        public override int GetHashCode() => val.GetHashCode();
        public override string ToString() => val.ToString();
        public static implicit operator Mod(long x) => new Mod(x);
        public static Mod operator +(Mod x, Mod y) => (x.val + y.val) % mod;
        public static Mod operator -(Mod x, Mod y) => x.val >= y.val ? (x.val - y.val) % mod : (x.val - y.val) % mod + mod;
        public static Mod operator *(Mod x, Mod y) => (x.val * y.val) % mod;
        public static Mod operator /(Mod x, Mod y) => x * y.Inverse();
        public static bool operator ==(Mod x, Mod y) => x.val == y.val;
        public static bool operator !=(Mod x, Mod y) => x.val != y.val;
        public Mod Inverse()
        {
            long a = val, b = mod, u = 1, v = 0;
            while (b > 0)
            {
                long t = a / b;
                var b2 = a - t * b;
                a = b; b = b2;
                var v2 = u - t * v;
                u = v; v = v2;
            }
            u %= mod;
            if (u < 0) u += mod;
            return u;
        }
        public static Mod Pow(Mod x, int y)
        {
            Mod res = 1;
            for (; y > 0; y >>= 1)
            {
                if ((y & 1) == 1) res *= x;
                x *= x;
            }
            return res;
        }

        public static Factor CreateFactor(int max) => new Factor(max);
        public class Factor
        {
            private readonly Mod[] fac, finv;
            public Factor(int max)
            {
                var inv = new Mod[max];
                fac = new Mod[max]; finv = new Mod[max];
                fac[0] = fac[1] = 1;
                finv[0] = finv[1] = 1;
                inv[1] = 1;
                for (int i = 2; i < max; i++)
                {
                    fac[i] = fac[i - 1] * i;
                    inv[i] = mod - inv[mod % i].val * (mod / i) % mod;
                    finv[i] = finv[i - 1] * inv[i];
                }
            }

            // 二項係数計算
            public Mod Combine(int n, int k)
            {
                if (n < k) return 0;
                if (n < 0 || k < 0) return 0;
                return fac[n] * finv[k] * finv[n - k];
            }

            public Mod Factorial(int n) => fac[n];
            public Mod FactorialInvers(int n) => finv[n];
        }
    }

#pragma warning disable
    public class ConsoleReader { public class SplitLine { string[] splited; public SplitLine(string line) { splited = line.Split(); } public string[] String => splited.ToArray(); public int[] Int => splited.Select(int.Parse).ToArray(); public long[] Long => splited.Select(long.Parse).ToArray(); public ulong[] ULong => splited.Select(ulong.Parse).ToArray(); public double[] Double => splited.Select(double.Parse).ToArray(); } public class RepeatLine { ConsoleReader reader; int num; public RepeatLine(ConsoleReader reader, int num) { this.reader = reader; this.num = num; } public IEnumerable<string> String { get { for (int i = 0; i < num; i++) yield return reader.String; } } public IEnumerable<int> Int { get { for (int i = 0; i < num; i++) yield return reader.Int; } } public IEnumerable<long> Long { get { for (int i = 0; i < num; i++) yield return reader.Long; } } public IEnumerable<string[]> Split { get { for (int i = 0; i < num; i++) yield return reader.Split.String; } } public IEnumerable<int[]> SplitInt { get { for (int i = 0; i < num; i++) yield return reader.Split.Int; } } } public virtual string String => Console.ReadLine(); public int Int => int.Parse(String); public long Long => long.Parse(String); public ulong ULong => ulong.Parse(String); public double Double => double.Parse(String); public SplitLine Split => new SplitLine(String); public RepeatLine Repeat(int count) => new RepeatLine(this, count); }
}