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.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using static System.Console;
using static System.Math;
using static MyIO;
using static MyUtil;

public class M { public static void Main() => (new Solver()).Solve(); }

public class Solver
{
	public void Solve()
	{
		long N = GetLong();
		long K = GetLong();
		var a = GetLongs(N);

		var dp = new MInt[N + 1, K + 1];
		var sum = new MInt[N + 1, K + 2];

		dp[0, 0] = new MInt(1);
		for(long i = 0; i <= K; i++)
			sum[0, i + 1] = dp[0, i] + sum[0, i];

		for(long i = 0; i < N; i++)
		{
			for(long j = 0; j <= K; j++)
			{
				dp[i + 1, j] = sum[i, j + 1] - sum[i, Max(0, j - a[i])];
				sum[i + 1, j + 1] = sum[i + 1, j] + dp[i + 1, j];
			}
		}
		WriteLine(dp[N, K].Val);
	}
}

public struct MInt
{
	private const long MOD = 1000000007;
	private static MInt[] f = null;
	private static MInt[] g = null;
	public long Val { get; private set; }

	public MInt(long v)
	{
		Val = (v + MOD) % MOD;
	}

	public static void InitTable(int N) => InitTable((long)N);
	public static void InitTable(long N)
	{
		f = new MInt[N + 1];
		f[0] = new MInt(1);
		for(long i = 1; i <= N; i++)
			f[i] = f[i - 1] * i;

		g = new MInt[N + 1];
		g[N] = MPow(f[N], MOD - 2);
		for(long i = N; i > 0; i--)
			g[i - 1] = g[i] * i;
	}

	public static MInt MComb(long n, long k)
	{
		if(n < 0 || n < k)
			return new MInt(0);
			
		return f[n] * g[k] * g[n - k];
	}

	public static MInt operator+(MInt x, long y) => x + new MInt(y);	
	public static MInt operator+(long x, MInt y) => new MInt(x) + y;
	public static MInt operator+(MInt x, MInt y)
	{
		long tmp = x.Val + y.Val;
		if(tmp >= MOD)
			tmp -= MOD;
		return new MInt(tmp);
	}

	public static MInt operator-(MInt x, long y) => x - new MInt(y);
	public static MInt operator-(long x, MInt y) => new MInt(x) - y;
	public static MInt operator-(MInt x, MInt y)
	{
		long tmp = x.Val;
		if(tmp < y.Val)
			tmp += MOD;
		tmp -= y.Val;
		return new MInt(tmp);
	}

	public static MInt operator*(MInt x, MInt y) => new MInt((x.Val * y.Val) % MOD);
	public static MInt operator*(MInt x, long y) => x * new MInt(y);
	public static MInt operator*(long x, MInt y) => new MInt(x) * y;

	public static MInt operator/(MInt x, MInt y) => x * MPow(y, MOD - 2);
	public static MInt operator/(MInt x, long y) => x * MPow(y, MOD - 2);
	public static MInt operator/(long x, MInt y) => x * MPow(y, MOD - 2);

	public static MInt MPow(MInt x, MInt y) => MPow(x.Val, y.Val);
	public static MInt MPow(MInt x, long y) => MPow(x.Val, y);
	public static MInt MPow(long x, MInt y) => MPow(x, y.Val);
	public static MInt MPow(long x, long y)
	{
		long exp = y;
		long xx = x;
		long res = 1;
		while(exp > 0)
		{
			if((exp & 1) == 1)
				res = (res * xx) % MOD;

			xx = (xx * xx) % MOD;
			exp >>= 1;
		}
		return new MInt(res);
	}
}

public static class MyIO
{
	private static string[] args = new string[0];
	private static int cnt = 0;
	private static IEnumerable<string> getArgs(long N) => Range(0, N).Select(_ => getArg());
	private static string getArg()
	{
		if(cnt == args.Length) { args = Console.ReadLine().Split(); cnt = 0; }
		return args[cnt++];
	}

	public static int GetInt() => int.Parse(getArg());
	public static long GetLong() => long.Parse(getArg());
	public static double GetDouble() => double.Parse(getArg());
	public static decimal GetDecimal() => decimal.Parse(getArg());
	public static string GetString() => getArg();
	public static char GetChar() => getArg()[0];
	public static int[] GetInts(long N) => getArgs(N).Select(int.Parse).ToArray();
	public static long[] GetLongs(long N) => getArgs(N).Select(long.Parse).ToArray();
	public static double[] GetDoubles(long N) => getArgs(N).Select(double.Parse).ToArray();
	public static decimal[] GetDecimals(long N) => getArgs(N).Select(decimal.Parse).ToArray();
	public static string[] GetStrings(long N) => getArgs(N).ToArray();
}

public static class MyUtil
{
	public static void chmax<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) < 0) x = y; }
	public static void chmin<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) > 0)	x = y; }
	public static void swap<T>(ref T x, ref T y) { T tmp = x; x = y; y = tmp; }
	public static long rightShift(long x, long y) => (x >> (int)y);
	public static long leftShift(long x, long y) => (x << (int)y);
	public static bool contains(long s, long x) => (rightShift(s, x) & 1L) == 1L;
	public static long gcd(long x, long y) => (y == 0) ? x : gcd(y, x % y);
	public static long lcm(long x, long y) => (x / gcd(x,y)) * y;	
	public static IEnumerable<int> Range(long start, long count) => Enumerable.Range((int)start, (int)count);
	public static void Sort(this List<string> source) => source.Sort(StringComparer.Ordinal);
}