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++ (GCC 9.2.1) (judge verdict: Accepted)

#define _CRT_SECURE_NO_WARNINGS
#include<bits/stdc++.h>

using namespace std;
typedef pair<int, int> pii;
typedef long long ll;
typedef vector<int> vi;
typedef vector<ll> vll;
typedef vector<pair<int, int>> vpi;
typedef vector<pair<ll, ll>> vpll;

#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define trav(a,x) for (auto& a: x)

#define pb push_back
#define mp make_pair
#define rsz resize
#define sz(x) int(x.size())
#define all(x) x.begin(),x.end()
#define f first
#define s second
#define cont continue
#define inf 1000000007
#define endl '\n'
#define ednl '\n'
#define test int t;cin>>t;while(t--)
#define vin(v,n) vi (v);(v).rsz((n));F0R(i,(n))cin>>(v)[i];
#define vlln(v,n) vector<ll> (v);(v).rsz((n));F0R(i,(n))cin>>(v)[i];
#define vvin(v,n,m) vector<vi>(v);(v).rsz((n));F0R(i,(n))(v)[i].rsz((m));F0R(i,(n))F0R(j,(m))cin>>(v)[i][j];
#define pi 3.1415926535
#define vvlln(v,n,m) vector<vector<ll>>(v);(v).rsz((n));F0R(i,(n))(v)[i].rsz((m));F0R(i,(n))F0R(j,(m))cin>>(v)[i][j];
#define pq priority_queue
#define linf 1000000000000000000LL


ll maximum(ll x, ll y)
{
	if (x > y)
		return x;
	return y;
}
ll minimum(ll x, ll y)
{
	if (x < y)
		return x;
	return y;
}
void print_vector(vector<int>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_vector(vector<ll>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_vector(vector<double>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_vector(vector<string>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_vector(vector<bool>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_vector(vector<pii>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).f << " ";
	cout << endl;
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).s << " ";
	cout << endl;
}
void print_vector(vector<pair<ll, ll>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).f << " ";
	cout << endl;
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).s << " ";
	cout << endl;
}
void print_vector(vector<pair<pii, int>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).f.f << " " << (*i).f.s << " " << (*i).s << endl;
	cout << endl;
}
void print_vector(vector<pair<int, pii>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << (*i).f << " " << (*i).s.f << " " << (*i).s.s << endl;
	cout << endl;
}
void print_vector(vector<vector<string>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<int>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<ll>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<pair<ll, ll>>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
	}
	cout << endl;
}
void print_vector(vector<vector<vi>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<vector<ll>>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<bool>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		print_vector(*i);
		cout << endl;
	}
	cout << endl;
}
void print_set(set<int>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_set(set<pii>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << i->f << " " << i->s << endl;
	cout << endl;
}
void print_vector(vector<set<int>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_set(*i);
		cout << endl;
	}
	cout << endl;
}
void print_vector(vector<vector<pii>>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
	{
		cout << i - amount.begin() << endl;
		print_vector(*i);
	}
}
void print_set(set<string>amount)
{
	for (auto i = amount.begin(); i != amount.end(); i++)
		cout << *i << " ";
	cout << endl;
}
void print_map(map<int, vector<int>> m)
{
	for (map<int, vector<int>>::iterator i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " ";
		print_vector(i->second);
	}
}
void print_map(map<string, vector<string>> m)
{
	for (map<string, vector<string>>::iterator i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " ";
		print_vector(i->second);
	}
}
void print_map(map<string, int> m)
{
	for (auto i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " " << i->s << endl;
	}
}
void print_map(map<string, vector<bool>> m)
{
	for (map<string, vector<bool>>::iterator i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " ";
		print_vector(i->second);
	}
}
void print_map(map<string, set<string>> m)
{
	for (auto i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " ";
		print_set(i->second);
	}
}
void print_map(map<int, int> m)
{
	for (map<int, int>::iterator i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << " " << i->second << endl;
	}
}
void print_map(map<char, vector<string>> m)
{
	for (auto i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << endl;
		cout << '\t';
		print_vector(i->second);
	}
}
void print_map(map<char, int> m)
{
	for (auto i = m.begin(); i != m.end(); i++)
	{
		cout << i->first << endl;
		cout << '\t' << i->second << endl;
	}
}
void print_map(map<pii, int> m)
{
	for (auto i = m.begin(); i != m.end(); i++)
	{
		cout << i->first.f << " " << i->first.s << " " << i->second << endl;
	}
}
bool comp(pii x, pii y)
{
	if (x.s == y.s)return x.f < y.f;
	return x.s < y.s;
}
struct Graph
{
	int V, E;
	vector<pair<int, pii>> edges; // node a, b, weight

	//constructor
	Graph(int a, int b)
	{
		V = a;
		E = b;
	}

	void addEdge(int a, int b, int w)
	{
		edges.push_back({ w,{a,b} });
	}

	// add function generate adj based on vertices and edges.

	vector<int> links, sizes;

	int find(int a) // return the root for node a
	{
		if (a == links[a])
			return a;
		return links[a] = find(links[a]);
	}

	bool same(int a, int b) // check if node a and b has the same root
	{
		return find(a) == find(b);
	}

	void unite(int a, int b)
	{
		a = find(a);
		b = find(b);
		if (sizes[a] < sizes[b]) swap(a, b);
		sizes[a] += sizes[b]; //a always bigger than b, move subtree b under a
		links[b] = a;
	}

	void sort_edges()
	{
		sort(edges.begin(), edges.end());
	}

	ll kruskalMST()
	{
		sort_edges();
		//initialize for MST
		links.resize(V);
		sizes.resize(V);

		for (int i = 0; i < V; i++)
		{
			links[i] = i;
			sizes[i] = 1;
		}
		ll sum = 0;
		int count = 0;
		F0R(i, E)
		{
			int start = edges[i].s.f;
			int end = edges[i].s.s;
			if (!same(start, end))
			{
				unite(start, end);
				sum += edges[i].f;
				count++;
			}
		}
		if (count != V - 1)return -inf;
		return sum;
	}
};

/*
void dijkstra(int i)
{//Preconditions for this to work: 1. no negative edges. 2. dist and parents are both initialized with size N and full of INT_MAX and 0 respectively while dist[i]=0 and parent[i]=-1
	priority_queue<pii>todo;
	vi finalized;
	finalized.rsz(N + 1, 0);//make sure that the size of adjacency is N+1
	todo.push(mp(0, i));
	while (!todo.empty())
	{//.s is index while .f is weight
		pii temp = todo.top();
		int index = temp.second;
		todo.pop();
		if (finalized[index])continue;
		finalized[index] = 1;
		trav(x, adjacency[index])
		{
			if (finalized[x.first])
				continue;
			if (dist[x.f] > x.s + dist[index])
			{
				dist[x.first] = x.second + dist[index];
				parents[x.f] = index;
				todo.push(mp(-dist[x.first], x.f));
			}
		}
	}
}
void build_primes()
{
	primes.reserve(50000);
	vi visited;
	visited.rsz(200000, 0);
	FOR(i, 2, 200000)
	{
		if (visited[i] == 0)
		{
			primes.pb(i);
			a = i;
			while (a < 200000)
			{
				visited[a] = 1;
				a += i;
			}
		}
	}
}
//Prim's Algorithm
// need vector<vpi>adjacency
ll prim()
{
	ll a = 0;
	vi visited;
	visited.rsz(n, 0);
	visited[0] = 1;
	pq<pii>todo;
	trav(x, adjacency[0])
		todo.push({ -x.s,x.f });
	F0R(i, n - 1)
	{
		pii temp = todo.top();
		todo.pop();
		int index = temp.s;
		if (visited[index])cont;
		a -= temp.f;
		visited[index] = 1;
		trav(x, adjacency[index])
			if (visited[x.f] == 0)
			{
				todo.push({ -x.s,x.f });
			}
	}
	return a;
}
bool is_ancestor(int anc,int node)
{
	return tin[anc]<=tin[node]&&tout[anc]>=tout[node];
}
int LCA(int node1, int node2)//using SQRT decomposition
{
	if (depth[node1] > depth[node2])swap(node1, node2);
	if (is_ancestor(node1, node2))return node1;
	int next = jump_parents[node1];
	while (next != -1 && !is_ancestor(next, node2))
	{
		node1 = next;
		next = jump_parents[node1];
	}
	next = parents[node1];
	while (next != -1 && !is_ancestor(next, node2))
	{
		node1 = next;
		next = parents[node1];
	}
	node1 = parents[node1];
	return node1;
}

//building the sparse table for LCA 

int h = *max_element(all(depth));
logh = log2(h - 1) + 1;
jump.rsz(n);
F0R(i, n)jump[i].rsz(logh + 1);
F0R(i, n)jump[i][0] = parents[i];
FOR(i, 1, logh + 1)F0R(j, n)
{
	a=jump[j][i - 1];
	if (a != -1)jump[j][i] = jump[a][i - 1];
	else jump[j][i] = -1;
}

int LCA(int x, int y)//LCA with Sparse Table
{
	if (depth[x] > depth[y])swap(x, y);
	if (is_ancestor(x, y))return x;
	int j = logh - 1;
	int next;
	while (!is_ancestor(parents[x], y))
	{
		next = jump[x][j];
		if (next == -1 || is_ancestor(next, y))
		{
			j--;
		}
		else
		{
			x = next;
		}
	}
	return parents[x];
}
*/
int modInverse(int a, int m)
{
	int m0 = m;
	int y = 0, x = 1;

	if (m == 1)
		return 0;

	while (a > 1)
	{
		// q is quotient 
		int q = a / m;
		int t = m;

		// m is remainder now, process same as 
		// Euclid's algo 
		m = a % m, a = t;
		t = y;

		// Update y and x 
		y = x - q * y;
		x = t;
	}

	// Make x positive 
	if (x < 0)
		x += m0;

	return x;
}
ll power(ll x, ll y)
{
	ll k = 1 << 30;
	ll z = 1;
	while (k != 0)
	{
		z *= z;
		z %= inf;
		if (y >= k)
		{
			z *= x;
			z %= inf;
			y -= k;
		}
		k >>= 1;
	}
	return z;
}
// remember that you need to take abs
long double area(pii x, pii y, pii z)
{
	return ((ll)x.s * y.f + (ll)y.s * z.f + (ll)z.s * x.f - (ll)x.f * y.s - (ll)y.f * z.s - (ll)z.f * x.s) / 2.0;
}
bool clockwise(pii x, pii y, pii z)
{
	return area(x, y, z) > 0;
}
int gcd(int a, int b)
{
	if (a > b)swap(a, b);
	if (a == 0)return b;
	return gcd(a, b % a);
}
//end of preprogrammed methods
ll a, b, c;
void setIO(string s) {
	ios_base::sync_with_stdio(0); cin.tie(0);
	//freopen((s + ".in").c_str(), "r", stdin); freopen((s + ".out").c_str(), "w", stdout);
}
int n, k;
vi A;
void init()
{
	setIO("");
	cin >> n >> k;
	A.rsz(n);
	F0R(i, n)cin >> A[i];
}
int main()
{
	init();

	vector<vll>dp;
	dp.rsz(n + 1);
	F0R(i, n + 1)dp[i].rsz(k + 1);
	dp[n][0] = 1;
	vll sums;
	sums.rsz(k + 2, 1);
	sums[0] = 0;
	R0F(i, n)
	{
		F0R(j, k + 1)
		{
			dp[i][j] = sums[j + 1] - sums[maximum(0, j - A[i])];		
			dp[i][j] %= inf;
		}	
		FOR(j, 1, k + 2)sums[j] = sums[j - 1] + dp[i][j - 1];
	//print_vector(sums);
	}
	//print_vector(dp);
	cout << dp[0][k] << endl;
}