We have a sequence A of N non-negative integers.
Compute the sum of \prod _{i = 1} ^N \dbinom{B_i}{A_i} over all sequences B of N non-negative integers whose sum is at most M, and print it modulo (10^9 + 7).
Here, \dbinom{B_i}{A_i}, the binomial coefficient, denotes the number of ways to choose A_i objects from B_i objects, and is 0 when B_i < A_i.

-----Constraints-----
 - All values in input are integers.
 - 1 \leq N \leq 2000
 - 1 \leq M \leq 10^9
 - 0 \leq A_i \leq 2000

-----Input-----
Input is given from Standard Input in the following format:
N M
A_1 A_2 \ldots A_N

-----Output-----
Print the sum of \prod _{i = 1} ^N \dbinom{B_i}{A_i}, modulo (10^9 + 7).

-----Sample Input-----
3 5
1 2 1

-----Sample Output-----
8

There are four sequences B such that \prod _{i = 1} ^N \dbinom{B_i}{A_i} is at least 1:
 - B = \{1, 2, 1\}, where \prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{2}{2} \times \dbinom{1}{1} = 1;
 - B = \{2, 2, 1\}, where \prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{2}{1} \times \dbinom{2}{2} \times \dbinom{1}{1} = 2;
 - B = \{1, 3, 1\}, where \prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{3}{2} \times \dbinom{1}{1} = 3;
 - B = \{1, 2, 2\}, where \prod _{i = 1} ^N \dbinom{B_i}{A_i} = \dbinom{1}{1} \times \dbinom{2}{2} \times \dbinom{2}{1} = 2.
The sum of these is 1 + 2 + 3 + 2 = 8.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'def mod_combination(n, k, mod = 10 ** 9 + 7):\n    def extended_gcd(a, b):\n        if b == 0:\n            return (a, 1, 0)\n        d, x, y = extended_gcd(b, a % b)\n        return (d, y, x - (a // b) * y)\n    p, q = 1, 1\n    for i in range(n - k + 1, n + 1):\n        p = (p * i) % mod\n    for i in range(2, k + 1):\n        q = (q * i) % mod\n    return p * (extended_gcd(q, mod)[1] % mod) % mod\n\n\nN, M, *A = map(int, open(0).read().split())\n\nS = sum(A)\nprint(mod_combination(M + N, S + N))'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 18657