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 = 'import bisect\nimport functools\nimport heapq\nimport itertools\nimport sys\nimport math\nimport random\nimport time\nfrom collections import Counter, deque, defaultdict\nfrom functools import reduce\nfrom operator import xor, itemgetter\nfrom pprint import pprint\nfrom types import FunctionType\nfrom typing import List, Any\nfrom sys import stdin\n\nmod = 10 ** 9 + 7\nsys.setrecursionlimit(10 ** 9)\n\ndef inp(): return stdin.readline().rstrip()\ndef lmi(): return list(map(int, stdin.readline().split()))\ndef narray(*shape, init: Any = 0):\n    if shape: return [narray(*shape[1:], init=init) for _ in range(shape[0])]\n    if callable(init): return init()\n    return init\n\n\ndef comb(N, k):\n    if (k > N) or (N < 0) or (k < 0):\n        return 0\n    N, k = list(map(int, (N, k)))\n    top = N\n    val = 1\n    while top > (N - k):\n        val *= top\n        top -= 1\n    n = 1\n    while n < k + 1:\n        val //= n\n        n += 1\n    return val\n\ndef cmb(x, y):\n    if x < y:\n        return 0\n    bunsi = 1\n    bunbo = 1\n    for i in range(1, y+1):\n        bunsi = (bunsi*(x+1-i)) % mod\n        bunbo = (bunbo*i) % mod\n    res = bunsi*pow(bunbo,mod-2,mod) % mod\n    return res\n\ndef main():\n    N, M = lmi()\n    A = lmi()\n    ans = cmb((M + N) % mod, (N + sum(A)) % mod) % mod\n    print(ans)\n\n\n\n\ndef __starting_point():\n    main()\n\n\n__starting_point()'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 1144017