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 = 'class Factorials:\n    def __init__(self, n=10**6, mod=10**9+7):\n        self.mod = mod\n\n        # self.fac[i] ≡ i! (factorial:階乗)\n        self.fac = [1] * (n+1)\n        for i in range(2, n+1):\n            self.fac[i] = self.fac[i-1] * i % mod\n\n        """\n        # self.rec[i] ≡ 1 / i! (reciprocal:逆数)\n        self.rec = [1] * (n+1)\n        self.rec[n] = pow(self.fac[n], mod-2, mod)\n        for i in range(n-1, 1, -1):\n            self.rec[i] = self.rec[i+1] * (i+1) % mod\n\n    # self.comb(n, r) ≡ nCr\n    def comb(self, n, r):\n        return self.fac[n] * self.rec[r] * self.rec[n-r] % self.mod\n    \n    # self.perm(n, r) ≡ nPr\n    def perm(self, n, r):\n        return self.fac[n] * self.rec[n-r] % self.mod\n\n    # self.inv(n) ≡ 1 / n ≡ pow(n, mod-2, mod)\n    def inv(self, n):\n        return self.fac[n-1] * self.rec[n] % self.mod\n    """\n\nmod = 10**9 + 7\n\ndef fact(n, r):\n    res = 1\n    for i in range(r):\n        res *= n - i\n        res %= mod\n    return res\n\nn, m = map(int, input().split())\na = list(map(int, input().split()))\n\nsum_a = sum(a)\nif m < sum_a:\n    print(0)\n    return\n\nfct = Factorials(sum_a + n)\n\nprint(fact(n+m, sum_a+n) * pow(fct.fac[sum_a+n], mod-2, mod) % mod)'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: