Permutation p is an ordered set of integers p_1,  p_2,  ...,  p_{n}, consisting of n distinct positive integers, each of them doesn't exceed n. We'll denote the i-th element of permutation p as p_{i}. We'll call number n the size or the length of permutation p_1,  p_2,  ...,  p_{n}.

We'll call position i (1 ≤ i ≤ n) in permutation p_1, p_2, ..., p_{n} good, if |p[i] - i| = 1. Count the number of permutations of size n with exactly k good positions. Print the answer modulo 1000000007 (10^9 + 7).


-----Input-----

The single line contains two space-separated integers n and k (1 ≤ n ≤ 1000, 0 ≤ k ≤ n).


-----Output-----

Print the number of permutations of length n with exactly k good positions modulo 1000000007 (10^9 + 7).


-----Examples-----
Input
1 0

Output
1

Input
2 1

Output
0

Input
3 2

Output
4

Input
4 1

Output
6

Input
7 4

Output
328



-----Note-----

The only permutation of size 1 has 0 good positions.

Permutation (1, 2) has 0 good positions, and permutation (2, 1) has 2 positions.

Permutations of size 3:



 (1, 2, 3) — 0 positions

 $(1,3,2)$ — 2 positions

 $(2,1,3)$ — 2 positions

 $(2,3,1)$ — 2 positions

 $(3,1,2)$ — 2 positions

 (3, 2, 1) — 0 positions

Source: codeforces | Difficulty: VERY_HARD
Tags: Dynamic programming

Combinatorics

Mathematics


Solutions:
mod = 10 ** 9 + 7
(n, k) = list(map(int, input().split()))
A = [0] * (n + 1)
B = [0] * (n + 1)
C = [0] * (n + 1)
F = [0] * (n + 1)
G = [0] * (n + 1)
F[0] = G[0] = 1
for i in range(1, n + 1):
	G[i] = F[i] = F[i - 1] * i % mod
	G[i] = pow(F[i], mod - 2, mod)
for i in range(0, n):
	if i * 2 > n:
		break
	B[i] = F[n - i] * G[i] * G[n - i * 2] % mod
for i in range(0, n // 2 + 1):
	for j in range(0, n // 2 + 1):
		A[i + j] = (A[i + j] + B[i] * B[j]) % mod
for i in range(0, n + 1):
	A[i] = A[i] * F[n - i] % mod
for i in range(0, n + 1):
	for j in range(0, i + 1):
		C[j] = (C[j] + A[i] * F[i] * G[j] * G[i - j] * (1 - (i - j) % 2 * 2)) % mod
print(C[k] % mod)