Problem Statement

You are given integers $N$ and $K$. Let us call an integer sequence $A=(A_1,A_2,\cdots,A_N)$ good when it satisfies all of the conditions below.

• $0 \leq A_i \leq i$ ($1 \leq i \leq N$)
• For every integer $v=1,2,\cdots,N$, there is at most one index $i$ such that $A_i=v$.

Find the sum, modulo $(10^9+7)$, of the answers to the following problem for all good sequences $A$.

• Find the number of length-$K$ (not necessarily contiguous) subsequences of $A$ consisting of positive integers that are decreasing. In other words, find the number of sequences of indices $1 \leq i_1 < i_2 < \cdots < i_K \leq N$ such that $A_{i_1}>A_{i_2}>\cdots>A_{i_K} \geq 1$.

Constraints

• $3 \leq N \leq 5000$
• $2 \leq K$
• $2K-1 \leq N$
• All values in input are integers.

Sample Input 1Copy

3 2

Sample Output 1Copy

1

For example, $A=(0,2,1)$ is a good sequence, with one subsequence satisfying the condition. There are other good sequences such as $A=(0,1,0),(1,2,3),(0,0,0)$, but none of them has subsequences satisfying the condition. In the end, no good sequence other than $A=(0,2,1)$ has subsequences satisfying the condition, so the answer is $1$.

Sample Input 2Copy

6 2

Sample Output 2Copy

660

Sample Input 3Copy

10 3

Sample Output 3Copy

242595

Sample Input 4Copy

100 10

Sample Output 4Copy

495811864