201384: [AtCoder]ARC138 E - Decreasing Subsequence

Memory Limit:1024 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:0 Solved:0

Description

Score : $1000$ points

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.

Input

Input is given from Standard Input in the following format:

$N$ $K$

Output

Print the answer.


Sample Input 1

3 2

Sample Output 1

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 2

6 2

Sample Output 2

660

Sample Input 3

10 3

Sample Output 3

242595

Sample Input 4

100 10

Sample Output 4

495811864

Input

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