Problem Description

You are practicing drawing magic circles on the ground to become the ultimate wizard. The method for drawing a magic circle is as follows:

• Draw \( N \) points equidistantly on a circle, with one of the points being exactly north.
• Connect the points with line segments to form several polygons with at least \( K \) vertices, ensuring no self-intersections.
• Each point must be included in exactly one polygon.
• The drawn polygons must be connected. By connected, we mean that for every pair of polygons \( (a, b) \), there exists a state where polygons \( a \) and \( b \) are connected, which can be defined as:
• Polygons \( a \) and \( b \) intersect at one or more edges.
• There exists a polygon \( c \) such that polygons \( a \) and \( c \) are connected, and polygons \( b \) and \( c \) are connected.

For example, when \( N = 10 \) and \( K = 3 \), you can draw a magic circle as shown below:

The following magic circle is incorrect because the polygons are not connected:

You decided to calculate how many such magic circles can be drawn.

Input Format

The input is given from the standard input in the following format:

\( N \) \( K \)

• The first line contains two integers \( N \) (\( 3 \leq N \leq 400 \)) and \( K \) (\( 3 \leq K \leq N \)) separated by a space.

Output Format

Output the number of magic circles that can be drawn using the described method, modulo \( 1,000,000,007 \) (\( 10^9 + 7 \)). End the output with a newline.

Sample Input and Output

Sample Input #1

6 3

Sample Output #1

8

Sample Input #2

6 4

Sample Output #2

1

Sample Input #3

50 5

Sample Output #3

665400552

Notes/Hints

Scoring

This problem has partial scoring:

• If you solve all test cases where \( N \leq 50 \) and \( K \) satisfies \( N/3 \), you get 30 points.
• If you solve all test cases where \( N \leq 50 \), you get an additional 50 points.
• If you solve all test cases, you get an additional 50 points.

Sample Explanation 1

You can draw 8 different magic circles as shown below:

Sample Explanation 2

Only the hexagon pattern can be drawn.

见题面 Input 部分

见题面 Output 部分