Counting the Number of Trees

A binary search tree (BST) is a tree in which every node has at most $2$ children. If the number written on a node is $X$, then the left subtree of that node may store only numbers smaller than $X$, and the right subtree only numbers larger than $X$.

The following pseudocode describes a function that inserts a number into a BST.

insert(number X, node N)
    if X is smaller than the number at node N
        if N has no left child
            create a new node containing X and make it the left child of N
        else
            insert(X, left child of N)
    else (X is larger than the number at node N)
        if N has no right child
            create a new node containing X and make it the right child of N
        else
            insert(X, right child of N)

The first number inserted becomes the root, and for every number X inserted afterward, insert(X, root) is called.

The height of a tree is the number of nodes on the longest path from the root node to a leaf node. A leaf node is a node with no children.

You are going to insert the numbers from $1$ to $N$ into a BST. When the insertion order can be chosen freely, find the number of BSTs with height at most $K$ that can be produced. The root node of a BST is assumed to have height 1.

Inserting $2$ $1$ $3$ and inserting $2$ $3$ $1$ produce the same BST, while $3$ $2$ $1$ $4$ and $2$ $1$ $3$ $4$ produce different BSTs.

The number of cases can be very large, so print the answer modulo $10^9+7$.

The first line contains two integers $N$ and $K$ separated by a space. ($1 \leq N \leq 3500,\, 1 \leq K \leq 12$)

Print the number of possible BSTs modulo $10^9+7$.