Expressions

Let $X$ be the smallest set defined by the following rules:

• the empty string belongs to $X$;
• if $A$ and $B$ belong to $X$, then both $(A)$ and $AB$ belong to $X$.

The elements of $X$ are called correctly built parenthesis expressions. For example, these strings are correctly built parenthesis expressions:

()(())()
(()(()))

while these are not:

(()))(()
())(()

Let $E$ be a correctly built parenthesis expression. The length of $E$ is the number of single parentheses in $E$. The depth $D(E)$ of $E$ is defined by:

$$ D(E) = \begin{cases} 0 & \text{if } E \text{ is empty} \ D(A) + 1 & \text{if } E = (A) \text{ with } A \in X \ \max(D(A), D(B)) & \text{if } E = AB \text{ with } A, B \in X \end{cases} $$

Given two positive integers $n$ and $d$, determine how many correctly built parenthesis expressions have length exactly $n$ and depth exactly $d$.

A single line containing two integers $n$ and $d$ separated by one space, with $2 \le n \le 38$ and $1 \le d \le 19$.

Print a single integer: the number of correctly built parenthesis expressions of length $n$ and depth $d$.

There are exactly three correctly built parenthesis expressions of length $6$ and depth $2$:

(())()
()(())
(()())