A11356. Balanced Subsequences

A sequence of brackets is called balanced if one can turn it into a valid math expression by adding characters '+' and '1'. For example, sequences '(())()', '()', and '(()(()))' are balanced, while ')(', '(()', and '(()))(' are not.

A subsequence is a sequence that can be derived from the given sequence by deleting zero or more elements without changing the order of the remaining elements.

You are given three integers $n$ , $m$ and $k$ . Find the number of sequences consisting of $n$ '(' and $m$ ')', such that the longest balanced subsequence is of length $2 \cdot k$ . Since the answer can be large calculate it modulo $1\,000\,000\,007$ ( $10^9 + 7$ ).

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 3 \cdot 10^3$ ). Description of the test cases follows.

The first line of each test case contains three integers $n$ , $m$ and $k$ ( $1 \le n, m, k \le 2 \cdot 10^3$ )

For each test case, print one integer — the answer to the problem.

For the first test case "()()", "(())" are the $2$ sequences

For the second test case no sequence is possible.

For the third test case ")((()", ")(()(", ")()((", "())((" are the $4$ sequences.