CF1747E.List Generation

For given integers $n$ and $m$ , let's call a pair of arrays $a$ and $b$ of integers good, if they satisfy the following conditions:

• $a$ and $b$ have the same length, let their length be $k$ .
• $k \ge 2$ and $a_1 = 0, a_k = n, b_1 = 0, b_k = m$ .
• For each $1 < i \le k$ the following holds: $a_i \geq a_{i - 1}$ , $b_i \geq b_{i - 1}$ , and $a_i + b_i \neq a_{i - 1} + b_{i - 1}$ .

Find the sum of $|a|$ over all good pairs of arrays $(a,b)$ . Since the answer can be very large, output it modulo $10^9 + 7$ .

The input consists of multiple test cases. The first line contains a single integer $t (1 \leq t \leq 10^4)$ — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $n$ and $m$ $(1 \leq n, m \leq 5 \cdot 10^6)$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^6$ and the sum of $m$ over all test cases does not exceed $5 \cdot 10^6$ .

For each test case, output a single integer — the sum of $|a|$ over all good pairs of arrays $(a,b)$ modulo $10^9 + 7$ .

In the first testcase, the good pairs of arrays are

• $([0, 1], [0, 1])$ , length = $2$ .
• $([0, 1, 1], [0, 0, 1])$ , length = $3$ .
• $([0, 0, 1], [0, 1, 1])$ , length = $3$ .

Hence the sum of the lengths would be ${2 + 3 + 3} = 8$ .