List Generation
Description
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 $10^9 + 7$.
题面翻译
给定 和 ，请问能构造满足以下条件的数组对 中， 长度之和是多少：
- 数组长度相等且长度大于 ，设数组长度都为 。
- ，，。
- 每个数组都单调不下降且对于所有 都满足 。
答案对 取模。
Input
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 $5 \cdot 10^6$.
Output
For each test case, output a single integer — the sum of $|a|$ over all good pairs of arrays $(a,b)$ modulo $10^9 + 7$.
4
1 1
1 2
2 2
100 100
8
26
101
886336572
Note
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 ${2 + 3 + 3} = 8$.
相关
在下列比赛中: