Counting Haybales

문제

As usual, Bessie the cow is causing trouble in Farmer John's barn. FJ has $NN$ ($1\le N\le 50001\backslash leq\; N\; \backslash leq\; 5000$) stacks of haybales. For each $i\in [1,N]i\backslash in\; [1,N]$, the $ii$th stack has $h_{i}h\_i$ ($1\le h_i\le {10}^{9}1\backslash le\; h\_i\backslash le\; 10^9$) haybales. Bessie does not want any haybales to fall, so the only operation she can perform is as follows:

• If two adjacent stacks' heights differ by exactly one, she can move the top haybale of the taller stack to the shorter stack.

How many configurations are obtainable after performing the above operation finitely many times, modulo ${10}^9+710^9+7$? Two configurations are considered the same if, for all $ii$, the $ii$th stack has the same number of haybales in both.

입력

The first line contains $TT$ ($1\le T\le 101\backslash le\; T\backslash le\; 10$), the number of independent test cases, all of which must be solved to solve one input correctly.

Each test case consists of $NN$, and then a sequence of $NN$ heights. It is guaranteed that the sum of $NN$ over all test cases does not exceed $50005000$.

출력

Please output $TT$ lines, one for each test case.

힌트

For the first test case, the four possible configurations are:

$$(2,2,2,3),(2,2,3,2),(2,3,2,2),(3,2,2,2)\mathrm{.}(2,2,2,3),\; (2,2,3,2),\; (2,3,2,2),\; (3,2,2,2).$$

For the second test case, the four possible configurations are:

$$(2,3,3,1),(3,2,3,1),(3,3,2,1),(3,3,1,2)\mathrm{.}(2,3,3,1),(3,2,3,1),(3,3,2,1),\; (3,3,1,2).$$

7
4
2 2 2 3
4
3 3 1 2
4
5 3 4 2
6
3 3 1 1 2 2
6
1 3 3 4 1 2
6
4 1 2 3 5 4
10
1 5 6 6 6 4 2 3 2 5

4
4
5
15
9
8
19