CQXYM Count Permutations

1000ms

262144K

Description:

CQXYM is counting permutations length of $$$2n$$$.

A permutation is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array) and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

A permutation $$$p$$$(length of $$$2n$$$) will be counted only if the number of $$$i$$$ satisfying $$$p_i<p_{i+1}$$$ is no less than $$$n$$$. For example:

Permutation $$$[1, 2, 3, 4]$$$ will count, because the number of such $$$i$$$ that $$$p_i<p_{i+1}$$$ equals $$$3$$$ ($$$i = 1$$$, $$$i = 2$$$, $$$i = 3$$$).
Permutation $$$[3, 2, 1, 4]$$$ won't count, because the number of such $$$i$$$ that $$$p_i<p_{i+1}$$$ equals $$$1$$$ ($$$i = 3$$$).

CQXYM wants you to help him to count the number of such permutations modulo $$$1000000007$$$ ($$$10^9+7$$$).

In addition, modulo operation is to get the remainder. For example:

$$$7 \mod 3=1$$$, because $$$7 = 3 \cdot 2 + 1$$$,
$$$15 \mod 4=3$$$, because $$$15 = 4 \cdot 3 + 3$$$.

Input:

The input consists of multiple test cases.

The first line contains an integer $$$t (t \geq 1)$$$ — the number of test cases. The description of the test cases follows.

Only one line of each test case contains an integer $$$n(1 \leq n \leq 10^5)$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$

Output:

For each test case, print the answer in a single line.

Sample Input:

Sample Output:

Note:

$$$n=1$$$, there is only one permutation that satisfies the condition: $$$[1,2].$$$

In permutation $$$[1,2]$$$, $$$p_1<p_2$$$, and there is one $$$i=1$$$ satisfy the condition. Since $$$1 \geq n$$$, this permutation should be counted. In permutation $$$[2,1]$$$, $$$p_1>p_2$$$. Because $$$0<n$$$, this permutation should not be counted.

$$$n=2$$$, there are $$$12$$$ permutations: $$$[1,2,3,4],[1,2,4,3],[1,3,2,4],[1,3,4,2],[1,4,2,3],[2,1,3,4],[2,3,1,4],[2,3,4,1],[2,4,1,3],[3,1,2,4],[3,4,1,2],[4,1,2,3].$$$

Informação

Provedor Codeforces

Código CF1581A