Count Special Matrices

All submissions for this problem are available.
Let N ≥ 3 be a fixed positive integer.
Further, let A[1..N][1..N] be N × N integer matrix.
The yth element in the xth row will be denoted as A[x][y].
This matrix is called special if it satisfies the following conditions:
 A[x][x] = 0 for 1 ≤ x ≤ N.
 A[x][y] = A[y][x] > 0 for 1 ≤ x < y ≤ N.
 A[x][y] ≤ max(A[x][z], A[z][y]) for 1 ≤ x, y, z ≤ N.
 A[x][y] ∈ {1, 2, ..., N − 2} for 1 ≤ x < y ≤ N.
 For each k ∈ {1, 2, ..., N − 2} there exist x, y ∈ {1, 2, ..., N} such that A[x][y] = k.
For example,
0 1 1
1 0 1
1 1 0
is the only special matrix of order 3.
The following matrices of order 3 are not special:
1 1 1 0 1 0 0 1 2
1 0 1 1 0 1 1 0 1
1 1 0 1 1 0 2 1 0
The first one violates the first condition, the second one  the second condition, the third one  the third condition (as well as the fourth condition).
It can be verified that there are 13 special matrices of order 4 in all.
The full list of special matrices of order 4 looks as follows:
0 1 2 2 0 2 1 2 0 2 2 2 0 1 1 2 0 2 2 1 0 2 2 1 0 2 2 2
1 0 2 2 2 0 2 2 2 0 1 2 1 0 1 2 2 0 2 2 2 0 1 2 2 0 2 1
2 2 0 2 1 2 0 2 2 1 0 2 1 1 0 2 2 2 0 2 2 1 0 2 2 2 0 2
2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 1 2 2 0 1 2 2 0 2 1 2 0
0 2 1 2 0 1 2 1 0 2 2 2 0 1 2 2 0 2 1 1 0 2 2 2
2 0 2 1 1 0 2 1 2 0 2 2 1 0 2 2 2 0 2 2 2 0 1 1
1 2 0 2 2 2 0 2 2 2 0 1 2 2 0 1 1 2 0 1 2 1 0 1
2 1 2 0 1 1 2 0 2 2 1 0 2 2 1 0 1 2 1 0 2 1 1 0
Your task is simple. For the given N count overall number of special matrices of order N.
Since the answer could be extremely large output it modulo 10^{9} + 7 = 1000000007.
Input
The first line of the input contains an integer T, denoting the number of test cases.
The description of T test cases follows.
The only line of each test case contains a single integer N.
Output
For each test case output the overall number of special matrices of order N modulo 1000000007 on the separate line.
Constraints
 1 ≤ T ≤ 100000 (10^{5})
 3 ≤ N ≤ 10000000 (10^{7})
Example
Input: 3 3 4 10 Output: 1 13 719666144
Explanation
Example cases 1 and 2 are explained above.
Author:  anton_lunyov 
Tester:  tuananh93 
Editorial  http://discuss.codechef.com/problems/SPMATRIX 
Tags  anton_lunyov, combinatorics, hard, june13 
Date Added:  21042013 
Time Limit:  0.666 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.5, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, SCALA, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, PERL6, TEXT, CLOJ, FS 
Comments
 Please login at the top to post a comment.
SUCCESSFUL SUBMISSIONS
Fetching successful submissions
HELP
If you are still having problems, see a sample solution here. 