Problem statement: A little girl whose name is Anne Spetring likes to play the following game. She draws a circle on paper. Then she draws another one and connects it to the first cicrle by a line. Then she draws another and connects it to one of the first two circles by a line. She continues this way until she has n circles drawn and each one connected to one of the previously drawn circles. Her circles never intersect and lines never cross. Finally, she numbers the circles from 1 to n in some random order.

How many different pictures can she draw that contain exactly n circles? Two pictures are different if one of them has a line connecting circle number i to circle number j, and the other picture does not.

INPUT: The first line of input gives the number of cases, N. N test cases follow. Each one is a line containing n (0 < n ≤ 100).

OUTPUT: For each test case, output one line containing ‘Case #x:’ followed by X, where X is the remainder after dividing the answer by 2000000011.

My logic: example: n=3, then 1-2-3, 1-3-2, 2-1-3 ('-' means line connecting circles) these 3 different pictures can be drawn. According to my observation, for n=4, it will be 12 different pictures. This way I find that answer will be (n!/2)%2000000011;

I want to know, why my logic is not correct(got WA), what is the formula to solve this problem and why it will work?

Search keyword: Prüfer Sequences

UPDATE: Got it!