4035: Games with Rectangle
Description
In this task Anna and Maria play the following game. Initially they have a checkered piece of paper with a painted n×m rectangle (only the border, no filling). Anna and Maria move in turns and Anna starts. During each move one should paint inside the last-painted rectangle a new lesser rectangle (along the grid lines). The new rectangle should have no common points with the previous one. Note that when we paint a rectangle, we always paint only the border, the rectangles aren't filled.
Nobody wins the game − Anna and Maria simply play until they have done k moves in total. Count the number of different ways to play this game.
The first and only line contains three integers: n,m,k (1≤n,m,k≤1000).
Print the single number − the number of the ways to play the game. As this number can be very big, print the value modulo 1000000007 (109+7).
3 3 1
1
4 4 1
9
6 7 2
75
Two ways to play the game are considered different if the final pictures are different. In other words, if one way contains a rectangle that is not contained in the other way.
In the first sample Anna, who performs her first and only move, has only one possible action plan − insert a 1×1 square inside the given 3×3 square.
In the second sample Anna has as much as 9 variants: 4 ways to paint a 1×1 square, 2 ways to insert a 1×2 rectangle vertically, 2 more ways to insert it horizontally and one more way is to insert a 2×2 square.