Games with Rectangle

题意翻译

一张网格纸（网格大小 $1 \times 1$），初始有一个 $n \times m$ 的矩形边框，每次绘制需要在上一次绘制的矩形内绘制（第一次则视 $n \times m$ 的矩形为上次绘制的），问绘制 $k$ 次的方案数量模 $10^9+7$。两个方案不同定义为结束以后在纸上的图像不同。

题目描述

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.

输入输出格式

输入格式

The first and only line contains three integers: $ n,m,k $ ( $ 1<=n,m,k<=1000 $ ).

输出格式

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 $ ( $ 10^{9}+7 $ ).

输入输出样例

输入样例 #1

3 3 1

输出样例 #1

1

输入样例 #2

4 4 1

输出样例 #2

9

输入样例 #3

6 7 2

输出样例 #3

75

说明

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.