CF128C.Games with Rectangle

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.

在本题中，安娜（Anna）与玛丽亚（Maria）玩如下游戏：初始时，她们拥有一张带方格的纸，纸上画有一个 $n \times m$ 的矩形（仅绘制边界，不填充）。安娜与玛丽亚轮流行动，安娜先手。每次行动时，玩家需在上一次所画矩形的内部，沿网格线绘制一个新的、更小的矩形；该新矩形与上一个矩形不能有任何公共点。注意：每次绘制矩形时，仅绘制其边界，矩形本身并不填充。

本游戏无胜负之分——安娜与玛丽亚仅持续进行，直至总共完成 $k$ 次行动为止。请计算完成此游戏的不同方式总数。

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

第一行且唯一一行包含三个整数：$n$、$m$、$k$（$1 \leq n, m, k \leq 1000$）。

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).

输出单个数字——即游戏的玩法数量。由于该数字可能非常大，请输出其对 $1000000007$（即 $10^9 + 7$）取模的结果。

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.

如果最终得到的图片不同，则认为这两种游戏方式不同。换言之，若其中一种方式包含某个矩形，而另一种方式不包含该矩形，则二者不同。

在第一个样例中，安娜仅执行一次操作（也是唯一一次操作），她只有一种可能的操作方案：在一个给定的 $3 \times 3$ 正方形内插入一个 $1 \times 1$ 的正方形。

在第二个样例中，安娜共有 9 种方案：4 种方式绘制 $1 \times 1$ 正方形，2 种方式将 $1 \times 2$ 矩形竖直插入，2 种方式将其水平插入，以及 1 种方式插入一个 $2 \times 2$ 正方形。