G. Cost of Coloringtime limit per test3 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputThere is a sheet of paper divided into $$$n$$$ rows and $$$m$$$ columns. Initially, no cell of this sheet is colored.In one operation, you can choose any column or row and color it (if some cells were previously colored, their color changes to the new one). During the first operation, cells are colored with color $$$1$$$; during operation $$$i > 1$$$, you can choose either color $$$c_{i-1}$$$ or $$$c_{i-1} + 1$$$, where $$$c_{i-1}$$$ is the color chosen during operation $$$(i-1)$$$.We call the final coloring beautiful if the following conditions are met: each cell is colored; for each color from $$$1$$$ to $$$k$$$, there is at least one cell colored in that color, and no other colors are used in the coloring. For a beautiful final coloring, we define its value as the minimum number of operations required to achieve it.For each $$$i$$$ from $$$\min(n, m)$$$ to $$$n + m - 1$$$, calculate the number of beautiful colorings with value $$$i$$$. Two colorings are considered different if the color of at least one cell differs in these colorings.InputThe input consists of a single line containing three integers $$$n, m, k$$$ ($$$2 \le n, m \le 2000$$$; $$$1 \le k \le n + m - 1$$$).OutputFor each $$$i$$$ from $$$\min(n, m)$$$ to $$$n + m - 1$$$, output a single integer — the number of beautiful colorings with value $$$i$$$, taken modulo $$$998244353$$$.ExamplesInput2 3 2Output2 12 6
Input2 3 3Output0 18 36
Input3 2 4Output0 0 36
Input2 2 3Output0 8
Input2 2 2Output4 4
Input2 2 1Output1 0
Input5 3 4Output0 90 1500 7830 8100
Input3 5 3Output6 120 750 1770 930
Input5 2 2Output2 15 30 35 10
Input2 5 2Output2 15 30 35 10