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.

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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.