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Problem 275

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Balanced Sculptures

Let us define a balanced sculpture of order n as follows:

• A polyomino made up of n+1 tiles known as the blocks (n tiles) and the plinth (remaining tile);
• the plinth has its centre at position (x = 0, y = 0);
• the blocks have y-coordinates greater than zero (so the plinth is the unique lowest tile);
• the centre of mass of all the blocks, combined, has x-coordinate equal to zero.

When counting the sculptures, any arrangements which are simply reflections about the y-axis, are not counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the y-axis) is counted as one sculpture:

There are 964 balanced sculptures of order 10 and 360505 of order 15.How many balanced sculptures are there of order 18?

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平衡雕塑

我们定义n阶平衡雕塑如下：

• 是由n+1个方块构成的多联骨牌，分为区块（n个方块）和基座（剩下的1个方块）；
• 基座的中心位于(x = 0, y = 0)；
• 区块的y坐标大于0（因此基座是唯一最低的方块）；
• 所有区块的总体质心的x坐标等于零。

在数雕塑的种类时，所有关于y轴翻转后相同的形状都不算作不同的形状。例如，如下展示了全部18种6阶的平衡雕塑；注意每一对镜像（关于y轴）都只算做一种雕塑：

10阶的平衡雕塑一共有964种，15阶平衡雕塑一共有360505种。18阶平衡雕塑一共有多少种？

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