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?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter