The 66 3-colored pentominoes:  JPEG Excel spreadsheet

The 55 4-colored pentominoes:  JPEG Excel spreadsheet

I've also counted the colored hexominoes. Instead of picturing all of them, I've given the number of each shape:  JPEG Excel spreadsheet

For n > 6, I will use this notion to compress the results: We say that two n-ominoes are GS-equivalent if (1) they have the same graph, and (2) the symmetry groups of the two pieces both induce the same set of automorphisms of this graph. It is not hard to show that two GS-equivalent pieces have the same number of k-colorings for any k (the graph isomorphism can be used to transfer colorings from one piece to the other.) So the coloring counts only have to be listed once for each GS-equivalence class. Here are the counts for the 108 heptominoes:  JPEG Excel spreadsheet
The spreadsheet shows all the pieces, but the JPEG only shows one from each class.

Notice that while there are 26 GS-equivalence classes, there are only 9 different counts. Please email me if you can explain why so many of the answers come out the same.

For n = 8 and 9, I've grouped all the pieces that have the same counts, and I show one example of each. For n = 8, there are about 60 GS-equivalence classes. I haven't tried to count them for n = 9.  JPEG Excel spreadsheet

Please email me if you find any errors in my calculations.

OEIS sequence

Matlab code I wrote to compute the sequence

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