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COMMENTS
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The Multiset Transform of this sequence generates a triangle with rows n >= 0, columns k >= 0:
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 4, 3, 1, 1;
0, 10, 6, 3, 1, 1;
0, 22, 17, 7, 3, 1, 1;
0, 56, 40, 19, 7, 3, 1, 1;
0, 136, 108, 47, 20, 7, 3, 1, 1;
0, 348, 276, 130, 49, 20, 7, 3, 1, 1;
0, 890, 739, 340, 137, 50, 20, 7, 3, 1, 1;
0, 2332, 1954, 929, 362, 139, 50, 20, 7, 3, 1, 1;
0, 6136, 5275, 2511, 998, 369, 140, 50, 20, 7, 3, 1, 1;
0, 16380, 14232, 6893, 2717, 1020, 371, 140, 50, 20, 7, 3, 1, 1;
0, 43988, 38808, 18911, 7520, 2786, 1027, 372, 140, 50, 20, 7, 3, 1, 1;
where a(n) defines the column k=1, and where the row sums are the Motzkin numbers, A001006. The question is: what set of or statistics on Motzkin paths of length n do the entries in row n of the triangle describe/refine?
a(n) is the number of Lyndon words of length n of a 3-letter alphabet {0,1,2} where the frequency of the first letter of the alphabet equals the frequency of the second letter of the alphabet (subset of the words in A027376). For n=1 this is (2), for n=2 this is (01), for n=3 these are (012), (021), for n=4 these are (0011) (0122) (0212) (0221), for n=5 these are (00112) (00121) (00211) (01012) (01021) (01102) (01222) (02122) (02212) (02221). - R. J. Mathar, Oct 26 2021
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