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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;

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 is 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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editing

editing

approved

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 is 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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editing

editing

approved

Conjecture: n*a(n) = Sum_{d|n} mobius(d)*A002426(n/d) where mobius=A008683. - R. J. Mathar, Nov 05 2021

approved

editing

editing

approved

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 equals the frequency of the second letter (subset of the words in A027376). For n=1 is 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=4 5 these are (00112) (00121) (00211) (01012) (01021) (01102) (01222) (02122) (02212) (02221). - R. J. Mathar, Oct 26 2021

read(transforms); # https://oeis.org/transforms.txt