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%I #23 Jan 07 2022 19:35:33
%S 1,1,2,4,10,22,56,136,348,890,2332,6136,16380,43988,119170,324720,
%T 890290,2452752,6789308,18868520,52635730,147323176,413618614,
%U 1164510896,3287073450,9300500508,26372968632,74937133488,213333642442,608400799010,1737954608280
%N Inverse Euler Transform of the Motzkin Numbers.
%C The Multiset Transform of this sequence generates a triangle with rows n >= 0, columns k >= 0:
%C 1;
%C 0, 1;
%C 0, 1, 1;
%C 0, 2, 1, 1;
%C 0, 4, 3, 1, 1;
%C 0, 10, 6, 3, 1, 1;
%C 0, 22, 17, 7, 3, 1, 1;
%C 0, 56, 40, 19, 7, 3, 1, 1;
%C 0, 136, 108, 47, 20, 7, 3, 1, 1;
%C 0, 348, 276, 130, 49, 20, 7, 3, 1, 1;
%C 0, 890, 739, 340, 137, 50, 20, 7, 3, 1, 1;
%C 0, 2332, 1954, 929, 362, 139, 50, 20, 7, 3, 1, 1;
%C 0, 6136, 5275, 2511, 998, 369, 140, 50, 20, 7, 3, 1, 1;
%C 0, 16380, 14232, 6893, 2717, 1020, 371, 140, 50, 20, 7, 3, 1, 1;
%C 0, 43988, 38808, 18911, 7520, 2786, 1027, 372, 140, 50, 20, 7, 3, 1, 1;
%C 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?
%C 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
%H Alois P. Heinz, <a href="/A290277/b290277.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 09 2019
%F Conjecture: n*a(n) = Sum_{d|n} mobius(d)*A002426(n/d) where mobius=A008683. - _R. J. Mathar_, Nov 05 2021
%p read(transforms); # https://oeis.org/transforms.txt
%p [seq(A001006(n),n=1..20)] ;
%p EULERi(%) ;
%Y Cf. A001006.
%K nonn
%O 1,3
%A _R. J. Mathar_, Jul 25 2017
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