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A290277 Inverse Euler Transform of the Motzkin Numbers.

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