A290277 Inverse Euler Transform of the Motzkin Numbers.

1, 1, 2, 4, 10, 22, 56, 136, 348, 890, 2332, 6136, 16380, 43988, 119170, 324720, 890290, 2452752, 6789308, 18868520, 52635730, 147323176, 413618614, 1164510896, 3287073450, 9300500508, 26372968632, 74937133488, 213333642442, 608400799010, 1737954608280

Offset

1,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Index entries for sequences related to Lyndon words

Formula

a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 09 2019

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

Maple

read(transforms); # https://oeis.org/transforms.txt
[seq(A001006(n), n=1..20)] ;
EULERi(%) ;

Crossrefs

Cf. A001006.

Sequence in context: A348009 A121285 A221536 * A030234 A205490 A221839

Adjacent sequences: A290274 A290275 A290276 * A290278 A290279 A290280

Keyword

nonn

Author

R. J. Mathar, Jul 25 2017

Status

approved