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

Author

R. J. Mathar, Jul 25 2017

Keywords

• nonn

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

Crossrefs

Cf. A001006.

Programs

Maple

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

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