A032253 -id:A032253 - cp's OEIS frontend

A185172 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).

3, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420, 11728123327840, 45035996273664, 173215367702370

Offset: 1

N. J. A. Sloane, Jan 23 2012

Maybe the definition should say: "Number of generators of degree n ...". The paper is a little unclear.
From Petros Hadjicostas, Jun 18 2019: (Start)
An unmarked cyclic composition of n >= 1 is an equivalence class of ordered partitions of n such that two such ordered partitions are equivalent iff one can be obtained from the other by rotation.
Here, a(n) is the number of aperiodic unmarked cyclic compositions of n where up to three colors can be used.
It is also the CHK (circular, identity, unlabeled) transform of the sequence 3, 3, 3, ... See the link by Bowers about such transforms.
If c = (c(n): n >= 1) is the input sequence with g.f. C(x) = Sum_{n >= 1} c(n)*x^n, then the g.f. of the output sequence ((CHK c)d: d >= 1) is -Sum{d >= 1} (mu(d)/d) * log(1 - C(x^d)). Here, c(n) = 3 for all n >= 1, and thus, C(x) = 3*x/(1 - x). It follows that the g.f. of the output sequence is -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)).
(End)

			From _Petros Hadjicostas_, Jun 18 2019: (Start)
Suppose we have three colors, say, A, B, C. Here, a(1) = 3 because we have the following aperiodic unmarked cyclic compositions of n = 1: 1_A, 1_B, 1_C.
We have a(2) = 6 because we have the following aperiodic unmarked cyclic compositions of n = 2: 2_A, 2_B, 2_C, 1_A + 1_B, 1_B + 1_C, 1_C + 1_A.
We have a(3) = 20 because we have the following aperiodic unmarked cyclic compositions of n = 3: 3_X, where X \in {A, B, C}; 1_X + 2_Y, where (X, Y) \in {(A, A), (A, B), (A, C), (B, A), (B, B), (B, C), (C, A), (C, B), (C, C)}; 1_A + 1_B + 1_C and 1_C + 1_B + 1_A; and 1_X + 1_Y + 1_Y, where (X, Y) \in {(A, B), (A, C), (B, A), (B, C), (C, A), (C, B)}.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1650
C. G. Bower, Transforms (2).
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008. See Eqs. (94) and (95).
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, Discrete Math. 310 (2010), no. 24, 3584-3606. See Eqs. (99) and (100).

Cf. A024036, A032253, A185171.

Essentially the same as A027377.

a[1] = 3; a[n_] := DivisorSum[n, MoebiusMu[#]*4^(n/#)&]/n; Array[a, 26] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

a(l=3,n) = if (n==1, l, sumdiv(n, d, moebius(d)*(1+l)^(n/d))/n);  \\ Michel Marcus, Feb 09 2013

From Petros Hadjicostas, Jun 17 2019: (Start)
a(1) = 3 and a(n) = (1/n) * Sum_{d|n} mu(d) * 4^(n/d) for n > 1 (from Eq. (95) in Novelli and Thibon (2008) or Eq. (100) in Novelli and Thibon (2010)).
a(n) = (1/n) * Sum_{d|n} mu(d) * (4^(n/d) - 1) = (1/n) * Sum_{d|n} mu(d) *A024036(n/d) for n >= 1.
G.f.: -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)) = -x - Sum_{d >= 1} (mu(d)/d) * log(1 - 4*x^d).
(End)

More terms from Michel Marcus, Feb 09 2013

Name edited by Petros Hadjicostas, Jun 17 2019

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A185172 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).

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