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A185171 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(2).

%I A185171 #51 Sep 03 2025 10:22:50
%S A185171 2,3,8,18,48,116,312,810,2184,5880,16104,44220,122640,341484,956576,
%T A185171 2690010,7596480,21522228,61171656,174336264,498111952,1426403748,
%U A185171 4093181688,11767874940,33891544368,97764009000,282429535752,817028131140,2366564736720
%N A185171 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(2).
%C A185171 Maybe the definition should say: "Number of generators of degree n ...". The paper is a little unclear.
%C A185171 From _Petros Hadjicostas_, Jun 18 2019: (Start)
%C A185171 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.
%C A185171 Here, a(n) is the number of aperiodic unmarked cyclic compositions of n where up to two colors can be used.
%C A185171 It is also the CHK (circular, identity, unlabeled) transform of the sequence 2, 2, 2, ... See the link by Bowers about such transforms.
%C A185171 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) = 2 for all n >= 1, and thus, C(x) = 2*x/(1 - x). It follows that the g.f. of the output sequence is -Sum_{d >= 1} (mu(d)/d) * log(1 - 2*x^d/(1 - x^d)).
%C A185171 (End)
%H A185171 G. C. Greubel, <a href="/A185171/b185171.txt">Table of n, a(n) for n = 1..2000</a>
%H A185171 J. Blumlein, <a href="http://doi.org/10.1016/j.cpc.2009.07.004">Structural relations of harmonic sums and Mellin transforms up to weight 5</a>, Comp. Phys. Com. 180 (2009) 2218-2249 eq. 4.2
%H A185171 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%H A185171 Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008. See Eqs. (93) and (95).
%H A185171 Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://doi.org/10.1016/j.disc.2010.09.008">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a>, Discrete Math. 310 (2010), no. 24, 3584-3606. See Eqs. (98) and (100).
%F A185171 From _Petros Hadjicostas_, Jun 18 2019: (Start)
%F A185171 a(1) = 2 and a(n) = (1/n) * Sum_{d|n} mu(d) * 3^(n/d) for n > 1 (from Eq. (95) in Novelli and Thibon (2008) or Eq. (100) in Novelli and Thibon (2010)).
%F A185171 a(n) = (1/n) * Sum_{d|n} mu(d) * (3^(n/d) - 1) = (1/n) * Sum_{d|n} mu(d) * A024023(n/d) for n >= 1.
%F A185171 G.f.: -Sum_{d >= 1} (mu(d)/d) * log(1 - 2*x^d/(1 - x^d)) = -x - Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d).
%F A185171 (End)
%e A185171 From _Petros Hadjicostas_, Jun 18 2019: (Start)
%e A185171 Suppose we have two colors, say, A and B. Here, a(1) = 2 because we have the following aperiodic unmarked cyclic compositions of n = 1: 1_A and 1_B.
%e A185171 We have a(2) = 3 because we have the following aperiodic unmarked cyclic compositions of n = 2: 2_A, 2_B, and 1_A + 1_B.
%e A185171 We have a(3) = 8 because we have the following aperiodic unmarked cyclic compositions of n = 3: 3_A and 3_B; 1_X + 2_Y, where (X, Y) \in {(A, A), (A, B), (B, A), (B, B)}; 1_A + 1_B + 1_B and 1_B + 1_A + 1_A.
%e A185171 (End)
%t A185171 a[1] = 2; a[n_] := DivisorSum[n, MoebiusMu[#]*3^(n/#)&]/n; Array[a, 29] (* _Jean-François Alcover_, Dec 07 2015, adapted from PARI *)
%o A185171 (PARI) a(l=2,n) = if (n==1, l, sumdiv(n, d, moebius(d)*(1+l)^(n/d))/n); \\ _Michel Marcus_, Feb 09 2013
%Y A185171 Cf. A024023, A032251, A185172.
%Y A185171 Essentially the same as A027376.
%K A185171 nonn,changed
%O A185171 1,1
%A A185171 _N. J. A. Sloane_, Jan 23 2012
%E A185171 More terms from _Michel Marcus_, Feb 09 2013
%E A185171 Name edited by _Petros Hadjicostas_, Jun 18 2019