%I A059966
%S A059966 1,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710,14532,27594,
%T A059966 52377,99858,190557,364722,698870,1342176,2580795,4971008,9586395,
%U A059966 18512790,35790267,69273666,134215680,260300986,505286415,981706806
%N A059966 [sum{ d divides n } mu(n/d) (2^d - 1)]/n.
%C A059966 Dimensions of the homogeneous parts of the free Lie algebra with one 
               generator in 1,2,3, etc. (Lie analogue of the partition numbers).
%C A059966 This sequence is the Lie analogue of the partition sequence (which gives 
               the dimensions of the homogeneous polynomials with one generator 
               in each degree) or similarly of the partitions into distinct (or 
               odd numbers) (which gives the dimensions of the homogeneous parts 
               of the exterior algebra with one generator in each dimension).
%C A059966 Contribution from David Pasino (davepasino(AT)yahoo.com), Jan 29 2009: 
               (Start)
%C A059966 The number of cycles of length n of rectangle shapes in the process of
%C A059966 repeatedly cutting a square off the end of the rectangle. For example,
%C A059966 the one cycle of length 1 is the golden rectangle. (End)
%D A059966 S. Kang, M. Kim, Free Lie Algebras, Generalized Witt Formula and the 
               Denominator Identity, Journal of Algebra 183, 560-594 (1996).
%D A059966 C. Reutenauer, Free Lie algebras, Clarendon press, Oxford (1993).
%H A059966 G. Niklasch, <a href="http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml">
               Some number theoretical constants: 1000-digit values</a>
%F A059966 G.f.: Product((1-q^k)^a(n), k = 1..infinity) = 1-q-q^2-q^3-q^4.. = 2-1/
               (1-q).
%e A059966 a(4)=3: the 3 elements [a,c], [a[a,b]] and d form a basis of all homogeneous 
               elements of degree 4 in the free Lie algebra with generators a of 
               degree 1, b of degree 2, c of degree 3 and d of degree 4.
%t A059966 Table[1/n Apply[Plus, Map[(MoebiusMu[n/# ](2^# - 1)) &, Divisors[n]]], 
               {n, 1, 20}].
%Y A059966 Apart from initial terms, same as A001037.
%Y A059966 Sequence in context: A066313 A018499 A107847 this_sequence A095718 A038751 
               A018518
%Y A059966 Adjacent sequences: A059963 A059964 A059965 this_sequence A059967 A059968 
               A059969
%K A059966 nonn,easy,nice
%O A059966 1,3
%A A059966 Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Mar 05 2001
%E A059966 Explicit formula from Paul Hanna (phanna(AT)ghs.org), Apr 15, 2002. Description 
               corrected by Axel Kleinschmidt, Sep 15 2002.