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A059966
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[sum{ d divides n } mu(n/d) (2^d - 1)]/n.
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8
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1, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Dimensions of the homogeneous parts of the free Lie algebra with one generator in 1,2,3, etc. (Lie analogue of the partition numbers).
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).
Contribution from David Pasino (davepasino(AT)yahoo.com), Jan 29 2009: (Start)
The number of cycles of length n of rectangle shapes in the process of
repeatedly cutting a square off the end of the rectangle. For example,
the one cycle of length 1 is the golden rectangle. (End)
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REFERENCES
| S. Kang, M. Kim, Free Lie Algebras, Generalized Witt Formula and the Denominator Identity, Journal of Algebra 183, 560-594 (1996).
C. Reutenauer, Free Lie algebras, Clarendon press, Oxford (1993).
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
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FORMULA
| G.f.: Product((1-q^k)^a(n), k = 1..infinity) = 1-q-q^2-q^3-q^4.. = 2-1/(1-q).
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EXAMPLE
| 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.
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MATHEMATICA
| Table[1/n Apply[Plus, Map[(MoebiusMu[n/# ](2^# - 1)) &, Divisors[n]]], {n, 1, 20}].
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PROG
| (Haskell)
a059966 n = sum (map (\x -> a008683 (n `div` x) * a000225 x)
[d | d <- [1..n], mod n d == 0]) `div` n
-- Reinhard Zumkeller, Nov 18 2011
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CROSSREFS
| Apart from initial terms, same as A001037.
Cf. A008683, A000225.
Sequence in context: A066313 A018499 A107847 * A095718 A038751 A018518
Adjacent sequences: A059963 A059964 A059965 * A059967 A059968 A059969
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Mar 05 2001
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EXTENSIONS
| Explicit formula from Paul Hanna (phanna(AT)ghs.org), Apr 15, 2002. Description corrected by Axel Kleinschmidt, Sep 15 2002.
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