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%I #96 May 07 2021 12:34:21
%S 1,2,5,20,109,784,6757,68240,787477,10224812,147512053,2340964372,
%T 40527565261,760095929840,15352212731933,332228417657960,
%U 7668868648772701,188085259070219000,4884294069438337429
%N Number of length n necklaces with integer entries that cover an initial interval of positive integers.
%C Original name: a(n) = number of necklaces of n beads with up to n unlabeled colors.
%C The Moebius transform of this sequence is A060223.
%H G. C. Greubel, <a href="/A019536/b019536.txt">Table of n, a(n) for n = 1..420</a>
%H M. Goebel, <a href="https://doi.org/10.1007/s002000050088">On the number of special permutation-invariant orbits and terms</a>, in: Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.); see p. 509 (stated as an open problem).
%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H Eric Weisstein's world of Mathematics, <a href="http://mathworld.wolfram.com/Necklace.html">Necklaces</a>.
%F See Mathematica code.
%F a(n) ~ (n-1)! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Jul 21 2019
%F From _Petros Hadjicostas_, Aug 19 2019: (Start)
%F The first formula is due to _Philippe Deléham_ from the Crossrefs (see also the programs below). The second one follows easily from the first one. The third one follows from the second one using the associative property of Dirichlet convolutions.
%F a(n) = Sum_{k = 1..n} (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n, k) = Stirling numbers of 2nd kind (A008277).
%F a(n) = (1/n) * Sum_{d|n} phi(d) * A000670(n/d).
%F a(n) = Sum_{d|n} A060223(d).
%F (End)
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F a(n) = (1/n)*Sum_{k=1..n} A000670(gcd(n,k)).
%F a(n) = (1/n)*Sum_{k=1..n} A000670(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%e a(3) = 5 since there are the following length 3 words up to rotation:
%e 111, 112, 122, 123, 132.
%e a(4) = 20 since there are the following length 4 words up to rotation:
%e 1111,
%e 1112, 1122, 1212, 1222,
%e 1123, 1132, 1213, 1223, 1232, 1233, 1322, 1323, 1332,
%e 1234, 1243, 1324, 1342, 1423, 1432.
%t Needs["DiscreteMath`Combinatorica`"];
%t mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]];
%t neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li];Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n];
%t Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
%t (* second program: *)
%t a[n_] := Sum[DivisorSum[n, EulerPhi[#]*StirlingS2[n/#, k] k! &]/n, {k, 1, n}];
%t Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Mar 31 2016, after _Philippe Deléham_ *)
%o (PARI) a(n) = sum(k=1, n, sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)*k!)/n); \\ _Michel Marcus_, Mar 31 2016
%Y Cf. A000670, A008277, A019537, A060223.
%Y Row sums of A087854. - _Philippe Deléham_
%K easy,nonn
%O 1,2
%A Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)
%E Edited by _Wouter Meeussen_, Aug 06 2002
%E Corrected by _T. D. Noe_, Oct 31 2006
%E Edited by _Andrew Howroyd_, Aug 19 2019
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