A254040 -id:A254040 - cp's OEIS frontend

A001037 Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

1, 2, 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, 1908866960, 3714566310, 7233615333, 14096302710, 27487764474

Offset: 0

N. J. A. Sloane

Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence.
This sequence also represents the number N of cycles of length L in a digraph under x^2 seen modulo a Mersenne prime M_q=2^q-1. This number does not depend on q and L is any divisor of q-1. See Theorem 5 and Corollary 3 of the Shallit and Vasiga paper: N=sum(eulerphi(d)/order(d,2)) where d is a divisor of 2^(q-1)-1 such that order(d,2)=L. - Tony Reix, Nov 17 2005
Except for a(0) = 1, Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 7th (rightmost) column. Other columns of the table include (but are not identified as) A006206-A006208. - Jonathan Vos Post, Jun 18 2007
"Number of binary Lyndon words" means: number of binary strings inequivalent modulo rotation (cyclic permutation) of the digits and not having a period smaller than n. This provides a link to A103314, since these strings correspond to the inequivalent zero-sum subsets of U_m (m-th roots of unity) obtained by taking the union of U_n (n|m) with 0 or more U_d (n | d, d | m) multiplied by some power of exp(i 2Pi/n) to make them mutually disjoint. (But not all zero-sum subsets of U_m are of that form.) - M. F. Hasler, Jan 14 2007
Also the number of dynamical cycles of period n of a threshold Boolean automata network which is a quasi-minimal positive circuit of size a multiple of n and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Feb 25 2009
Also, the number of periodic points with (minimal) period n in the iteration of the tent map f(x):=2min{x,1-x} on the unit interval. - Pietro Majer, Sep 22 2009
Number of distinct cycles of minimal period n in a shift dynamical system associated with a totally disconnected hyperbolic iterated function system (see Barnsley link). - Michel Marcus, Oct 06 2013
From Jean-Christophe Hervé, Oct 26 2014: (Start)
For n > 0, a(n) is also the number of orbits of size n of the transform associated to the Kolakoski sequence A000002 (and this is true for any map with 2^n periodic points of period n). The Kolakoski transform changes a sequence of 1's and 2's by the sequence of the lengths of its runs. The Kolakoski sequence is one of the two fixed points of this transform, the other being the same sequence without the initial term. A025142 and A025143 are the periodic points of the orbit of size 2. A027375(n) = n*a(n) gives the number of periodic points of minimal period n.
For n > 1, this sequence is equal to A059966 and to A060477, and for n = 1, a(1) = A059966(1)+1 = A060477(1)-1; this because the n-th term of all 3 sequences is equal to (1/n)*sum_{d|n} mu(n/d)*(2^d+e), with e = -1/0/1 for resp. A059966/this sequence/A060477, and sum_{d|n} mu(n/d) equals 1 for n = 1 and 0 for all n > 1. (End)
Warning: A000031 and A001037 are easily confused, since they have similar formulas.
From Petros Hadjicostas, Jul 14 2020: (Start)
Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.
Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) = A001221(N) being the number of distinct primes of N).
For N = 3, a = phi(3)/2 + omega(3) = 2/2 + 1 = 2 and b = omega(3) - 1 = 0. It follows that D(w, N=3) = A000079(w) = 2^w.
For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.
For N = 3, we get that c(w, N=3) = A000079(w)/w = 2^w/w.
He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)
Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).
For N = 3, we saw that a = 2 and b = 0, and hence d*(w, N=3) = a(w) = Sum_{k | w} (mu(k)/k) * 2^(w/k) / (w/k) = (1/w) * Sum_{k | w} mu(k) * 2^(w/k) for w >= 1. See Table 1 on p. 6 in Kam Cheong Au (2020). (End)

			Binary strings (Lyndon words, cf. A102659):
a(0) = 1 = #{ "" },
a(1) = 2 = #{ "0", "1" },
a(2) = 1 = #{ "01" },
a(3) = 2 = #{ "001", "011" },
a(4) = 3 = #{ "0001", "0011", "0111" },
a(5) = 6 = #{ "00001", "00011", "00101", "00111", "01011", "01111" }.

Michael F. Barnsley, Fractals Everywhere, Academic Press, San Diego, 1988, page 171, Lemma 3.
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie. On the digraph defined by squaring mod m, when m has primitive roots. Congr. Numer. 82 (1991), 167-177.
P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. See 1.925.
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983, pp. 65, 79.
Robert M. May, "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019
Guy Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in entries N0046 and N0287).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (terms 0..200 from T. D. Noe)
Per Alexandersson and Joakim Uhlin, Cyclic sieving, skew Macdonald polynomials and Schur positivity, arXiv:1908.00083 [math.CO], 2019.
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, When are Hopf algebras determined by integer sequences?, arXiv:2505.06941 [math.CO], 2025. See p. 17.
Joerg Arndt, Matters Computational (The Fxtbook), pp.379-383, pp.843-845.
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 3rd line of Table 1 (p. 6).
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie, On a digraph defined by squaring modulo n, Fibonacci Quart. 30 (1992), 322-333.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Émilie Charlier, Manon Philibert, and Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. Also J. Comb. Thy. A, 167 (2019), 60-90.
R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209.
J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive and negative threshold Boolean automata circuits, 2010 IEEE 24th Intl. Conf. Advan. Inf. Network. Appl. Workshops (WAINA), p 782-789.
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 6.
Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.
S. V. Duzhin and D. V. Pasechnik, Groups acting on necklaces and sandpile groups, 2014. See page 85. - _N. J. A. Sloane_, Jun 30 2014
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
M. A. Harrison, On the classification of Boolean functions by the general linear and affine groups, J. Soc. Indust. Appl. Math. 12 (1964) 285-299.
A. Knopfmacher and M. E. Mays, Graph Compositions I: Basic enumeration, Integers 1 (2001), A4, eq. (1).
T. Laarhoven and B de Weger, The Collatz conjecture and De Bruijn graphs, arXiv preprint arXiv:1209.3495 [math.NT], 2012. - From _N. J. A. Sloane_, Dec 27 2012
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arithmetica, LVI (1990), pp. 33-53.
Piotr Laskawiec, Joint Spectral Radius with focus on pairs of 2 X 2 matrices, Ph. D. Dissertation, Penn. State Univ. (2025). See p. 55.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence gamma_{2,j}^(T), arXiv:0903.2514 [math.NT], 2009-2011.
Ueli M. Maurer, Asymptotically-tight bounds on the number of cycles in generalized de Bruijn-Good graphs, Discrete applied mathematics 37 (1992): 421-436. See Table 1.
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
H. Meyn and W. Götz, Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
J.-F. Michon and P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.
Hans Z. Munthe-Kaas and Jonatan Stava, Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces, arXiv:2306.15582 [math.DG], 2023.
Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444, ArXiv 1011.3930 [cs.DM]. - _N. J. A. Sloane_, Jul 07 2012
Cormac O'Sullivan, Topographs for binary quadratic forms and class numbers, arXiv:2408.14405 [math.NT], 2024. See p. 30.
George Petrides and Johannes Mykkeltveit, On the Classification of Periodic Binary Sequences into Nonlinear Complexity Classes, in Sequences and Their Applications SETA 2006, Lecture Notes in Computer Science, Volume 4086/2006, pp 209-222. [From _N. J. A. Sloane_, Jul 09 2009]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Primitive and Irreducible Polynomials
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Troy Vasiga and Jeffrey Shallit, On the iteration of certain quadratic maps over GF(p), Discrete Mathematics, Volume 277, Issues 1-3, 2004, pages 219-240.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Lyndon Word
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words
Index entries for "core" sequences

Column 2 of A074650.
Row sums of A051168, which gives the number of Lyndon words with fixed number of zeros and ones.
Euler transform is A000079.
See A058943 and A102569 for initial terms. See also A058947, A011260, A059966.
Irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058943, A058944, A058948, A058945, A058946. Primitive irreducible over GF(2), GF(3), GF(4), GF(5), GF(7): A058947, A058949, A058952, A058950, A058951.
Cf. A000031 (n-bead necklaces but may have period dividing n), A014580, A046211, A046209, A006206-A006208, A038063, A060477, A103314.
Cf. A027750, A008683, A254040.
See also A102659 for the list of binary Lyndon words themselves.
Cf. A000010, A008683.

a001037 0 = 1
a001037 n = (sum $ map (\d -> (a000079 d) * a008683 (n `div` d)) $
                       a027750_row n) `div` n
-- Reinhard Zumkeller, Feb 01 2013

with(numtheory): A001037 := proc(n) local a,d; if n = 0 then RETURN(1); else a := 0: for d in divisors(n) do a := a+mobius(n/d)*2^d; od: RETURN(a/n); fi; end;

f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[n/d]*2^d/n)]; Array[f, 32]

A001037(n)=if(n>1,sumdiv(n,d,moebius(d)*2^(n/d))/n,n+1) \\ Edited by M. F. Hasler, Jan 11 2016

{a(n)=polcoeff(1-sum(k=1,n,moebius(k)/k*log(1-2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

a(n)=if(n>1,my(s);forstep(i=2^n+1,2^(n+1),2,s+=polisirreducible(Mod(1,2) * Pol(binary(i))));s,n+1) \\ Charles R Greathouse IV, Jan 26 2012

from sympy import divisors, mobius
def a(n): return sum(mobius(d) * 2**(n//d) for d in divisors(n))/n if n>1 else n + 1 # Indranil Ghosh, Apr 26 2017

For n >= 1:
a(n) = (1/n)*Sum_{d | n} mu(n/d)*2^d.
A000031(n) = Sum_{d | n} a(d).
2^n = Sum_{d | n} d*a(d).
a(n) = A027375(n)/n.
a(n) = A000048(n) + A051841(n).
For n > 1, a(n) = A059966(n) = A060477(n).
G.f.: 1 - Sum_{n >= 1} moebius(n)*log(1 - 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
From Richard L. Ollerton, May 10 2021: (Start)
For n >= 1:
a(n) = (1/n)*Sum_{k=1..n} mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
a(n) ~ 2^n / n. - Vaclav Kotesovec, Aug 11 2021

Revised by N. J. A. Sloane, Jun 10 2012

A060223 Number of orbits of length n under the map whose periodic points are counted by A000670.

Original entry on oeis.org

1, 1, 1, 4, 18, 108, 778, 6756, 68220, 787472, 10224702, 147512052, 2340963570, 40527565260, 760095923082, 15352212731820, 332228417589720, 7668868648772700, 188085259069430744, 4884294069438337428, 133884389812214097774, 3863086904690670182596

Offset: 0

Thomas Ward, Mar 21 2001

From Gus Wiseman, Oct 14 2016: (Start)
A finite sequence is normal if it spans an initial interval of positive integers. The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, (2 2 1) * (2 1 3) = (2 1 2 2 1 3). If Q is the set of compositions (finite sequences of positive integers) then (Q,*) is an Abelian group freely generated by a set P of prime sequences. The number of normal prime sequences of length n is equal to a(n). See example 2 and Mathematica program 2.
If N is the species (endofunctor over the category of finite sets and permutations) of unlabeled necklaces and N(S) represents the set of all non-isomorphic primitive necklaces of length n=|S|, then the numbers |N(S)| are equal to the numbers a(|S|) for any finite set S. This is because the number of orderless *-factorizations (see A034691 and A269134) of any finite sequence q is equal to the number of multiset partitions (see A007716 and A255906) of the multiset of prime factors of q. (End)

			a(5) = 108 since A000670(5) is 541 and A000670(1) is 1, so there must be (541-1)/5 = 108 orbits of length 5.
From _Gus Wiseman_, Oct 14 2016: (Start)
The a(4) = 18 normal prime sequences are the columns:
[2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4]
[1 2 2 1 1 1 2 2 2 2 3 3 1 1 2 2 3 3]
[1 1 2 1 2 2 1 1 2 3 1 2 2 3 1 3 1 2]
[1 1 1 2 1 2 1 2 1 1 2 1 3 2 3 1 2 1].
The symmetric function A(x_1,x_2,x_3,...) expanded in terms of monomial symmetric functions m(y) (indexed by integer partitions y) is equal to:
A = m(1) +
    m(11) +
    (2*m(21) + 2*m(111) +
    (m(22) + 2*m(31) + 9*m(211) + 6*m(1111)) +
    (4*m(32) + 2*m(41) + 18*m(221) + 12*m(311) + 48*m(2111) + 24*m(11111)) +
    (3*m(33) + 4*m(42) + 2*m(51) + 14*m(222) + 60*m(321) + 15*m(411) + 180*m(2211) + 80*m(3111) + 300*m(21111) + 120*m(111111)) + ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences

Cf. A000670, A034691 (multisets of compositions), A269134, A007716, A277427, A215474, A255906.

Row sums of A254040.

a[n_] := DivisorSum[n, MoebiusMu[#] HurwitzLerchPhi[1/2, -n/#, 0]/2 &] / n; a[0] = 1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2016 *)
thufbin[{},b_List]:=b;thufbin[a_List,{}]:=a;thufbin[a_List]:=a;
thufbin[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{thufbin[{a},{x,b}],thufbin[{x,a},{b}]}]],{1,2},Prepend[thufbin[{a},{y,b}],x],{2,1},Prepend[thufbin[{x,a},{b}],y]];
thufbin[a_List,b_List,c__List]:=thufbin[a,thufbin[b,c]];
priseqs[n_]:=Fold[Select,Tuples[Range[n],n],{Union[#]===Range[First[#]]&,Function[q,Select[Table[List[Take[q,{1,j}],Take[q,{j+1,n}]],{j,1,n-1}],thufbin@@Sort[#]===q&,1]==={}]}];
Table[Length[priseqs[n]],{n,1,7}] (* Gus Wiseman, Oct 14 2016 *)

\\ here b(n) is A000670
b(n)={polcoeff(serlaplace(1/(2-exp(x+O(x*x^n)))), n)}
a(n)={if(n<1, n==0, sumdiv(n, d, moebius(d)*b(n/d))/n)} \\ Andrew Howroyd, Dec 12 2017

a(n) = (1/n)* Sum_{d|n} mu(d)*A000670(n/d) for n > 0, where mu is A008683, the Moebius function. - Edited by Michel Marcus, Mar 30 2016
Let A = Sum_{q in P} Prod_i x_{q_i} = Sum_y c_y m(y) be the symmetric function whose coefficient of m(y) is equal to the number of permutations of the normal multiset [k]^y that belong to P, where the multiplicity of i in [k]^y is defined to be y_i. Then a(n) is the sum of c_y taken over all integer partitions of n. See example 3. - Gus Wiseman, Oct 14 2016
a(n) = Sum_{d|n} mu(d) * A019536(n/d) for n >= 1. - Petros Hadjicostas, Aug 19 2019

More terms from Alois P. Heinz, Jan 23 2015

A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 9, 6, 1, 6, 30, 48, 24, 1, 12, 91, 260, 300, 120, 1, 18, 258, 1200, 2400, 2160, 720, 1, 34, 729, 5106, 15750, 23940, 17640, 5040, 1, 58, 2018, 20720, 92680, 211680, 258720, 161280, 40320, 1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880

Offset: 1

Philippe Deléham

Equivalently, T(n,k) is the number of sequences (words) of length n on an alphabet of k letters where each letter of the alphabet occurs at least once in the sequence. Two sequences are considered equivalent if one can be obtained from the other by a cyclic shift of the letters. Cf. A054631 where the surjective restriction is removed. - Geoffrey Critzer, Jun 18 2013

Robert A. Russell's g.f. for column k >= 1 (in the Formula section below) can be proved by integrating both sides of the formula Sum_{n>=1} S2(n, k)*x^(n-1) = x^(k-1)/((1 - x)* (1 - 2*x) * (1 - 3*x) * ... * (1 - k*x)) w.r.t. x. A variation of this identity (valid for |x| < 1/k) can be found in the Formula section of A008277. - Petros Hadjicostas, Aug 20 2019

			The triangle begins with T(1,1):
  1;
  1,   1;
  1,   2,    2;
  1,   4,    9,     6;
  1,   6,   30,    48,     24;
  1,  12,   91,   260,    300,     120;
  1,  18,  258,  1200,   2400,    2160,     720;
  1,  34,  729,  5106,  15750,   23940,   17640,    5040;
  1,  58, 2018, 20720,  92680,  211680,  258720,  161280,   40320;
  1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880;
  ...
For T(4,2) = 4, the necklaces are AAAB, AABB, ABAB, and ABBB.
For T(4,4) = 6, the necklaces are ABCD, ABDC, ACBD, ACDB, ADBC, and ADCB.

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-6: A057427, A052823, A056283, A056284, A056285, A056286.
Diagonals: A000142 and A074143.
Row sums: A019536.
Cf. A000010 (Euler totient phi function), A008277 (Stirling2 numbers), A075195 (table of Jablonski).
Cf. A254040, A273891.

with(numtheory):
T:= (n, k)-> (k!/n) *add(phi(d) *Stirling2(n/d, k), d=divisors(n)):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Jun 19 2013

Table[Table[Sum[EulerPhi[d]*StirlingS2[n/d,k]k!,{d,Divisors[n]}]/n,{k,1,n}],{n,1,10}]//Grid (* Geoffrey Critzer, Jun 18 2013 *)

T(n, k) = (k!/n) * sumdiv(n, d, eulerphi(d) * stirling(n/d, k, 2)); \\ Joerg Arndt, Sep 25 2020

T(n,k) = Sum_{i=0..k-1} (-1)^i * C(k,i) * A075195(n,k-i); A075195 = Jablonski's table.
T(n,k) = (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n,k) = Stirling numbers of 2nd kind A008277.
G.f. for column k: -Sum_{d>0} (phi(d)/d) * Sum_{j = 1..k} (-1)^(k-j) * C(k,j) * log(1 - j * x^d). - Robert A. Russell, Sep 26 2018
T(n,k) = Sum_{d|n} A254040(d, k) for n, k >= 1. - Petros Hadjicostas, Aug 19 2019

Formula section edited by Petros Hadjicostas, Aug 20 2019

A074143 a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, 18144000, 219542400, 2874009600, 40475635200, 610248038400, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 24329020081766400000, 536454892802949120000

Offset: 1

Amarnath Murthy, Aug 28 2002

nonn

a(n) is also the number of elements of the alternating semigroup (A^c_n) for F(n, p) if p = n - 1 (cf. A001710). - Bakare Gatta Naimat, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets
Stephen Lipscomb, Symmetric inverse semigroups, Mathematical surveys and monographs, Vol. 46 Amer. Math. Soc. (1996).
Michael Penn, Australian Mathematical Olympiad 2018 Question 5, Youtube video, 2020.

A diagonal of A254040.
Cf. A001563, A001710.
Cf. A007808, A082425, A082427, A082428, A082430, A383436, A383437.
Cf. A001620, A091725, A099285.

[Numerator(Factorial(n)/2*n): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014

seq(sum(mul(j,j=3..n), k=1..n), n=1..19); # Zerinvary Lajos, Jun 01 2007
a := n -> `if`(n=1,1,n!*n/2): seq(a(n), n=1..19); # Peter Luschny, Jan 22 2016

A074143[1] = 1; A074143[n_] := A074143[n] = n * Sum[a[k], {k, n - 1}]; Array[A074143, 20] (* T. D. Noe, Apr 05 2011 *)
Table[Numerator[n!/2 n], {n, 21}] (* Vincenzo Librandi, Apr 15 2014 *)

def b(n): return 1/2 if (n==1) else n^2*b(n-1)/(n-1)
def A074143(n): return b(n) + int(n==1)/2
[A074143(n) for n in range(1,41)] # G. C. Greubel, Nov 29 2022

a(n) = n^2 * a(n-1)/(n-1) for n > 2.
a(n) = n*ceiling(n!/2) = n*A001710(n) = ceiling(A001563(n)/2). - Henry Bottomley, Nov 27 2002
a(n) = ((n+1)!-n!)/2 for n > 1. - Vladimir Joseph Stephan Orlovsky, Apr 03 2011
G.f.: (U(0) + x)/(2*x) where U(k) = 1 - 1/(k+1 - x*(k+1)^2*(k+2)/(x*(k+1)*(k+2) - 1/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2012
G.f.: 1/2 + Q(0), where Q(k)= 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*binomial(n, j)*(j)^(n+1) / (n+1), n > 1, a(1) = 1. - Vladimir Kruchinin, Jun 01 2013
a(n) = numerator(n!/2*n). - Vincenzo Librandi, Apr 15 2014
a(n) is F(n;p) = n^2(n-1)!/2 if p = n-1 in A^c_n. For instance for n=4 and p=n-1: F(4; 4-1)= 4^2(4-1)!/2 = 16*6/2 = 48. - Bakare Gatta Naimat, Nov 18 2015
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: x/2 * (1 + 1/(1-x)^2).
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)
From Amiram Eldar, May 04 2025: (Start)
Sum_{n>=1} 1/a(n) = 2*ExpIntegralEi(1) - 2*gamma - 1 = 2*A091725 - 2*A001620 - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*gamma - 1 - 2*ExpIntegralEi(-1) = 2*A001620 - 1 + 2*A099285. (End)

More terms from Henry Bottomley, Nov 27 2002

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811

Offset: 1

Christian G. Bower

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A029768, A038037, A055340.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703

Offset: 1

Christian G. Bower

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 10 locally Lyndon plane trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  ((oo(o)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Lyndon word
Index entries for sequences related to mobiles

Cf. A032200, A055363.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

T[n_, k_] := Module[{A}, A[, ] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-François Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
lynplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lynplane/@c],LyndonQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lynplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CHK" (necklace, identity, unlabeled) transform.
From Petros Hadjicostas, Dec 03 2017: (Start)
a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...
(End)

A309651 T(n,k) is the number of non-equivalent distinguishing colorings of the cycle on n vertices with exactly k colors (k>=1). Regular triangle read by rows, n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 34, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 315, 2484, 7845, 11970, 8820, 2520, 0, 14, 933, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2622, 40464, 254664, 821592, 1481760, 1512000, 816480, 181440

Offset: 1

Bahman Ahmadi, Aug 11 2019

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called equivalent if there is an automorphism of the graph which preserves the colors of the vertices.  Given a graph G, we use the notation phi_k(G) to denote the number of non-equivalent distinguishing colorings of G with exactly k colors. The sequence here, displays T(n,k)=phi_k(C_n), i.e., the number of non-equivalent distinguishing colorings of the cycle C_n on n vertices with exactly k colors.
From Andrew Howroyd, Aug 15 2019: (Start)
First differs from A305541 at n = 6.
Also the number of n-bead asymmetric bracelets with exactly k different colored beads. More precisely the number of chiral pairs of primitive (aperiodic) color loops of length n with exactly k different colors. For example, for n=4 and k = 3, there are 3 achiral loops (1213, 1232, 1323) and 3 pairs of chiral loops (1123/1132, 1223/1322, 1233/1332).
(End)

			The triangle begins:
  0
  0,  0;
  0,  0,    1;
  0,  0,    3,     3;
  0,  0,   12,    24,     12;
  0,  1,   34,   124,    150,     60;
  0,  2,  111,   588,   1200,   1080,     360;
  0,  6,  315,  2484,   7845,  11970,    8820,    2520;
  0, 14,  933, 10240,  46280, 105840,  129360,   80640,  20160;
  0, 30, 2622, 40464, 254664, 821592, 1481760, 1512000, 816480, 181440;
  ...
For n=4, we can color the vertices of the cycle C_4 with exactly 3 colors, in 3 ways, such that all the colorings distinguish the graph (i.e., no non-identity automorphism of C_4 preserves the coloring) and that all the three colorings are non-equivalent. The color classes are as follows:
{ { 1 }, { 2 }, { 3, 4 } }
{ { 1 }, { 2, 3 }, { 4 } }
{ { 1, 2 }, { 3 }, { 4 } }

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Columns k=2..4 are A032239(n>=3), A326660, A326789.
Row sums are A326888.
Cf. A001710, A254040, A309528, A305541.

\\ U(n,k) is A309528
U(n,k)={sumdiv(n, d, moebius(n/d)*(k^d/n - if(d%2, k^((d+1)/2), (k+1)*k^(d/2)/2)))/2}
T(n,k)={sum(i=2, k, (-1)^(k-i)*binomial(k,i)*U(n,i))} \\ Andrew Howroyd, Aug 12 2019

Let n>2. For any k >= floor(n/2) we have phi_k(C_n)=k! * Stirling2(n,k)/2n.
T(n, k) = Sum_{i=2..k} (-1)^(k-i)*binomial(k,i)*A309528(n, i). - Andrew Howroyd, Aug 12 2019
Column k is the Moebius transform of column k of A305541. - Andrew Howroyd, Sep 13 2019

cp's OEIS Frontend

A001037 Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

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A060223 Number of orbits of length n under the map whose periodic points are counted by A000670.

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A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

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A074143 a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).

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A032200 Number of rooted compound windmills (mobiles) of n nodes.

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A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

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