A060165 -id:A060165 - cp's OEIS frontend

A022553 Number of binary Lyndon words containing n letters of each type; periodic binary sequences of period 2n with n zeros and n ones in each period.

1, 1, 1, 3, 8, 25, 75, 245, 800, 2700, 9225, 32065, 112632, 400023, 1432613, 5170575, 18783360, 68635477, 252085716, 930138521, 3446158600, 12815663595, 47820414961, 178987624513, 671825020128, 2528212128750, 9536894664375, 36054433807398, 136583760011496

Offset: 0

David Broadhurst

Also number of asymmetric rooted plane trees with n+1 nodes. - Christian G. Bower
Conjecturally, number of irreducible alternating Euler sums of depth n and weight 3n.
a(n+1) is inverse Euler transform of A000108. Inverse Witt transform of A006177.
Dimension of the degree n part of the primitive Lie algebra of the Hopf algebra CQSym (Catalan Quasi-Symmetric functions). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 22 2006
For n>0, 2*a(n) is divisible by n (cf. A268619), 12*a(n) is divisible by n^2 (cf. A268592). - Max Alekseyev, Feb 09 2016

			a(3)=3 counts 6-periodic 000111, 001011 and 001101. a(4)=8 counts 00001111, 00010111, 00011011, 00011101, 00100111, 00101011, 00101101, and 00110101. - _R. J. Mathar_, Oct 20 2021

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 336 (4.4.64)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. J. H. Al-Kaabi, Monomial Bases for Free Post-Lie Algebras, IOP Conf. Ser.: Mater. Sci. Eng. (2020) Vol. 871, 012048.
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 pp. 3, 6.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
Gilbert Labelle and Pierre Leroux, Enumeration of (uni- or bicolored) plane trees according to their degree distribution, Disc. Math. 157 (1996) 227-240, Eq. (1.20).
Hans Munthe-Kaas and Alexander Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames, arXiv preprint arXiv:1203.4738 [math.NA], 2012. - From _N. J. A. Sloane_, Sep 20 2012
Jean-Christophe Novelli and Jean-Yves Thibon, Hopf algebras and dendriform structures arising from parking functions, arXiv:math/0511200 [math.CO], 2005.
Tilman Piesk, List of 2n-bead balanced binary Lyndon words for n=1...8
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for sequences related to rooted trees
Index entries for sequences related to Lyndon words

Cf. A003239, A005354, A000740, A007727, A086655, A289978 (multiset trans.), A001037 (binary Lyndon wds.), A074655 (3 letters), A074656 (4 letters).

A diagonal of the square array described in A051168.

with(numtheory):
a:= n-> `if`(n=0, 1,
        add(mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 21 2011

a[n_] := Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n); a[0] = 1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 02 2015 *)

a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*binomial(2*d,d))/2/n)

from sympy import mobius, binomial, divisors
def a(n):
    return 1 if n == 0 else sum(mobius(n//d)*binomial(2*d, d) for d in divisors(n))//(2*n)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 05 2017

def a(n):
    return 1 if n ==0 else sum(moebius(n//d)*binomial(2*d, d) for d in divisors(n))//(2*n)
# F. Chapoton, Apr 23 2020

a(n) = A060165(n)/2 = A007727(n)/(2*n) = A045630(n)/n.
Product_n (1-x^n)^a(n) = 2/(1+sqrt(1-4*x));  a(n) = 1/(2*n) * Sum_{d|n} mu(n/d)*C(2*d,d). Also Moebius transform of A003239. - Christian G. Bower
a(n) ~ 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2014
G.f.: 1 + Sum_{k>=1} mu(k)*log((1 - sqrt(1 - 4*x^k))/(2*x^k))/k. - Ilya Gutkovskiy, May 18 2019

A007727 Number of 2n-bead black-white strings with n black beads and fundamental period 2n.

Original entry on oeis.org

1, 2, 4, 18, 64, 250, 900, 3430, 12800, 48600, 184500, 705430, 2703168, 10400598, 40113164, 155117250, 601067520, 2333606218, 9075085776, 35345263798, 137846344000, 538257870990, 2104098258284, 8233430727598, 32247600966144

Offset: 0

Doug Bowman, bowman(AT)math.uiuc.edu

nonn

For n>0, a(n) is divisible by n^2 (cf. A268619) and 6*a(n) is divisible by n^3 (cf. A268592). - Max Alekseyev, Feb 07 2016

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A045630, A060165, A022553.

A007727 := proc(n)
    if n = 0 then
        1;
    else
        add(numtheory[mobius](n/d)*binomial(2*d,d), d =numtheory[divisors](n)) ;
    end if ;
end proc:
seq(A007727(n),n=0..10) ; # R. J. Mathar, Nov 10 2021

a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 05 2023 *)

{ a(n) = if(n>0,sumdiv(n, d, moebius(n/d)*binomial(2*d, d)),0); }

For n>0, a(n) = Sum_{d|n} A008683(n/d)*A000984(d).
For n>0, a(n) = 2 * A045630(n).
a(0)=1, a(n) = n * A060165(n) = 2n * A022553(n). - Ralf Stephan, Sep 01 2003

Edited by Max Alekseyev, Feb 09 2016

A115141 Convolution of A115140 with itself.

Original entry on oeis.org

1, -2, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, -58786, -208012, -742900, -2674440, -9694845, -35357670, -129644790, -477638700, -1767263190, -6564120420, -24466267020, -91482563640, -343059613650, -1289904147324, -4861946401452, -18367353072152

Offset: 0

Wolfdieter Lang, Jan 13 2006

This is the so-called A-sequence for the Riordan triangles A053122, A110162, A129818, A158454 and signed A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. Wolfdieter Lang, Dec 20 2010. [Revised, Nov 13 2012, Nov 22 2012 and Oct 22 2019]

a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - Wolfdieter Lang, Jun 26 2011

			G.f. = 1 - 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A115140, A115142 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

a[n_] := -First[ ListConvolve[ cc = Array[ CatalanNumber, n-1, 0], cc]]; a[0] = 1; a[1] = -2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 21 2011 *)
CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

{a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */

((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x) )/(2*x) of A000108 (Catalan numbers).
a(0)=1, a(1)=-2, a(n) = -C(n-1), n>=2, with C(n):=A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
The convolution inverse is A000108(x)^2. - Michael Somos, Mar 28 2012
REVERT transform is A069271. - Michael Somos, Mar 28 2012
EULER transform of -A060165. - Michael Somos, Mar 28 2012
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A060170 Number of orbits of length n under the map whose periodic points are counted by A005809.

Original entry on oeis.org

3, 6, 27, 120, 600, 3078, 16611, 91872, 520749, 3004200, 17594247, 104304888, 624801957, 3775722342, 22991161500, 140928011136, 868886416866, 5384796881850, 33525472069563, 209592223788000, 1315211209630794, 8281053081282894, 52301607644921259, 331260902534858976, 2103541885645955625, 13389670112374830378

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A005809 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

a(n) is divisible by n (cf. A268617), 2*a(n) is divisible by n^2 (cf. A268618).

			a(3) = 27 since a map whose periodic points are counted by A005809 has 3 fixed points and 84 points of period 3, hence 27 orbits of length 3.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A005809, A060164, A060165, A060166, A060167, A060168, A060179, A060171, A060171, A060172, A060173.

a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d, d))/n; \\ Michel Marcus, Sep 10 2017

a(n) = (1/n)* Sum_{d|n} A008683(n/d)*A005809(d).

Edited by Max Alekseyev, Feb 09 2016

A060171 Number of orbits of length n under a map whose periodic points seem to be counted by A006953.

Original entry on oeis.org

12, 54, 80, 30, 24, 5400, 0, 990, 1568, 636, 24, 2720, 0, 240, 5704, 510, 0, 3835776, 0, 26724, 3600, 108, 24, 89760, 0, 240, 1064, 120, 24, 113569300, 0, 510, 11752, 0, 264, 278281640

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A006953 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(3) = 80 since a map whose periodic points are counted by A006953 has 12 fixed points and 252 points of period 3, hence 80 orbits of length 3.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A006953, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060172, A060173.

a(n) = sumdiv(n, d, moebius(d)*denominator(bernfrac(2*n/d)/(2*n/d)))/n; \\ Michel Marcus, Sep 10 2017

a(n) = (1/n)* Sum_{d|n} mu(d)*A006953(n/d).

A060164 Number of orbits of length n under the map whose periodic points are counted by A000364.

Original entry on oeis.org

1, 2, 20, 345, 10104, 450450, 28480140, 2423938845, 267208852820, 37037118818700, 6304443126648900, 1292877846962865230, 314390193022547991720, 89447117243116404721950, 29436259549934873636908816, 11094961973721205588579579845, 4748429366816935180127543967840

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A000364 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(3) = 20 since the conjectured map whose periodic points are counted by A000364 would have 1 fixed point and 61 points of period 3, so it must have 20 orbits of length 3.

Alois P. Heinz, Table of n, a(n) for n = 1..243
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A000364, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060172, A060173.

a(n) = (1/n)* Sum_{d|n} mu(d)*A000364(n/d).

A006173 Witt vector *2!.

Original entry on oeis.org

2, 1, 4, 13, 44, 135, 472, 1492, 5324, 17405, 63944, 215096, 799416, 2752909, 10310384, 36443256, 137263244, 489166324, 1860249448, 6739795717, 25596173800, 93596253769, 357974884304, 1319325363658, 5056389932088

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

The Somos transform sends sequence {a(n)} to sequence with g.f. Product_{i=1..n} 1/(1-a(i)*x^i).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Cf. A006177, A006973.

Inverse Somos transform of A000108 shifted left. - Wouter Meeussen, Aug 20 2002

Witt transform of A060165. - Sean A. Irvine, Jan 15 2017

Edited by Christian G. Bower, Aug 20 2002

A060166 Number of orbits of length n under the map whose periodic points are counted by A001641.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 10, 17, 26, 44, 68, 115, 184, 306, 500, 835, 1374, 2301, 3822, 6409, 10718, 18028, 30280, 51077, 86130, 145641, 246370, 417600, 708246, 1203069, 2045010, 3480408, 5927660, 10105819, 17241140, 29439580, 50302162, 86012630, 147166248, 251963055, 431633348

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001641 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			a(7) = 7 since a map whose periodic points are counted by A001641 would have 1 fixed point and 50 points of period 7, hence 7 orbits of length 7.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001641, A060164, A060165, A060167, A060168, A060169, A060170, A060171, A060171.

a001641(n)=if(n<0, 0, polcoeff(x*(1+2*x+4*x^3)/(1-x-x^2-x^4)+x*O(x^n), n))
a(n) = sumdiv(n, d, moebius(d)*a001641(n/d))/n; \\ Michel Marcus, Sep 10 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001641(n/d).

More terms from Michel Marcus, Sep 10 2017

A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 9, 13, 23, 36, 63, 101, 175, 290, 497, 840, 1445, 2460, 4247, 7293, 12619, 21805, 37856, 65695, 114401, 199280, 347944, 607959, 1064130, 1864083, 3269948, 5740840, 10090148, 17748870, 31250297, 55063603, 97102485, 171355485, 302605780, 534729160, 945513850

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001642 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(7) = 9 since a map whose periodic points are counted by A001642 would have 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060168, A060169, A060170, A060171, A060171.

a001642(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4)/(1-x-x^2-x^4-x^5)+x*O(x^n), n));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001642(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001642(n/d).

More terms from Michel Marcus, Sep 11 2017

A060168 Number of orbits of length n under the map whose periodic points are counted by A001643.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 10, 15, 26, 42, 74, 121, 212, 357, 620, 1064, 1856, 3209, 5618, 9794, 17192, 30153, 53114, 93554, 165308, 292250, 517802, 918207, 1630932, 2899434, 5161442, 9196168, 16402764, 29281168, 52319364, 93555601, 167427844, 299841117, 537357892, 963641588, 1729192432

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001643 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(7) = 10 since a map whose periodic points are counted by A001643 would have 1 fixed point and 71 points of period 7, hence 10 orbits of length 7.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060167, A060169, A060170, A060171, A060171.

a001643(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4+6*x^5)/(1-x-x^2-x^4-x^5-x^6)+x*O(x^n), n))
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001643(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001643(n/d).

More terms from Michel Marcus, Sep 11 2017

cp's OEIS Frontend

A022553 Number of binary Lyndon words containing n letters of each type; periodic binary sequences of period 2n with n zeros and n ones in each period.

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A007727 Number of 2n-bead black-white strings with n black beads and fundamental period 2n.

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A115141 Convolution of A115140 with itself.

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A060170 Number of orbits of length n under the map whose periodic points are counted by A005809.

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A060171 Number of orbits of length n under a map whose periodic points seem to be counted by A006953.

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A060164 Number of orbits of length n under the map whose periodic points are counted by A000364.

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A006173 Witt vector *2!.

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