A007727 -id:A007727 - 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

A060165 Number of orbits of length n under the map whose periodic points are counted by A000984.

Original entry on oeis.org

2, 2, 6, 16, 50, 150, 490, 1600, 5400, 18450, 64130, 225264, 800046, 2865226, 10341150, 37566720, 137270954, 504171432, 1860277042, 6892317200, 25631327190, 95640829922, 357975249026, 1343650040256, 5056424257500, 19073789328750, 72108867614796

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A000984 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.
The number of n-cycles in the graph of overlapping m-permutations where n <= m. - Richard Ehrenborg, Dec 10 2013
a(n) is divisible by n (cf. A268619), 6*a(n) is divisible by n^2 (cf. A268592). - Max Alekseyev, Feb 09 2016
Apparently the number of Lyndon words of length n with a 4-letter alphabet (see A027377) where the first letter of the alphabet appears with the same frequency as the second of the alphabet. E.g a(1)=2 counts the words (2), (3), a(2)= 2 counts (01) (23), a(3)=6 counts (021) (031) (012) (013) (223) (233). R. J. Mathar, Nov 04 2021

			a(5) = 50 because if a map has A000984 as its periodic points, then it would have 2 fixed points and 252 points of period 5, hence 50 orbits of length 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1669
R. Ehrenborg, S. Kitaev and E. Steingrimsson, Number of cycles in the graph of 312-avoiding permutations, arXiv:1310.1520 [math.CO], 2013.
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. A000984, A007727, A060164, A060166, A060167, A060168, A060169, A060170, A060171, A060172, A060173.

with(numtheory):
a:= n-> add(mobius(n/d)*binomial(2*d, d), d=divisors(n))/n:
seq(a(n), n=1..30); # Alois P. Heinz, Dec 10 2013

a[n_] := (1/n)*Sum[MoebiusMu[d]*Binomial[2*n/d, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jul 16 2015 *)

a(n)=sumdiv(n,d,moebius(n/d)*binomial(2*d,d))/n \\ Charles R Greathouse IV, Dec 10 2013

from sympy import mobius, binomial, divisors
def a(n): return sum(mobius(n//d) * binomial(2*d, d) for d in divisors(n))//n
print([a(n) for n in range(1, 31)])  # Indranil Ghosh, Jul 24 2017

a(n) = (1/n) * Sum_{d|n} mu(d) A000984(n/d) with mu = A008683.
a(n) = 2*A022553(n).
a(n) = A007727(n)/n. - R. J. Mathar, Jul 24 2017
G.f.: 2 * Sum_{k>=1} mu(k)*log((1 - sqrt(1 - 4*x^k))/(2*x^k))/k. - Ilya Gutkovskiy, May 18 2019
a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 04 2022

A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 5, 10, 10, 5, 0, 0, 6, 12, 18, 12, 6, 0, 0, 7, 21, 35, 35, 21, 7, 0, 0, 8, 24, 56, 64, 56, 24, 8, 0, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Clark Kimberling

			For example, T(4,2) counts 1100,1001,0011,0110; T(2,1) counts 10, 01 (hence also counts 1010, 0101).
Rows:
  1;
  1,  1;
  0,  2,  0;
  0,  3,  3,  0;
  0,  4,  4,  4,  0;
  0,  5, 10, 10,  5,  0;

M. F. Hasler, A050186, rows 0..50, Sep 27 2018
M. F. Hasler, A050186, rows 0..100, Sep 27 2018
M. F. Hasler, A050186, rows 0..200, Sep 27 2018
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Same triangle as A053727 except this one includes column 0.

T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

T[n_, k_] := If[n == 0, 1, DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#]&]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 16 2022 *)

A050186(n,k)=sumdiv(gcd(n+!n,k),d,moebius(d)*binomial(n/d,k/d)) \\ M. F. Hasler, Sep 27 2018

MOEBIUS transform of A007318 Pascal's Triangle.

If rows n > 1 are divided by n, this yields the triangle A051168, which equals A245558 surrounded by 0's (except for initial terms). This differs from A011847 from row n = 9 on. - M. F. Hasler, Sep 29 2018

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

Original entry on oeis.org

12, 3, 4, 6, 12, 25, 60, 150, 400, 1107, 3180, 9386, 28404, 87711, 275764, 880470, 2849916, 9336508, 30918732, 103384758, 348725540, 1185630123, 4060210764, 13996354586, 48541672872, 169293988125, 593488622344, 2090567755278, 7396924802052, 26281018091013, 93738717046476, 335563502259798

Offset: 1

Max Alekseyev, Feb 07 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1500
R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632

Cf. A254593, A268618, A268619.

a[n_] := (6/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[2 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))*6/n^3; }

a(n) = A007727(n)*6/n^3 = A045630(n)*12/n^3 = A060165(n)*6/n^2 = A022553(n)*12/n^2 = A268619(n)*6/n.

For n == 0, 1, or 3 (mod 4), a(n) = 2*A254593(n); for n == 2 (mod 4), a(n) = 2*A254593(n) - A254593(n/2)/2.

A045630 Number of 2n-bead black-white complementable strings with n black beads and fundamental period 2n.

Original entry on oeis.org

1, 1, 2, 9, 32, 125, 450, 1715, 6400, 24300, 92250, 352715, 1351584, 5200299, 20056582, 77558625, 300533760, 1166803109, 4537542888, 17672631899, 68923172000, 269128935495, 1052049129142, 4116715363799, 16123800483072

Offset: 0

David W. Wilson

nonn

For n>0, a(n) is divisible by n (cf. A022553), 2*a(n) is divisible by n^2 (cf. A268619), and 12*a(n) is divisible by n^3 (cf. A268592). - Max Alekseyev, Feb 07 2016

Cf. A007727.

Moebius transform of A001700(n-1). - Christian G. Bower.

For n>0, a(n) = A007727(n)/2 = n * A022553(n).

A045662 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse.

Original entry on oeis.org

1, 2, 4, 6, 32, 50, 204, 266, 1024, 1224, 4900, 5522, 21600, 23998, 95508, 102750, 409600, 437546, 1747152, 1847522, 7380000, 7758870, 31027876, 32449826, 129752064, 135207500, 540783100, 561628620, 2246337184, 2326762742

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A007727, A045663, A045664, A045680.

a[n_] := If[n == 0, 1, 2n Sum[MoebiusMu[n/d] Binomial[d - Mod[d, 2], Quotient[d, 2]], {d, Divisors[n]}]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, from PARI *)

a(n) = if(n<1, n==0, 2*n*sumdiv(n, d, moebius(n/d) * binomial(d-d%2, d\2))); \\ Andrew Howroyd, Sep 14 2019

a(n) = 2*n*A045680(n) for n > 0.

a(n) = 2*n*Sum_{d|n} mu(n/d) * binomial(2*floor(d/2), floor(d/2)) for n > 0. - Andrew Howroyd, Sep 14 2019

A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

Original entry on oeis.org

1, 2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4080, 8190, 16380, 32730, 65536, 131070, 262080, 524286, 1048560, 2097018, 4194300, 8388606, 16776960, 33554400, 67108860, 134217216, 268435440, 536870910, 1073740740, 2147483646

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000048, A007727, A045662, A045664, A064355.

a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 23 2019 *)

a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ Andrew Howroyd, Sep 14 2019

from sympy import mobius, divisors
def A045663(n): return sum(mobius(d)<>(~n&n-1).bit_length(),generator=True)) if n else 1 # Chai Wah Wu, Jul 22 2024

a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.

a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - Andrew Howroyd, Sep 14 2019

A045664 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement.

Original entry on oeis.org

1, 2, 4, 18, 48, 150, 324, 882, 1920, 4536, 9900, 22506, 48240, 106470, 227556, 490950, 1044480, 2228190, 4708368, 9961434, 20950800, 44037378, 92229588, 192937938, 402549120, 838860000, 1744617420, 3623864832, 7515733680

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000740, A001037, A007727, A027375, A045662, A045663.

a[n_] := If[n == 0, 1, 2n Sum[MoebiusMu[n/d] 2^(d-1), {d, Divisors[n]}]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, from PARI *)

a(n)={if(n<1, n==0, 2*n*sumdiv(n, d, moebius(n/d)*2^(d-1)))} \\ Andrew Howroyd, Sep 14 2019

For n >= 1, a(n) = n*A027375(n) = 2*n*A000740(n) = n^2*A001037(n).

a(n) = 2*n*Sum_{d|n} mu(n/d)*2^(d-1) for n > 0. - Andrew Howroyd, Sep 14 2019

A045665 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse, complement and reversed complement.

Original entry on oeis.org

1, 2, 4, 6, 16, 30, 36, 98, 128, 252, 300, 682, 720, 1638, 1764, 3690, 4096, 8670, 9072, 19418, 20400, 42630, 45012, 94162, 97920, 204600, 212940, 441504, 458640, 950214, 981900, 2031554, 2097152, 4323198, 4456380, 9174270, 9434880

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A007727, A045662, A045663, A045664, A045674, A045683.

a(n)={if(n<1, n==0, n*sumdiv(n, d, if(d%2, moebius(d)*2^((n/d+1)\2))))} \\ Andrew Howroyd, Oct 01 2019

a(n) = 2*n*A045683(n) for n > 0.

a(n) = n * Sum_{d|n, d odd} mu(d) * 2^ceiling(n/(2*d)) for n > 0.

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).

Original entry on oeis.org

2, 1, 2, 4, 10, 25, 70, 200, 600, 1845, 5830, 18772, 61542, 204659, 689410, 2347920, 8074762, 28009524, 97909318, 344615860, 1220539390, 4347310451, 15564141262, 55985418344, 202256970300, 733607281875, 2670698800548, 9755982857964, 35751803209918, 131405090455065, 484316704740126, 1789672012052256

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

6*a(n) is divisible by n (cf. A268592).

Cf. A000984, A007727, A008683, A022553, A045630, A060165, A268592, A268617.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[2*#, #] &] / n^2; Array[a, 35] (* Amiram Eldar, Aug 24 2023 *)

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

a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A000984(d).

a(n) = A007727(n)/n^2 = A045630(n)*2/n^2 = A060165(n)/n = A022553(n)*2/n.

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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A060165 Number of orbits of length n under the map whose periodic points are counted by A000984.

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A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

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A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

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A045630 Number of 2n-bead black-white complementable strings with n black beads and fundamental period 2n.

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A045662 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reverse.

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A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

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Python

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A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).