A022553 -id:A022553 - cp's OEIS frontend

A292287 Number of multisets of exactly n nonempty balanced binary Lyndon words with a total of 4n letters (2n zeros and 2n ones).

1, 1, 4, 12, 43, 142, 508, 1781, 6414, 23124, 84296, 308613, 1137129, 4207456, 15636927, 58322808, 218272766, 819319778, 3083913810, 11636761924, 44010780075, 166802192488, 633420816341, 2409731688860, 9182826866499, 35048239457878, 133965833871427

Offset: 0

Alois P. Heinz, Sep 20 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1667
Index entries for sequences related to Lyndon words

Cf. A022553, A289978.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(
      mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

g[n_] := g[n] = If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2023, after Alois P. Heinz *)

G.f.: Product_{j>=1} 1/(1-x^j)^A022553(j+1).

a(n) = A289978(2n,n).

A066316 Number of identity (asymmetric) bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

Original entry on oeis.org

1, 5, 24, 91, 340, 1224, 4365, 15521, 55311, 197964, 712243, 2577105, 9375360, 34284971, 125977464, 464938189, 1722817415, 6407307543, 23909159417, 89491715105, 335908316844, 1264097675775, 4768430557020, 18027183349395

Offset: 4

Christian G. Bower, Dec 13 2001

nonn

Index entries for sequences related to bracelets

a(n)=A022553(n)-A066315(n)

A131764 Inverse Euler transform of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 3, 10, 30, 102, 335, 1170, 4080, 14560, 52377, 190650, 698870, 2581110, 9586395, 35791358, 134215680, 505290270, 1908866960, 7233629130, 27487764474, 104715392730, 399822314775, 1529755308210, 5864061663920, 22517998136832, 86607683851185, 333599972392960, 1286742745883790, 4969489243995030, 19215358392200893, 74382032555280450, 288230376084602880

Offset: 0

F. Chapoton, Oct 04 2007

nonn

This is the sequence of dimensions of a free Lie algebra on some specific set of generators.

			2*x + 3*x^2 + 10*x^3 + 30*x^4 + 102*x^5 + 335*x^6 + 1170*x^7 + 4080*x^8 + ...
(1-x)^(-2)*(1-x^2)^(-3)*(1-x^3)^(-10)*(1-x^4)^(-30)*(1-x^5)^(-102) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 + ... .

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A022553, A000984, A000108.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> binomial(2*n, n)):
seq(a(n), n = 0..32); # Peter Luschny, Nov 21 2022

a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 20 2017 *)

a(n):=proc(n) begin 1/n*_plus(moebius(n/d)*2^(2*d-1)$d in divisors(n)) end;

a(n)=sumdiv(n,d,1/n*moebius(n/d)*2^(d*2-1)); /* Joerg Arndt, Jul 06 2011 */

{a(n) = local(A); if( n<1, 0, A = sqrt(1 - 4*x + x * O(x^n)); for( k=1, n-1, A *= (1 - x^k + x * O(x^n))^ polcoeff( A, k)); -polcoeff( A, n))} /* Michael Somos, Apr 01 2012 */

a(n) = (1/n) * Sum_{d|n} moebius(n/d)*2^(2*d-1) for n > 0, a(0) = 1.

a(n) ~ 2^(2*n-1) / n. - Vaclav Kotesovec, Oct 09 2019

More explicit definition from Michael Somos, Apr 01 2012. - N. J. A. Sloane, Feb 20 2017

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.

Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Lyndon words

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023

A130513 Subtriangle of triangle in A051168: remove central column of A051168 and all columns to the right; now read by upwards diagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 2, 1, 0, 14, 7, 3, 1, 0, 42, 20, 9, 3, 1, 0, 132, 66, 30, 12, 4, 1, 0, 429, 212, 99, 40, 15, 4, 1, 0, 1430, 715, 333, 143, 55, 18, 5, 1, 0, 4862, 2424, 1144, 497, 200, 70, 22, 5, 1, 0, 16796, 8398, 3978, 1768, 728, 273, 91, 26, 6, 1, 0, 58786, 29372, 13995

Offset: 1

Philippe Deléham, Aug 08 2007

			Triangle T(n,k), 1<=k<=n, begins:
1;
1, 0;
2, 1, 0;
5, 2, 1, 0;
14, 7, 3, 1, 0;
42, 20, 9, 3, 1, 0;
132, 66, 30, 12, 4, 1, 0;
429, 212, 99, 40, 15, 4, 1, 0;

A. Errera, Analysis situs: Un problème d'énumération, Memoires Acad. Bruxelles (1931), Serie 2, Vol. 11, No. 6, 26pp.

Cf. A000108, A000150, A050181, A050182, A050183, A050184, A050185.

Table[1/(2n-k) Plus@@ (MoebiusMu[ # ]Binomial[(2n-k)/#,(n-k)/# ]&/@ Divisors[GCD[2n-k,n-k]]),{n,12},{k,n}] (* Wouter Meeussen, Jul 20 2008 *)

Sum_{k, 1<=k<=n} T(n,k) = A022553(n); Sum_{k, 1<=k<=n}k*T(n,k) = A002996(n).

T(n,k) = 1/(2n-k) Sum( d | gcd(2n-k,n-k) = mu(d) C((2n-k)/d,(n-k)/d) ). - Wouter Meeussen, Jul 20 2008

Edited by N. J. A. Sloane, Oct 08 2007

cp's OEIS Frontend

A292287 Number of multisets of exactly n nonempty balanced binary Lyndon words with a total of 4n letters (2n zeros and 2n ones).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A066316 Number of identity (asymmetric) bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

Views

Author

Keywords

Links

Formula

A131764 Inverse Euler transform of central binomial coefficients A000984.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

MuPAD

PARI

PARI

Formula

Extensions

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A130513 Subtriangle of triangle in A051168: remove central column of A051168 and all columns to the right; now read by upwards diagonals.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

Extensions