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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. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

D. 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.

H. Munthe-Kaas and A. 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

J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, arXiv:math/0511200 [math.CO], 2005.

Y. Puri and T. 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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

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

(Python)

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 map(a, xrange(31)) # Indranil Ghosh, Aug 05 2017

CROSSREFS

Cf. A003239, A005354, A000740, A007727, A086655.

A diagonal of the square array described in A051168.

Sequence in context: A213439 A006177 A148788 * A148789 A088327 A148790

Adjacent sequences:  A022550 A022551 A022552 * A022554 A022555 A022556

KEYWORD

nonn

AUTHOR

David Broadhurst

STATUS

approved

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Last modified September 20 01:09 EDT 2017. Contains 292251 sequences.