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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. 24
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)



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


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

G. Labelle, P. Leroux, Enumeration of (uni- or bicolored) plane trees according to their degree distribution, Disc. Math. 157 (1996) 227-240, Eq. (1.20)


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


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



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-Fran├žois Alcover, Feb 02 2015 *)


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


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

A diagonal of the square array described in A051168.

Sequence in context: A006177 A148788 A292778 * A292884 A148789 A088327

Adjacent sequences:  A022550 A022551 A022552 * A022554 A022555 A022556




David Broadhurst



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Last modified March 17 18:33 EDT 2018. Contains 300573 sequences. (Running on oeis4.)