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A022553
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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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24
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 336 (4.4.64)
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LINKS
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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.
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).
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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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([a(n) for n in range(31)]) # Indranil Ghosh, Aug 05 2017
(Sage)
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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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David Broadhurst
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STATUS
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approved
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