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A022553
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Number of 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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16
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Also number of asymmetric rooted plane trees with n+1 nodes (Christian 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
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REFERENCES
| 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
| D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory
J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions
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
| prod_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 Bower).
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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);
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PROG
| (PARI) a(n)=if(n<1, n==0, sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/2/n)
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CROSSREFS
| Cf. A003239, A005354, A000740. a(n)=A060165(n)/2.
Cf. A007727, A086655.
A diagonal of the square array described in A051168.
Sequence in context: A093969 A006177 A148788 * A148789 A088327 A148790
Adjacent sequences: A022550 A022551 A022552 * A022554 A022555 A022556
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KEYWORD
| nonn
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AUTHOR
| David Broadhurst (D.Broadhurst(AT)open.ac.uk)
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