Search: id:A022553
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%I A022553
%S A022553 1,1,1,3,8,25,75,245,800,2700,9225,32065,112632,400023,1432613,5170575,
%T A022553 18783360,68635477,252085716,930138521,3446158600,12815663595,
%U A022553 47820414961,178987624513,671825020128,2528212128750,9536894664375
%N A022553 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.
%C A022553 Also number of asymmetric rooted plane trees with n+1 nodes (Christian 
               Bower).
%C A022553 Conjecturally, number of irreducible alternating Euler sums of depth 
               n and weight 3n.
%C A022553 a(n+1) is inverse Euler transform of A000108. Inverse Witt transform 
               of A006177.
%C A022553 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
%D A022553 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Cambridge, 1998, p. 336 (4.4.64)
%H A022553 D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the 
               enumeration of irreducible k-fold Euler sums and their roles in knot 
               theory and field theory</a>
%H A022553 J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/abs/math.CO/
               0511200">Hopf algebras and dendriform structures arising from parking 
               functions</a>
%H A022553 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A022553 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A022553 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to 
               Lyndon words</a>
%F A022553 prod_n (1-x^n)^{a[ n ]} = 2/(1+\sqrt{1-4x}); a[ n ] = (1/2n) sum_{d|n} 
               \mu(n/d) {2d choose d}. Also Moebius transform of A003239 (Christian 
               Bower).
%o A022553 (PARI) a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*binomial(2*d,d))/2/n)
%Y A022553 Cf. A003239, A005354, A000740. a(n)=A060165(n)/2.
%Y A022553 Cf. A007727, A086655.
%Y A022553 A diagonal of the square array described in A051168.
%Y A022553 Sequence in context: A093969 A006177 A148788 this_sequence A148789 A088327 
               A148790
%Y A022553 Adjacent sequences: A022550 A022551 A022552 this_sequence A022554 A022555 
               A022556
%K A022553 nonn
%O A022553 0,4
%A A022553 David Broadhurst (D.Broadhurst(AT)open.ac.uk)

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