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%I #13 Oct 08 2017 13:12:32
%S 1,1,1,3,6,16,45,128,371,1106,3343,10230,31635,98714,310366,982437,
%T 3128051,10011848,32193840,103955571,336946034,1095873115,3575319049,
%U 11697938232,38374479841,126190075741,415889689954,1373506798548
%N Number of rooted ordered trees where no branch is identical to its adjacent neighbor.
%H Alois P. Heinz, <a href="/A106361/b106361.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Shifts left under Carlitz transform.
%F Carlitz transform T(A(x)) has g.f. 1/(1-sum(k>0, (-1)^(k+1)*A(x^k))).
%F From _Petros Hadjicostas_, Sep 17 2017: (Start)
%F The following results are simple consequences of the fact that the sequence shifts left under the Carlitz transform.
%F a(n) = Sum_{1 <= s <= n-1} a(n-s)*b(s) for n>=2, where b(n) = Sum_{m|n} (-1)^{1+(n/m)} a(m), with a(1) = 1.
%F If A(x) = Sum_{n>=1} a(n)*x^n, then A(x)-x = A(x)*Sum_{n>=1} a(n)*x^n/(1+x^n).
%F (End)
%Y Cf. A000081, A106362, A106363.
%K nonn,eigen
%O 1,4
%A _Christian G. Bower_, Apr 29 2005
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