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%I #50 Feb 08 2020 20:41:49
%S 1,1,1,3,8,26,76,247,783,2565,8447,28256,95168,323720,1108415,3821144,
%T 13246307,46158480,161574043,567925140,2003653016,7092953340,
%U 25186731980,89690452750,320221033370,1146028762599,4110596336036,14774346783745,53203889807764,191934931634880
%N Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.
%C Also the number of plane trees with n nodes where the sequence of branches directly under any given node has relatively prime run-lengths.
%H Andrew Howroyd, <a href="/A317852/b317852.txt">Table of n, a(n) for n = 1..200</a>
%e The a(5) = 8 locally aperiodic plane trees:
%e ((((o)))),
%e (((o)o)), ((o(o))), (((o))o), (o((o))),
%e ((o)oo), (o(o)o), (oo(o)).
%e The a(6) = 26 locally aperiodic plane trees:
%e (((((o))))) ((((o)o))) (((o)oo)) ((o)ooo)
%e (((o(o)))) ((o(o)o)) (o(o)oo)
%e ((((o))o)) ((oo(o))) (oo(o)o)
%e ((o((o)))) (((o)o)o) (ooo(o))
%e ((((o)))o) ((o(o))o)
%e (o(((o)))) (o((o)o))
%e (((o))(o)) (o(o(o)))
%e ((o)((o))) (((o))oo)
%e (o((o))o)
%e (oo((o)))
%e ((o)(o)o)
%e ((o)o(o))
%e (o(o)(o))
%t aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
%t aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[aperplane[n]],{n,10}]
%o (PARI)
%o Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
%o seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ _Andrew Howroyd_, Feb 08 2020
%Y Cf. A000108, A000837, A007853, A032171, A032200, A254040, A301700, A303386, A303431, A304173, A304175, A317708, A317852.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 05 2018
%E a(16)-a(17) from _Robert Price_, Sep 15 2018
%E Terms a(18) and beyond from _Andrew Howroyd_, Feb 08 2020
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