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%I A032027 #15 Nov 15 2022 17:52:07
%S A032027 1,1,1,3,5,13,35,95,255,715,2081,6003,17645,52127,155863,468129,
%T A032027 1415521,4301055,13134789,40275109,123970669,382919917,1186475687,
%U A032027 3686899725,11487023793,35876838669,112304155021,352276801491
%N A032027 Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different.
%H A032027 Andrew Howroyd, <a href="/A032027/b032027.txt">Table of n, a(n) for n = 1..200</a>
%H A032027 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032027 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A032027 Shifts left under "AGK" (ordered, elements, unlabeled) transform.
%e A032027 From _Gus Wiseman_, Nov 15 2022: (Start)
%e A032027 The a(1) = 1 through a(6) = 13 ordered rooted identity trees (ranked by A358374):
%e A032027   o  (o)  ((o))  ((o)o)   (((o))o)   (((o)o)o)
%e A032027                  (o(o))   (((o)o))   ((o(o))o)
%e A032027                  (((o)))  ((o(o)))   (o((o)o))
%e A032027                           (o((o)))   (o(o(o)))
%e A032027                           ((((o))))  ((((o)))o)
%e A032027                                      ((((o))o))
%e A032027                                      ((((o)o)))
%e A032027                                      (((o))(o))
%e A032027                                      (((o(o))))
%e A032027                                      ((o)((o)))
%e A032027                                      ((o((o))))
%e A032027                                      (o(((o))))
%e A032027                                      (((((o)))))
%e A032027 (End)
%t A032027 aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A032027 Table[Length[Select[aot[n],FreeQ[#,_[__]?(!UnsameQ@@#&)]&]],{n,1,10}] (* _Gus Wiseman_, Nov 15 2022 *)
%o A032027 (PARI)
%o A032027 AGK(v)={apply(p->subst(serlaplace(y^0*p),y,1), Vec(prod(k=1, #v, (1 + x^k*y + O(x*x^#v))^v[k])-1, -#v))}
%o A032027 seq(n)={my(v=[1]); for(i=2, n, v=concat([1], AGK(v))); v} \\ _Andrew Howroyd_, Sep 20 2018
%Y A032027 The unordered version is A004111, ranked by A276625.
%Y A032027 These trees (ordered rooted identity) are ranked by A358374.
%Y A032027 Cf. A000081, A001190, A005043, A003238, A063895, A358377.
%K A032027 nonn,eigen
%O A032027 1,4
%A A032027 _Christian G. Bower_

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