%I #15 Nov 15 2022 17:52:07

%S 1,1,1,3,5,13,35,95,255,715,2081,6003,17645,52127,155863,468129,

%T 1415521,4301055,13134789,40275109,123970669,382919917,1186475687,

%U 3686899725,11487023793,35876838669,112304155021,352276801491

%N Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node are different.

%H Andrew Howroyd, <a href="/A032027/b032027.txt">Table of n, a(n) for n = 1..200</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F Shifts left under "AGK" (ordered, elements, unlabeled) transform.

%e From _Gus Wiseman_, Nov 15 2022: (Start)

%e The a(1) = 1 through a(6) = 13 ordered rooted identity trees (ranked by A358374):

%e o (o) ((o)) ((o)o) (((o))o) (((o)o)o)

%e (o(o)) (((o)o)) ((o(o))o)

%e (((o))) ((o(o))) (o((o)o))

%e (o((o))) (o(o(o)))

%e ((((o)))) ((((o)))o)

%e ((((o))o))

%e ((((o)o)))

%e (((o))(o))

%e (((o(o))))

%e ((o)((o)))

%e ((o((o))))

%e (o(((o))))

%e (((((o)))))

%e (End)

%t aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];

%t Table[Length[Select[aot[n],FreeQ[#,_[__]?(!UnsameQ@@#&)]&]],{n,1,10}] (* _Gus Wiseman_, Nov 15 2022 *)

%o (PARI)

%o 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 seq(n)={my(v=[1]); for(i=2, n, v=concat([1], AGK(v))); v} \\ _Andrew Howroyd_, Sep 20 2018

%Y The unordered version is A004111, ranked by A276625.

%Y These trees (ordered rooted identity) are ranked by A358374.

%Y Cf. A000081, A001190, A005043, A003238, A063895, A358377.

%K nonn,eigen

%O 1,4

%A _Christian G. Bower_