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A358590 Number of square ordered rooted trees with n nodes.

%I A358590 #9 Jan 02 2023 00:06:12
%S A358590 1,0,1,0,6,5,36,84,309,890,3163,9835,32979,108252,360696,1192410,
%T A358590 3984552,13276769,44371368,148402665,497072593,1665557619,5586863093,
%U A358590 18750662066,62968243731,211565969511,711187790166,2391640404772,8045964959333,27077856222546
%N A358590 Number of square ordered rooted trees with n nodes.
%C A358590 We say that a tree is square if it has the same height as number of leaves.
%H A358590 Andrew Howroyd, <a href="/A358590/b358590.txt">Table of n, a(n) for n = 1..200</a>
%e A358590 The a(1) = 1 through a(6) = 5 ordered trees:
%e A358590   o  .  (oo)  .  ((o)oo)  ((o)(o)o)
%e A358590                  ((oo)o)  ((o)(oo))
%e A358590                  ((ooo))  ((o)o(o))
%e A358590                  (o(o)o)  ((oo)(o))
%e A358590                  (o(oo))  (o(o)(o))
%e A358590                  (oo(o))
%t A358590 aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A358590 Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,1,10}]
%o A358590 (PARI) \\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
%o A358590 R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1  + 1/(1 - A + O(x^n))); Z += f(h, A-p)); Z}
%o A358590 seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ _Andrew Howroyd_, Jan 01 2023
%Y A358590 For internals instead of height we have A000891, unordered A185650 aerated.
%Y A358590 For internals instead of leaves we have A358588, unordered A358587.
%Y A358590 The unordered version is A358589, ranked by A358577.
%Y A358590 A000108 counts ordered rooted trees, unordered A000081.
%Y A358590 A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
%Y A358590 A080936 counts ordered rooted trees by nodes and height, unordered A034781.
%Y A358590 A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
%Y A358590 Cf. A065097, A109129, A358371, A358552, A358579, A358586.
%K A358590 nonn
%O A358590 1,5
%A A358590 _Gus Wiseman_, Nov 25 2022
%E A358590 Terms a(16) and beyond from _Andrew Howroyd_, Jan 01 2023