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%I #12 Jan 01 2023 12:34:46
%S 1,0,1,0,3,2,11,17,55,107,317,720,1938,4803,12707,32311,85168,220879,
%T 581112,1522095,4014186,10568936,27934075,73826753,195497427,
%U 517927859,1373858931,3646158317,9684878325,25737819213,68439951884,182070121870,484583900955,1290213371950
%N Number of square rooted trees with n nodes.
%C We say that a tree is square if it has the same height as number of leaves.
%H Andrew Howroyd, <a href="/A358589/b358589.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 1 through a(7) = 11 trees:
%e o . (oo) . ((ooo)) ((o)(oo)) (((oooo)))
%e (o(oo)) (o(o)(o)) ((o(ooo)))
%e (oo(o)) ((oo(oo)))
%e ((ooo(o)))
%e (o((ooo)))
%e (o(o(oo)))
%e (o(oo(o)))
%e (oo((oo)))
%e (oo(o(o)))
%e (ooo((o)))
%e ((o)(o)(o))
%t art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[Select[art[n],Count[#,{},{0,Infinity}]==Depth[#]-1&]],{n,1,10}]
%o (PARI) \\ R(n,f) enumerates trees by height(h), nodes(x) and leaves(y).
%o R(n,f) = {my(A=O(x*x^n), Z=0); for(h=1, n, my(p = A); A = x*(y - 1 + exp( sum(i=1, n-1, 1/i * subst( subst( A + O(x*x^((n-1)\i)), x, x^i), y, y^i) ) )); Z += f(h, A-p)); Z}
%o seq(n) = {Vec(R(n, (h,p)->polcoef(p,h,y)), -n)} \\ _Andrew Howroyd_, Jan 01 2023
%Y For internals instead of height we have A185650 aerated, ranked by A358578.
%Y These trees are ranked by A358577.
%Y For internals instead of leaves we have A358587, ranked by A358576.
%Y The ordered version is A358590.
%Y A000081 counts rooted trees, ordered A000108.
%Y A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
%Y Cf. A000891, A065097, A109129, A358552, A358588, A358591.
%K nonn
%O 1,5
%A _Gus Wiseman_, Nov 23 2022
%E Terms a(19) and beyond from _Andrew Howroyd_, Jan 01 2023
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