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%I #10 Jan 01 2023 14:46:05
%S 0,1,0,2,1,6,7,26,43,135,276,755,1769,4648,11406,29762,75284,195566,
%T 503165,1310705,3402317,8892807,23231037,60906456,159786040,420144405,
%U 1105673058,2914252306,7688019511,20304253421,53667498236,141976081288,375858854594,995728192169
%N Number of n-node rooted trees of edge-height equal to their number of leaves.
%C Edge-height (A109082) is the number of edges in the longest path from root to leaf.
%H Andrew Howroyd, <a href="/A358723/b358723.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 0 through a(7) = 7 trees:
%e . (o) . ((oo)) ((o)(o)) (((ooo))) (((o))(oo))
%e (o(o)) ((o(oo))) (((o)(oo)))
%e ((oo(o))) ((o)((oo)))
%e (o((oo))) ((o)(o(o)))
%e (o(o(o))) ((o(o)(o)))
%e (oo((o))) (o((o)(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[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]
%o (PARI) \\ Needs R(n,f) defined in A358589.
%o seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ _Andrew Howroyd_, Jan 01 2023
%Y For internals instead of leaves: A011782, ranked by A209638.
%Y For internals instead of edge-height: A185650 aerated, ranked by A358578.
%Y For node-height: A358589 (square trees), ranked by A358577, ordered 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 internals, ordered A090181.
%Y Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.
%K nonn
%O 1,4
%A _Gus Wiseman_, Nov 29 2022
%E Terms a(19) and beyond from _Andrew Howroyd_, Jan 01 2023
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