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%I #8 Jan 01 2023 14:45:41
%S 0,0,0,1,1,5,10,30,76,219,582,1662,4614,13080,36903,105098,298689,
%T 852734,2434660,6964349,19931147,57100177,163647811,469290004,
%U 1346225668,3863239150,11089085961,31838349956,91430943515,262615909503,754439588007,2167711283560
%N Number of n-node rooted trees whose node-height is less than their number of leaves.
%C Node-height is the number of nodes in the longest path from root to leaf.
%H Andrew Howroyd, <a href="/A358728/b358728.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 0 through a(7) = 10 trees:
%e . . . (ooo) (oooo) (ooooo) (oooooo)
%e ((oooo)) ((ooooo))
%e (o(ooo)) (o(oooo))
%e (oo(oo)) (oo(ooo))
%e (ooo(o)) (ooo(oo))
%e (oooo(o))
%e ((o)(ooo))
%e ((oo)(oo))
%e (o(o)(oo))
%e (oo(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],Depth[#]-1<Count[#,{},{-2}]&]],{n,1,10}]
%o (PARI) \\ Needs R(n,f) defined in A358589.
%o seq(n) = {Vec(R(n, (h,p)->sum(j=h+1, n-1, polcoef(p,j,y))), -n)} \\ _Andrew Howroyd_, Jan 01 2023
%Y These trees are ranked by A358727.
%Y For internals instead of node-height we have A358581, ordered A358585.
%Y The case of equality is A358589 (square trees), ranked by A358577.
%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 Cf. A109082, A109129, A185650, A358552, A358582-A358586, A358587, A358591.
%K nonn
%O 1,6
%A _Gus Wiseman_, Nov 29 2022
%E Terms a(19) and beyond from _Andrew Howroyd_, Jan 01 2023
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