A358728 - OEIS

%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