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A358581 Number of rooted trees with n nodes, most of which are leaves.

%I A358581 #15 Dec 31 2022 15:05:08
%S A358581 1,0,1,1,4,5,20,28,110,169,663,1078,4217,7169,27979,49191,191440,
%T A358581 345771,1341974,2477719,9589567,18034670,69612556,132984290,511987473,
%U A358581 991391707,3807503552,7460270591,28585315026,56595367747,216381073935,432396092153
%N A358581 Number of rooted trees with n nodes, most of which are leaves.
%H A358581 Andrew Howroyd, <a href="/A358581/b358581.txt">Table of n, a(n) for n = 1..200</a>
%F A358581 A358581(n) + A358584(n) = A000081(n).
%F A358581 A358582(n) + A358583(n) = A000081(n).
%F A358581 a(n) = Sum_{k=floor(n/2)+1..n} A055277(n, k). - _Andrew Howroyd_, Dec 31 2022
%e A358581 The a(1) = 1 through a(7) = 20 trees:
%e A358581   o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
%e A358581                      ((ooo))  ((oooo))  ((ooooo))
%e A358581                      (o(oo))  (o(ooo))  (o(oooo))
%e A358581                      (oo(o))  (oo(oo))  (oo(ooo))
%e A358581                               (ooo(o))  (ooo(oo))
%e A358581                                         (oooo(o))
%e A358581                                         (((oooo)))
%e A358581                                         ((o)(ooo))
%e A358581                                         ((o(ooo)))
%e A358581                                         ((oo)(oo))
%e A358581                                         ((oo(oo)))
%e A358581                                         ((ooo(o)))
%e A358581                                         (o((ooo)))
%e A358581                                         (o(o)(oo))
%e A358581                                         (o(o(oo)))
%e A358581                                         (o(oo(o)))
%e A358581                                         (oo((oo)))
%e A358581                                         (oo(o)(o))
%e A358581                                         (oo(o(o)))
%e A358581                                         (ooo((o)))
%t A358581 art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A358581 Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>Count[#,_[__],{0,Infinity}]&]],{n,0,10}]
%o A358581 (PARI) \\ See A358584 for R(n).
%o A358581 seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\2+1..#u]))} \\ _Andrew Howroyd_, Dec 31 2022
%Y A358581 For equality we have A185650 aerated, ranked by A358578.
%Y A358581 The opposite version is A358582, non-strict A358584.
%Y A358581 The non-strict version is A358583.
%Y A358581 The ordered version is A358585, odd-indexed terms A065097.
%Y A358581 A000081 counts rooted trees, ordered A000108.
%Y A358581 A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A358581 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358581 A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
%Y A358581 A358589 counts square trees, ranked by A358577, ordered A358590.
%Y A358581 Cf. A000891, A109129, A342507, A358579, A358580, A358586, A358591.
%K A358581 nonn
%O A358581 1,5
%A A358581 _Gus Wiseman_, Nov 23 2022
%E A358581 Terms a(19) and beyond from _Andrew Howroyd_, Dec 31 2022