cp's OEIS Frontend

A358583 Number of rooted trees with n nodes, at least half of which are leaves.

%I A358583 #9 Dec 31 2022 15:05:20
%S A358583 1,1,1,3,4,13,20,67,110,383,663,2346,4217,15118,27979,101092,191440,
%T A358583 695474,1341974,4893067,9589567,35055011,69612556,254923825,511987473,
%U A358583 1877232869,3807503552,13972144807,28585315026,104955228432,216381073935,794739865822
%N A358583 Number of rooted trees with n nodes, at least half of which are leaves.
%H A358583 Andrew Howroyd, <a href="/A358583/b358583.txt">Table of n, a(n) for n = 1..200</a>
%F A358583 A358581(n) + A358584(n) = A000081(n).
%F A358583 A358582(n) + A358583(n) = A000081(n).
%F A358583 a(n) = Sum_{k=floor((n-1)/2)+1..n} A055277(n, k). - _Andrew Howroyd_, Dec 31 2022
%e A358583 The a(1) = 1 through a(6) = 13 trees:
%e A358583   o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
%e A358583                 ((oo))  ((ooo))  ((oooo))
%e A358583                 (o(o))  (o(oo))  (o(ooo))
%e A358583                         (oo(o))  (oo(oo))
%e A358583                                  (ooo(o))
%e A358583                                  (((ooo)))
%e A358583                                  ((o)(oo))
%e A358583                                  ((o(oo)))
%e A358583                                  ((oo(o)))
%e A358583                                  (o((oo)))
%e A358583                                  (o(o)(o))
%e A358583                                  (o(o(o)))
%e A358583                                  (oo((o)))
%t A358583 art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A358583 Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>=Count[#,_[__],{0,Infinity}]&]],{n,1,10}]
%o A358583 (PARI) \\ See A358584 for R(n).
%o A358583 seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[(n-1)\2+1..#u]))} \\ _Andrew Howroyd_, Dec 31 2022
%Y A358583 For equality we have A185650 aerated, ranked by A358578.
%Y A358583 The strict case is A358581.
%Y A358583 The opposite version is A358584, strict A358582.
%Y A358583 The ordered version is A358586, strict A358585.
%Y A358583 A000081 counts rooted trees, ordered A000108.
%Y A358583 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358583 A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
%Y A358583 A358589 counts square rooted trees, ranked by A358577, ordered A358590.
%Y A358583 Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358586.
%K A358583 nonn
%O A358583 1,4
%A A358583 _Gus Wiseman_, Nov 23 2022
%E A358583 Terms a(19) and beyond from _Andrew Howroyd_, Dec 31 2022