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

1, 1, 1, 3, 4, 13, 20, 67, 110, 383, 663, 2346, 4217, 15118, 27979, 101092, 191440, 695474, 1341974, 4893067, 9589567, 35055011, 69612556, 254923825, 511987473, 1877232869, 3807503552, 13972144807, 28585315026, 104955228432, 216381073935, 794739865822

Offset: 1

Author

Gus Wiseman, Nov 23 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((oo))  ((ooo))  ((oooo))
                (o(o))  (o(oo))  (o(ooo))
                        (oo(o))  (oo(oo))
                                 (ooo(o))
                                 (((ooo)))
                                 ((o)(oo))
                                 ((o(oo)))
                                 ((oo(o)))
                                 (o((oo)))
                                 (o(o)(o))
                                 (o(o(o)))
                                 (oo((o)))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

For equality we have A185650 aerated, ranked by A358578.

The strict case is A358581.

The opposite version is A358584, strict A358582.

The ordered version is A358586, strict A358585.

A000081 counts rooted trees, ordered A000108.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

A358575 counts rooted trees by nodes and internal nodes, ordered A090181.

A358589 counts square rooted trees, ranked by A358577, ordered A358590.

Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358586.

Programs

Mathematica

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]

PARI

\\ See A358584 for R(n).
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

Formula

A358581(n) + A358584(n) = A000081(n).

A358582(n) + A358583(n) = A000081(n).

a(n) = Sum_{k=floor((n-1)/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022