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

1, 0, 1, 1, 4, 5, 20, 28, 110, 169, 663, 1078, 4217, 7169, 27979, 49191, 191440, 345771, 1341974, 2477719, 9589567, 18034670, 69612556, 132984290, 511987473, 991391707, 3807503552, 7460270591, 28585315026, 56595367747, 216381073935, 432396092153

Offset: 1

Author

Gus Wiseman, Nov 23 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(7) = 20 trees:
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)
                     ((ooo))  ((oooo))  ((ooooo))
                     (o(oo))  (o(ooo))  (o(oooo))
                     (oo(o))  (oo(oo))  (oo(ooo))
                              (ooo(o))  (ooo(oo))
                                        (oooo(o))
                                        (((oooo)))
                                        ((o)(ooo))
                                        ((o(ooo)))
                                        ((oo)(oo))
                                        ((oo(oo)))
                                        ((ooo(o)))
                                        (o((ooo)))
                                        (o(o)(oo))
                                        (o(o(oo)))
                                        (o(oo(o)))
                                        (oo((oo)))
                                        (oo(o)(o))
                                        (oo(o(o)))
                                        (ooo((o)))
		

Links

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

Crossrefs

For equality we have A185650 aerated, ranked by A358578.

The opposite version is A358582, non-strict A358584.

The non-strict version is A358583.

The ordered version is A358585, odd-indexed terms A065097.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height, ordered A080936.

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

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

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

Cf. A000891, A109129, A342507, A358579, A358580, A358586, A358591.

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,0,10}]

PARI

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[n\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/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Extensions

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