A304175 Number of leaf-balanced rooted plane trees with n nodes.

1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543

Offset

1,3

Comments

A rooted plane tree is leaf-balanced if every branch of the root has the same number of leaves, and every branch of the root is itself leaf-balanced.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(5) = 12 leaf-balanced plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list are ((oo)o) and (o(oo)).

Mathematica

lbplane[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[lbplane/@c], SameQ@@(Count[#, {}, {0, Infinity}]&/@#)&], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]], {n, 10}]

Prog

(PARI) seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(d<k, v[d]^(k/d)))/(1-x) ); Vec(vecsum(v) + O(x*x^n))} \\ Andrew Howroyd, Dec 13 2020

Crossrefs

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304173.

Sequence in context: A083378 A116712 A000102 * A086589 A299270 A190171

Adjacent sequences: A304172 A304173 A304174 * A304176 A304177 A304178

Keyword

nonn

Author

Gus Wiseman, Aug 16 2018

Extensions

Terms a(17) and beyond from Andrew Howroyd, Dec 13 2020

Status

approved