A343663 Number of unlabeled binary rooted semi-identity plane trees with 2*n - 1 nodes.

1, 1, 2, 4, 12, 34, 108, 344, 1136, 3796, 12920, 44442, 154596, 542336, 1917648, 6825464, 24439008, 87962312, 318087216, 1155090092, 4210494616, 15400782912, 56508464736, 207935588586, 767162495940, 2837260332472, 10516827106016, 39063666532784, 145378611426512

Offset: 1

Author

Gus Wiseman, May 05 2021

Keywords

• nonn

Comments

In a semi-identity tree, only the non-leaf branches of any given vertex are required to be distinct. Alternatively, a rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.

Examples

			The a(1) = 1 through a(5) = 12 trees:
  o  (oo)  ((oo)o)  (((oo)o)o)  ((((oo)o)o)o)
           (o(oo))  ((o(oo))o)  (((o(oo))o)o)
                    (o((oo)o))  (((oo)o)(oo))
                    (o(o(oo)))  ((o((oo)o))o)
                                ((o(o(oo)))o)
                                ((o(oo))(oo))
                                ((oo)((oo)o))
                                ((oo)(o(oo)))
                                (o(((oo)o)o))
                                (o((o(oo))o))
                                (o(o((oo)o)))
                                (o(o(o(oo))))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
• Samuele Giraudo, Mockingbird lattices, Séminaire Lotharingien de Combinatoire XX, Proceedings of the 34th Conf. on Formal Power, Series and Algebraic Combinatorics (Bangalore, India, 2022).

Crossrefs

The not necessarily semi-identity version is A000108.

The non-plane version is A063895, ranked by A339193.

The Matula-Goebel numbers in the non-plane case are A339193.

The not-necessarily binary version is A343937.

A000081 counts unlabeled rooted trees with n nodes.

2*A001190 - 1 counts binary trees, ranked by A111299.

A001190 counts semi-binary trees, ranked by A292050.

A004111 counts identity trees, ranked by A276625.

A306200 counts semi-identity trees, ranked by A306202.

A306201 counts balanced semi-identity trees, ranked by A306203.

A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.

Cf. A001678, A320230, A331934, A331963, A331964.

Programs

Mathematica

crsiq[n_]:=Join@@Table[Select[Union[Tuples[crsiq/@ptn]],#=={}||#=={{},{}}||Length[#]==2&&(UnsameQ@@DeleteCases[#,{}])&],{ptn,Join@@Permutations/@IntegerPartitions[n-1]}];
Table[Length[crsiq[n]],{n,1,11,2}]
(* Second program: *)
m = 29; p[_] = 1;
Do[p[x_] = 1 + x + x (p[x]^2 - p[x^2]) + O[x]^m // Normal, {m}];
CoefficientList[p[x], x] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

PARI

seq(n)={my(p=O(1)); for(n=1, n, p=1 + x + x*(p^2-subst(p,x,x^2))); Vec(p)} \\ Andrew Howroyd, May 07 2021

Formula

G.f.: x*A(x) where A(x) satisfies A(x) = 1 + x + x*(A(x)^2 - A(x^2)). - Andrew Howroyd, May 07 2021

Extensions

Terms a(13) and beyond from Andrew Howroyd, May 07 2021