A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926

Offset: 1

Author

Gus Wiseman, Feb 05 2020

Keywords

• nonn

Comments

Lone-child-avoiding means there are no unary branchings.

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

Examples

			The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        (o(o(oo)))  (oooo(oo))   (oooo(ooo))
                                                    ((oo)(ooo))  (ooooo(oo))
                                                    (o(o(ooo)))  ((oo)(oooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo)(ooo))
                                                                 (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 ((oo)(o(oo)))
                                                                 (o(o(o(oo))))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Crossrefs

The non-semi case is A000007.

Lone-child-avoiding rooted trees are A001678.

The locally disjoint case is A212804.

Not requiring lone-child-avoidance gives A306200.

Matula-Goebel numbers of these trees are A331965.

The semi-lone-child-avoiding version is A331993.

Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.

Programs

Mathematica

ssb[n_]:=If[n==1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[ssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]],{n,10}]

PARI

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

Extensions

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020