proposed

approved

editing

proposed

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

Andrew Howroyd, <a href="/A331966/b331966.txt">Table of n, a(n) for n = 1..1000</a>

(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

nonn,more,new

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

approved

editing

proposed

approved

editing

proposed

Matula-Goebel numbers of these trees are A331965.

Matula-Goebel numbers of these trees are A331965.

The a(1) = 1 through a(9) = 16 trees (empty column show shown as dot):

allocated for Gus WisemanNumber 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

1,5

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

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

Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>

The a(1) = 1 through a(9) = 16 trees (empty column show 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))))

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}]

The non-semi case is A000007.

Lone-child-avoiding rooted trees are A001678.

The locally disjoint case is A212804.

Matula-Goebel numbers of these trees are A331965.

Not requiring lone-child-avoidance gives A306200.

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

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

allocated

nonn,more

Gus Wiseman, Feb 05 2020

approved

editing

allocated for Gus Wiseman

allocated

approved