OFFSET
1,5
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
EXAMPLE
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))))
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}]
PROG
(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
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 05 2020
EXTENSIONS
Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020
STATUS
approved