OFFSET
1,4
COMMENTS
Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
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(7) = 11 trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
(o(o)) (o(oo)) (o(ooo)) (o(oooo))
(oo(o)) (oo(oo)) (oo(ooo))
(ooo(o)) (ooo(oo))
((o)(oo)) (oooo(o))
(o(o(o))) ((o)(ooo))
(o(o)(oo))
(o(o(oo)))
(o(oo(o)))
(oo(o(o)))
((o)(o(o)))
MATHEMATICA
sssb[n_]:=Switch[n, 1, {{}}, 2, {{{}}}, _, Join@@Function[c, Select[Union[Sort/@Tuples[sssb/@c]], UnsameQ@@DeleteCases[#, {}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sssb[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]); for(n=1, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020
CROSSREFS
Not requiring any lone-child-avoidance gives A306200.
Matula-Goebel numbers of these trees are A331994.
Lone-child-avoiding rooted identity trees are A000007.
Semi-lone-child-avoiding rooted trees are A331934.
Semi-lone-child-avoiding rooted identity trees are A331964.
Lone-child-avoiding rooted semi-identity trees are A331966.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 05 2020
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, Feb 09 2020
STATUS
approved