A301700 Number of aperiodic rooted trees with n nodes.

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset

1,4

Comments

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Example

The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

Mathematica

arut[n_]:=arut[n]=If[n===1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[arut/@c]], GCD@@Length/@Split[#]===1&]]/@IntegerPartitions[n-1]];
Table[Length[arut[n]], {n, 20}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n, d, moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Crossrefs

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

Sequence in context: A165136 A165137 A065023 * A123445 A104431 A130666

Adjacent sequences: A301697 A301698 A301699 * A301701 A301702 A301703

Keyword

nonn

Author

Gus Wiseman, Apr 23 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

Status

approved