A301700 - OEIS

%I #31 Sep 01 2018 21:27:52

%S 1,1,1,2,4,10,21,52,120,290,697,1713,4200,10446,26053,65473,165257,

%T 419357,1068239,2732509,7013242,18059960,46641983,120790324,313593621,

%U 816046050,2128101601,5560829666,14557746453,38177226541,100281484375,263815322761,695027102020

%N Number of aperiodic rooted trees with n nodes.

%C 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.

%H Andrew Howroyd, <a href="/A301700/b301700.txt">Table of n, a(n) for n = 1..500</a>

%e 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)).

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

%t Table[Length[arut[n]],{n,20}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}

%o seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ _Andrew Howroyd_, Sep 01 2018

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Apr 23 2018

%E Terms a(21) and beyond from _Andrew Howroyd_, Sep 01 2018