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A316795 Number of aperiodic rooted trees on n nodes with locally distinct multiplicities. 5


%S 1,1,1,1,2,5,8,17,30,55,101,194,352,663,1227,2275,4225,7877,14600,

%T 27158,50414,93666,173972,323286,600353,1115407,2071843,3848794,

%U 7149196,13280874,24669606,45827047,85126845,158131764,293742200,545655290,1013598733

%N Number of aperiodic rooted trees on n nodes with locally distinct multiplicities.

%C An aperiodic rooted tree is an unlabeled rooted tree in which the multiplicities of branches under any given node are relatively prime. A rooted tree has locally distinct multiplicities if the multiset of branches under any given node has all distinct multiplicities.

%H Andrew Howroyd, <a href="/A316795/b316795.txt">Table of n, a(n) for n = 1..100</a>

%H Gus Wiseman, <a href="/A316795/a316795.png">The a(9) = 30 aperiodic trees with locally distinct multiplicities.</a>

%e The a(7) = 8 trees:

%e ((((((o))))))

%e (((oo(o))))

%e ((oo((o))))

%e ((o(o)(o)))

%e ((ooo(o)))

%e (oo(((o))))

%e (ooo((o)))

%e (oooo(o))

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

%t Table[Length[strut[n]],{n,15}]

%o (PARI)

%o C(v,n)={my(recurse(r,b,g,p,k)=if(!r, g==1, sum(m=1, r, if(!bittest(b,m), sum(i=1, min(r\m, p), my(f=if(i==p, k+1, 1)); if(v[i]>=f, (v[i]-f+1)*self()(r-m*i, bitor(b, 1<<m), gcd(g,m), i, f)/f)))))); recurse(n, 0, 0, #v, 0)}

%o seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=C(v[1..n-1], n-1)); v} \\ _Andrew Howroyd_, Feb 08 2020

%Y Cf. A000081, A000837, A004111, A301700, A303431, A316793, A316794, A316796.

%K nonn

%O 1,5

%A _Gus Wiseman_, Jul 14 2018

%E Terms a(26) and beyond from _Andrew Howroyd_, Feb 08 2020

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Last modified April 23 12:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)