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A316795 Number of aperiodic rooted trees on n nodes with locally distinct multiplicities. 5
1, 1, 1, 1, 2, 5, 8, 17, 30, 55, 101, 194, 352, 663, 1227, 2275, 4225, 7877, 14600, 27158, 50414, 93666, 173972, 323286, 600353, 1115407, 2071843, 3848794, 7149196, 13280874, 24669606, 45827047, 85126845, 158131764, 293742200, 545655290, 1013598733 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..100

Gus Wiseman, The a(9) = 30 aperiodic trees with locally distinct multiplicities.

EXAMPLE

The a(7) = 8 trees:

((((((o))))))

(((oo(o))))

((oo((o))))

((o(o)(o)))

((ooo(o)))

(oo(((o))))

(ooo((o)))

(oooo(o))

MATHEMATICA

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]&]];

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

PROG

(PARI)

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)}

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

CROSSREFS

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

Sequence in context: A034445 A285459 A259580 * A054754 A054755 A093331

Adjacent sequences:  A316792 A316793 A316794 * A316796 A316797 A316798

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 14 2018

EXTENSIONS

Terms a(26) and beyond from Andrew Howroyd, Feb 08 2020

STATUS

approved

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Last modified February 28 10:42 EST 2021. Contains 341703 sequences. (Running on oeis4.)