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A318689 Number of powerful uniform rooted trees with n nodes. 3
1, 1, 2, 3, 5, 6, 11, 12, 19, 23, 35, 36, 63, 64, 98, 112, 173, 174, 291, 292, 473, 509, 791, 792, 1345, 1356, 2158, 2257, 3634, 3635, 6053, 6054, 9807, 10091, 16173, 16216, 26783, 26784, 43076, 43880, 70631, 70632, 114975, 114976, 184665, 186996, 298644, 298645, 481978, 482011 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A powerful uniform rooted tree with n nodes is either a single powerful uniform branch with n-1 nodes, or a powerful uniform multiset (all multiplicities are equal to the same number > 1) of powerful uniform rooted trees with a total of n-1 nodes.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Gus Wiseman, All 35 powerful uniform rooted trees with 11 nodes.

EXAMPLE

The a(8) = 12 powerful uniform rooted trees:

  (((((((o)))))))

  ((((((oo))))))

  (((((o)(o)))))

  ((((o))((o))))

  (((((ooo)))))

  (((o)(o)(o)))

  ((((oooo))))

  (((oo)(oo)))

  ((oo(o)(o)))

  (((ooooo)))

  ((oooooo))

  (ooooooo)

MATHEMATICA

rurt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]], Or[Length[#]==1, And[Min@@Length/@Split[#]>=2, SameQ@@Length/@Split[#]]]&], {ptn, IntegerPartitions[n-1]}]];

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

PROG

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

seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n, d, u[d]) - u[n] + v[n]); v} \\ Andrew Howroyd, Dec 09 2020

CROSSREFS

Cf. A000081, A003238, A072774, A317705, A317707, A317710, A317717, A317718, A318611, A318612, A318690, A318691, A318692.

Sequence in context: A033159 A199366 A332275 * A083710 A127524 A117086

Adjacent sequences:  A318686 A318687 A318688 * A318690 A318691 A318692

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 31 2018

EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Dec 09 2020

STATUS

approved

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Last modified December 1 20:25 EST 2021. Contains 349435 sequences. (Running on oeis4.)