A317712 Number of uniform rooted trees with n nodes.

1, 1, 2, 4, 8, 15, 35, 72, 169, 388, 934, 2234, 5508, 13557, 33883, 85017, 215091, 546496, 1396524, 3582383, 9228470, 23852918, 61857180, 160871716, 419516462, 1096671326, 2873403980, 7544428973, 19847520789, 52308750878, 138095728065, 365153263313, 966978876376

Offset

1,3

Comments

An unlabeled rooted tree is uniform if the multiplicities of the branches directly under any given node are all equal.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..2250 (terms 1..200 from Andrew Howroyd)

Gus Wiseman, All 72 uniform rooted trees with 8 nodes.

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.774067238136373782458114960391469140405537808253... and c = 0.43338208953061974806801546569720246018271214... - Vaclav Kotesovec, Sep 07 2019

Example

The a(5) = 8 uniform rooted trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  ((ooo))
  (o((o)))
  (o(oo))
  ((o)(o))
  (oooo)

Mathematica

purt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]], SameQ@@Length/@Split[#]&], {ptn, IntegerPartitions[n-1]}];
Table[Length[purt[n]], {n, 10}]

Prog

(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sumdiv(n-1, d, t[d]))); v} \\ Andrew Howroyd, Aug 28 2018

Crossrefs

Cf. A000081, A001190, A004111, A072774, A301700, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317717, A317718.

Sequence in context: A036661 A145671 A316522 * A108693 A371351 A026096

Adjacent sequences: A317709 A317710 A317711 * A317713 A317714 A317715

Keyword

nonn

Author

Gus Wiseman, Aug 05 2018

Extensions

Term a(21) and beyond from Andrew Howroyd, Aug 28 2018

Status

approved