A318691 Number of series-reduced powerful uniform rooted trees with n nodes.

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 6, 1, 5, 4, 8, 1, 11, 1, 15, 6, 13, 1, 26, 3, 24, 9, 36, 1, 50, 1, 58, 14, 67, 7, 107, 1, 105, 25, 160, 1, 213, 1, 245, 45, 291, 1, 443, 5, 492, 68, 644, 1, 851, 15, 1019, 106, 1263, 1, 1785, 1, 1986, 189, 2592, 26, 3426, 1, 4071, 292

Offset

1,7

Comments

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

Links

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

Formula

a(p+1) = 1 for prime p. - Andrew Howroyd, Dec 09 2020

Example

The a(19) = 11 series-reduced powerful uniform rooted trees with 19 nodes:
  (((ooo)(ooo))((ooo)(ooo)))
  ((oo(oo)(oo))(oo(oo)(oo)))
  ((oo)(oo)(oo)(oo)(oo)(oo))
  ((oo)(oo)(ooooo)(ooooo))
  ((ooo)(ooo)(oooo)(oooo))
  (oo(oo)(oo)(oooo)(oooo))
  ((ooooo)(ooooo)(ooooo))
  (ooo(oooo)(oooo)(oooo))
  ((oooooooo)(oooooooo))
  (oo(ooooooo)(ooooooo))
  (oooooooooooooooooo)

Mathematica

rurt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]], And[Min@@Length/@Split[#]>=2, SameQ@@Length/@Split[#]]&], {ptn, IntegerPartitions[n-1]}]];
Table[Length[rurt[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=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} \\ Andrew Howroyd, Dec 09 2020

Crossrefs

Cf. A000081, A001190, A001678, A001694, A003238, A072774, A317705, A317707, A317710, A318611, A318612, A318689, A318692.

Sequence in context: A245328 A060162 A026730 * A075256 A334523 A001480

Adjacent sequences: A318688 A318689 A318690 * A318692 A318693 A318694

Keyword

nonn

Author

Gus Wiseman, Aug 31 2018

Extensions

Terms a(51) and beyond from Andrew Howroyd, Dec 09 2020

Status

approved