A318689 Number of powerful uniform rooted trees with n nodes.

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

Offset: 1

Author

Gus Wiseman, Aug 31 2018

Keywords

• nonn

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.

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Gus Wiseman, All 35 powerful uniform rooted trees with 11 nodes.

Crossrefs

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

Programs

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}]

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

Extensions

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