A318691 -id:A318691 - cp's OEIS frontend

A318611 Number of series-reduced powerful rooted trees with n nodes.

1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 4, 4, 8, 5, 11, 10, 14, 14, 24, 18, 34, 32, 46, 45, 72, 60, 103, 96, 138, 137, 212, 184, 296, 282, 403, 397, 591, 539, 830, 798, 1125, 1119, 1624, 1519, 2253, 2195, 3067, 3056, 4341, 4158, 6004, 5897, 8145, 8164, 11397, 11090

Offset: 1

Gus Wiseman, Aug 30 2018

nonn

A series-reduced rooted tree is powerful if either it is a single node, or the branches of the root all appear with multiplicities greater than 1 and are themselves series-reduced powerful rooted trees.

			The a(13) = 8 series-reduced powerful rooted trees:
  ((oo)(oo)(oo)(oo))
  ((ooo)(ooo)(ooo))
  (ooo(oo)(oo)(oo))
  ((ooooo)(ooooo))
  (oo(oooo)(oooo))
  (oooo(ooo)(ooo))
  (oooooo(oo)(oo))
  (oooooooooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000081, A001190, A001678, A001694, A004111, A167865, A291636, A317102, A317705, A317707, A318612, A318691.

h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2)):
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 31 2018

purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Min@@Length/@Split[#]>1&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,20}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t],
     If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1]*h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n - 1, n - 1]];
Array[a, 60] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(41)-a(56) from Alois P. Heinz, Aug 31 2018

A318689 Number of powerful uniform rooted trees with n nodes.

Original entry on oeis.org

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

Gus Wiseman, Aug 31 2018

nonn

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.

			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)

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

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

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}]

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

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

A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 512, 1024, 1444, 2048, 2401, 2744, 2809, 4096, 6859, 8192, 11236, 16384, 16807, 17161, 17689, 32768, 38416, 51529, 54872, 65536, 68644, 70756, 96721, 117649, 130321, 131072, 137641, 148877, 206116, 262144

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A positive integer n is a Matula-Goebel number of a series-reduced powerful uniform rooted tree iff either n = 1 or n is a squarefree number, whose prime indices are all Matula-Goebel numbers of series-reduced powerful uniform rooted trees, taken to a power > 1.

			The sequence of all series-reduced powerful uniform rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  512: (ooooooooo)

Cf. A000081, A001694, A061775, A072774, A214577, A291636, A317705, A317707, A317710, A318611, A318612, A318690, A318691.

srpowunQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],Min@@FactorInteger[n][[All,2]]>1,And@@srpowunQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100000],srpowunQ]

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A318611 Number of series-reduced powerful rooted trees with n nodes.

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A318689 Number of powerful uniform rooted trees with n nodes.

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A318692 Matula-Goebel numbers of series-reduced powerful uniform rooted trees.

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