A324738 -id:A324738 - cp's OEIS frontend

A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

1, 0, 1, 1, 2, 3, 8, 16, 35, 74, 166, 367, 831, 1878, 4299, 9857, 22775, 52777, 122957, 287337

Offset: 1

Gus Wiseman, Mar 18 2019

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
Also the number of finitary sets with n brackets where no element is also a subset. For example, the a(7) = 8 sets are (o = {}):
  {{{{{{o}}}}}}
  {{{{o,{o}}}}}
  {{{o,{{o}}}}}
  {{o,{{{o}}}}}
  {{o,{o,{o}}}}
  {{{o},{{o}}}}
  {{o},{{{o}}}}
  {{o},{o,{o}}}

			The a(1) = 1 through a(8) = 16 rooted identity trees:
  o  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o)(o(o)))    (((o))(o(o)))
                     ((((o))))  ((o((o))))   ((o(o(o))))    (((o)(o(o))))
                                (((((o)))))  ((((o(o)))))   (((o(o(o)))))
                                             (((o)((o))))   ((o)((o(o))))
                                             (((o((o)))))   ((o)(o((o))))
                                             ((o)(((o))))   ((o((o(o)))))
                                             ((o(((o)))))   ((o(o)((o))))
                                             ((((((o))))))  ((o(o((o)))))
                                                            (((((o(o))))))
                                                            ((((o)((o)))))
                                                            ((((o((o))))))
                                                            (((o)(((o)))))
                                                            (((o(((o))))))
                                                            ((o)((((o)))))
                                                            ((o((((o))))))
                                                            (((((((o)))))))

Cf. A000081, A290760, A304360, A306844, A317787.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324767, A324770, A324771, A324838, A324840, A324844, A324846.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]

A324845 Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of terms together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))

Gus Wiseman, The sequence of rooted trees whose Matula-Goebel numbers belong to A324845.

Cf. A000081, A007097, A290822, A306844, A318186.

Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324842, A324844, A324846, A324847, A324849.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]

cp's OEIS Frontend

A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

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