This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330471 #9 Apr 30 2020 09:34:15
%S A330471 1,1,2,9,69,623,7803,110476,1907428
%N A330471 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.
%C A330471 A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
%C A330471 A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).
%e A330471 The a(0) = 1 through a(3) = 9 trees:
%e A330471 () (1) (11) (111)
%e A330471 (12) (112)
%e A330471 (123)
%e A330471 ((1)(11))
%e A330471 ((1)(12))
%e A330471 ((1)(23))
%e A330471 ((2)(11))
%e A330471 ((2)(13))
%e A330471 ((3)(12))
%e A330471 The a(4) = 69 trees, with singleton leaves (x) replaced by just x:
%e A330471 (1111) (1112) (1122) (1123) (1234)
%e A330471 (1(111)) (1(112)) (1(122)) (1(123)) (1(234))
%e A330471 (11(11)) (11(12)) (11(22)) (11(23)) (12(34))
%e A330471 ((11)(11)) (12(11)) (12(12)) (12(13)) (13(24))
%e A330471 (1(1(11))) (2(111)) (2(112)) (13(12)) (14(23))
%e A330471 ((11)(12)) (22(11)) (2(113)) (2(134))
%e A330471 (1(1(12))) ((11)(22)) (23(11)) (23(14))
%e A330471 (1(2(11))) (1(1(22))) (3(112)) (24(13))
%e A330471 (2(1(11))) ((12)(12)) ((11)(23)) (3(124))
%e A330471 (1(2(12))) (1(1(23))) (34(12))
%e A330471 (2(1(12))) ((12)(13)) (4(123))
%e A330471 (2(2(11))) (1(2(13))) ((12)(34))
%e A330471 (1(3(12))) (1(2(34)))
%e A330471 (2(1(13))) ((13)(24))
%e A330471 (2(3(11))) (1(3(24)))
%e A330471 (3(1(12))) ((14)(23))
%e A330471 (3(2(11))) (1(4(23)))
%e A330471 (2(1(34)))
%e A330471 (2(3(14)))
%e A330471 (2(4(13)))
%e A330471 (3(1(24)))
%e A330471 (3(2(14)))
%e A330471 (3(4(12)))
%e A330471 (4(1(23)))
%e A330471 (4(2(13)))
%e A330471 (4(3(12)))
%t A330471 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330471 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330471 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A330471 mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],Length[#]>1&&Length[#]<Length[m]&]}],m];
%t A330471 Table[Sum[Length[mtot[s]],{s,strnorm[n]}],{n,0,5}]
%Y A330471 The case with all atoms different is A000311.
%Y A330471 The case with all atoms equal is A196545.
%Y A330471 The orderless version is A316652.
%Y A330471 The unlabeled version is A330470.
%Y A330471 The case where the leaves are sets is A330628.
%Y A330471 The version for just normal (not strongly normal) is A330654.
%Y A330471 Cf. A000669, A004114, A005121, A005804, A281118, A318812, A318848, A319312, A330465, A330467, A330475.
%K A330471 nonn,more
%O A330471 0,3
%A A330471 _Gus Wiseman_, Dec 23 2019