cp's OEIS Frontend

A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

%I A330470 #15 Dec 21 2020 19:33:06
%S A330470 1,1,2,7,39,236,1836,16123,162008,1802945,22012335,291290460,
%T A330470 4144907830,62986968311,1016584428612,17344929138791,311618472138440,
%U A330470 5875109147135658,115894178676866576,2385755803919949337,51133201045333895149,1138659323863266945177,26296042933904490636133
%N A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.
%C A330470 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).
%e A330470 Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
%e A330470   (1111)      (1112)      (1122)      (1123)      (1234)
%e A330470   (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
%e A330470   (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
%e A330470   ((11)(11))  (12(11))    (12(12))    (12(13))    ((12)(34))
%e A330470   (1(1(11)))  (2(111))    ((11)(22))  (2(113))    (1(2(34)))
%e A330470               ((11)(12))  (1(1(22)))  (23(11))
%e A330470               (1(1(12)))  ((12)(12))  ((11)(23))
%e A330470               (1(2(11)))  (1(2(12)))  (1(1(23)))
%e A330470               (2(1(11)))              ((12)(13))
%e A330470                                       (1(2(13)))
%e A330470                                       (2(1(13)))
%e A330470                                       (2(3(11)))
%o A330470 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A330470 cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
%o A330470 InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 11 2020
%Y A330470 The case with all atoms equal or all atoms different is A000669.
%Y A330470 Not requiring singleton-reduction gives A330465.
%Y A330470 Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).
%Y A330470 The case where the leaves are sets is A330626.
%Y A330470 Row sums of A339645.
%Y A330470 Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.
%K A330470 nonn
%O A330470 0,3
%A A330470 _Gus Wiseman_, Dec 22 2019
%E A330470 Terms a(7) and beyond from _Andrew Howroyd_, Dec 11 2020