%I #15 Dec 21 2020 19:33:06

%S 1,1,2,7,39,236,1836,16123,162008,1802945,22012335,291290460,

%T 4144907830,62986968311,1016584428612,17344929138791,311618472138440,

%U 5875109147135658,115894178676866576,2385755803919949337,51133201045333895149,1138659323863266945177,26296042933904490636133

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

%C 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 Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:

%e (1111) (1112) (1122) (1123) (1234)

%e (1(111)) (1(112)) (1(122)) (1(123)) (1(234))

%e (11(11)) (11(12)) (11(22)) (11(23)) (12(34))

%e ((11)(11)) (12(11)) (12(12)) (12(13)) ((12)(34))

%e (1(1(11))) (2(111)) ((11)(22)) (2(113)) (1(2(34)))

%e ((11)(12)) (1(1(22))) (23(11))

%e (1(1(12))) ((12)(12)) ((11)(23))

%e (1(2(11))) (1(2(12))) (1(1(23)))

%e (2(1(11))) ((12)(13))

%e (1(2(13)))

%e (2(1(13)))

%e (2(3(11)))

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o 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 InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 11 2020

%Y The case with all atoms equal or all atoms different is A000669.

%Y Not requiring singleton-reduction gives A330465.

%Y Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).

%Y The case where the leaves are sets is A330626.

%Y Row sums of A339645.

%Y Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 22 2019

%E Terms a(7) and beyond from _Andrew Howroyd_, Dec 11 2020