OFFSET
0,3
COMMENTS
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).
EXAMPLE
Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
(1111) (1112) (1122) (1123) (1234)
(1(111)) (1(112)) (1(122)) (1(123)) (1(234))
(11(11)) (11(12)) (11(22)) (11(23)) (12(34))
((11)(11)) (12(11)) (12(12)) (12(13)) ((12)(34))
(1(1(11))) (2(111)) ((11)(22)) (2(113)) (1(2(34)))
((11)(12)) (1(1(22))) (23(11))
(1(1(12))) ((12)(12)) ((11)(23))
(1(2(11))) (1(2(12))) (1(1(23)))
(2(1(11))) ((12)(13))
(1(2(13)))
(2(1(13)))
(2(3(11)))
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
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)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020
CROSSREFS
The case with all atoms equal or all atoms different is A000669.
Not requiring singleton-reduction gives A330465.
The case where the leaves are sets is A330626.
Row sums of A339645.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 22 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020
STATUS
approved