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A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n. 12
1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..22.

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.

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

The case where the leaves are sets is A330626.

Row sums of A339645.

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

Sequence in context: A119602 A121752 A054133 * A266310 A032118 A125660

Adjacent sequences:  A330467 A330468 A330469 * A330471 A330472 A330473

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 22 2019

EXTENSIONS

Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020

STATUS

approved

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Last modified July 25 19:28 EDT 2021. Contains 346291 sequences. (Running on oeis4.)