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A320173
Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.
8
1, 2, 3, 12, 23, 84, 204, 830, 2940, 13397, 58794, 283132, 1377302, 7087164, 37654377, 209943842, 1226495407, 7579549767, 49541194089, 341964495985, 2476907459261, 18703210872343, 146284738788714, 1179199861398539, 9760466433602510, 82758834102114911, 717807201648148643
OFFSET
1,2
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.
EXAMPLE
Inequivalent representatives of the a(1) = 1 through a(5) = 23 colorings:
1 (11) (111) (1111) (11111)
(12) (112) (1112) (11112)
(123) (1122) (11122)
(1123) (11123)
(1234) (11223)
((11)(11)) (11234)
((11)(12)) (12345)
((11)(22)) ((11)(111))
((11)(23)) ((11)(112))
((12)(12)) ((11)(122))
((12)(13)) ((11)(123))
((12)(34)) ((11)(223))
((11)(234))
((12)(111))
((12)(112))
((12)(113))
((12)(123))
((12)(134))
((12)(345))
((13)(122))
((22)(111))
((23)(111))
((23)(114))
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=x*sv(1) + O(x*x^n), q=0); while(p, q+=p; p=sEulerT(p)-1-p); q}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020
STATUS
approved