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%I #8 Dec 14 2020 14:18:33
%S 1,2,3,12,23,84,204,830,2940,13397,58794,283132,1377302,7087164,
%T 37654377,209943842,1226495407,7579549767,49541194089,341964495985,
%U 2476907459261,18703210872343,146284738788714,1179199861398539,9760466433602510,82758834102114911,717807201648148643
%N Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.
%C 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.
%e Inequivalent representatives of the a(1) = 1 through a(5) = 23 colorings:
%e 1 (11) (111) (1111) (11111)
%e (12) (112) (1112) (11112)
%e (123) (1122) (11122)
%e (1123) (11123)
%e (1234) (11223)
%e ((11)(11)) (11234)
%e ((11)(12)) (12345)
%e ((11)(22)) ((11)(111))
%e ((11)(23)) ((11)(112))
%e ((12)(12)) ((11)(122))
%e ((12)(13)) ((11)(123))
%e ((12)(34)) ((11)(223))
%e ((11)(234))
%e ((12)(111))
%e ((12)(112))
%e ((12)(113))
%e ((12)(123))
%e ((12)(134))
%e ((12)(345))
%e ((13)(122))
%e ((22)(111))
%e ((23)(111))
%e ((23)(114))
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o cycleIndexSeries(n)={my(p=x*sv(1) + O(x*x^n), q=0); while(p, q+=p; p=sEulerT(p)-1-p); q}
%o InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 11 2020
%Y Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.
%Y Cf. A320154, A320155, A320160, A320174, A320175, A320179, A339645.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 11 2020
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