%I #10 Dec 13 2020 17:26:35
%S 1,0,2,3,9,23,73,229,796,2891,11118,44695,187825,820320,3716501,
%T 17413308,84209071,419461933,2148673503,11301526295,60956491070,
%U 336744177291,1903317319015,10995856040076,64873456288903,390544727861462,2397255454976268,14993279955728851
%N Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.
%C In a series-reduced rooted tree, every non-leaf node has at least two branches.
%e Inequivalent representatives of the a(6) = 23 leaf-colorings:
%e (11(11)) (1(111)) (11111)
%e (11(12)) (1(112)) (11112)
%e (11(22)) (1(122)) (11122)
%e (11(23)) (1(123)) (11123)
%e (12(11)) (1(222)) (11223)
%e (12(12)) (1(223)) (11234)
%e (12(13)) (1(234)) (12345)
%e (12(33))
%e (12(34))
%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(concat(v[1..n-2], [0]))), n-1 )); x*Ser(v)}
%o InequivalentColoringsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 11 2020
%Y Cf. A000081, A001190, A001678, A003238, A004111, A290689, A291636, A304486.
%Y Cf. A318226, A318227, A318228, A318229, A318230, A318234, A339645, A339648.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 21 2018
%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 11 2020