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%I #22 Jan 22 2021 20:27:15
%S 1,1,1,2,3,2,5,17,12,5,12,73,95,44,12,33,369,721,512,168,33,90,1795,
%T 5487,5480,2556,625,90,261,9192,41945,58990,36711,12306,2342,261,766,
%U 47324,321951,625088,516952,224241,57155,8702,766,2312,249164,2483192,6593103,7141755,3965673,1283624,258887,32313,2312
%N Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.
%C Only the leaves are colored. Equivalence is up to permutation of the colors.
%C Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.
%H Andrew Howroyd, <a href="/A339645/a339645_1.txt">PARI Functions for Combinatorial Species</a>, v2, Dec 2020.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Combinatorial_species">Combinatorial species</a>
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 3, 2;
%e 5, 17, 12, 5;
%e 12, 73, 95, 44, 12;
%e 33, 369, 721, 512, 168, 33;
%e 90, 1795, 5487, 5480, 2556, 625, 90;
%e 261, 9192, 41945, 58990, 36711, 12306, 2342, 261;
%e 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
%e ...
%e From _Gus Wiseman_, Jan 02 2021: (Start)
%e Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
%e (1111) (1112) (1123) (1234)
%e (1(111)) (1122) (1(123)) (1(234))
%e (11(11)) (1(112)) (11(23)) (12(34))
%e ((11)(11)) (11(12)) (12(13)) ((12)(34))
%e (1(1(11))) (1(122)) (2(113)) (1(2(34)))
%e (11(22)) (23(11))
%e (12(11)) ((11)(23))
%e (12(12)) (1(1(23)))
%e (2(111)) ((12)(13))
%e ((11)(12)) (1(2(13)))
%e (1(1(12))) (2(1(13)))
%e ((11)(22)) (2(3(11)))
%e (1(1(22)))
%e (1(2(11)))
%e ((12)(12))
%e (1(2(12)))
%e (2(1(11)))
%e (End)
%o (PARI) \\ See link above for combinatorial species functions.
%o cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
%o {my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n,1..n]))}
%Y The case with only one color is A000669.
%Y Counting by nodes gives A318231.
%Y A labeled version is A319376.
%Y Row sums are A330470.
%Y A000311 counts singleton-reduced phylogenetic trees.
%Y A001678 counts unlabeled lone-child-avoiding rooted trees.
%Y A005121 counts chains of set partitions, with maximal case A002846.
%Y A005804 counts phylogenetic rooted trees with n labels.
%Y A060356 counts labeled lone-child-avoiding rooted trees.
%Y A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
%Y A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
%Y A316651 counts lone-child-avoiding rooted trees with normal leaves.
%Y A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
%Y A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
%Y Cf. A196545, A213427, A281118, A289501, A292504, A318812, A319312, A330627.
%K nonn,tabl
%O 1,4
%A _Andrew Howroyd_, Dec 11 2020
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