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A339645
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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.
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45
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1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312
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OFFSET
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1,4
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COMMENTS
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Only the leaves are colored. Equivalence is up to permutation of the colors.
Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.
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LINKS
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 3, 2;
5, 17, 12, 5;
12, 73, 95, 44, 12;
33, 369, 721, 512, 168, 33;
90, 1795, 5487, 5480, 2556, 625, 90;
261, 9192, 41945, 58990, 36711, 12306, 2342, 261;
766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
...
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
(1111) (1112) (1123) (1234)
(1(111)) (1122) (1(123)) (1(234))
(11(11)) (1(112)) (11(23)) (12(34))
((11)(11)) (11(12)) (12(13)) ((12)(34))
(1(1(11))) (1(122)) (2(113)) (1(2(34)))
(11(22)) (23(11))
(12(11)) ((11)(23))
(12(12)) (1(1(23)))
(2(111)) ((12)(13))
((11)(12)) (1(2(13)))
(1(1(12))) (2(1(13)))
((11)(22)) (2(3(11)))
(1(1(22)))
(1(2(11)))
((12)(12))
(1(2(12)))
(2(1(11)))
(End)
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PROG
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(PARI) \\ See link above for combinatorial species functions.
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)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n, 1..n]))}
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CROSSREFS
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The case with only one color is A000669.
A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
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KEYWORD
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AUTHOR
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STATUS
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approved
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