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 A318227 Number of inequivalent leaf-colorings of rooted identity trees with n nodes. 7
 1, 1, 1, 3, 5, 14, 38, 114, 330, 1054, 3483, 11841, 41543, 149520, 552356, 2084896, 8046146, 31649992, 127031001, 518434863, 2153133594, 9081863859, 38909868272, 169096646271, 745348155211, 3329032020048, 15063018195100, 68998386313333, 319872246921326, 1500013368166112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS In a rooted identity tree, all branches directly under any given branch are different. The leaves are colored after selection of the tree. Since all trees are asymmetric, the symmetry group of the leaves will be the identity group and a tree with k leaves will have Bell(k) inequivalent leaf-colorings. - Andrew Howroyd, Dec 10 2020 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 FORMULA a(n) = Sum_{k=1..n} A055327(n,k) * A000110(k). - Andrew Howroyd, Dec 10 2020 EXAMPLE Inequivalent representatives of the a(6) = 14 leaf-colorings:   (1(1(1)))  ((1)((1)))  (1(((1))))  ((1((1))))  (((1(1))))  (((((1)))))   (1(1(2)))  ((1)((2)))  (1(((2))))  ((1((2))))  (((1(2))))   (1(2(1)))   (1(2(2)))   (1(2(3))) MATHEMATICA idt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[idt/@c]], UnsameQ@@#&], {c, IntegerPartitions[n-1]}]]; Table[Sum[BellB[Count[tree, {}, {0, Infinity}]], {tree, idt[n]}], {n, 16}] PROG (PARI) \\ bell(n) is A000110(n). WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)} bell(n)={sum(k=1, n, stirling(n, k, 2))} seq(n)={my(v=[y], b=vector(n, k, bell(k))); for(n=2, n, v=concat(v, WeighMT(v))); vector(n, k, sum(i=1, k, polcoef(v[k], i)*b[i]))} \\ Andrew Howroyd, Dec 10 2020 CROSSREFS Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486. Cf. A318226, A318228, A318229, A318230, A318231, A318234. Cf. A000110 (Bell numbers), A055327, A301342. Sequence in context: A295064 A052974 A284415 * A230585 A006395 A078718 Adjacent sequences:  A318224 A318225 A318226 * A318228 A318229 A318230 KEYWORD nonn AUTHOR Gus Wiseman, Aug 21 2018 EXTENSIONS Terms a(17) and beyond from Andrew Howroyd, Dec 10 2020 STATUS approved

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Last modified January 16 06:48 EST 2021. Contains 340204 sequences. (Running on oeis4.)