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 A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices. 5
 1, 1, 1, 2, 3, 6, 11, 22, 43, 90, 185, 393, 835, 1802, 3904, 8540, 18756, 41463, 92022, 205179, 459086, 1030917, 2321949, 5245104, 11878750, 26967957, 61359917, 139902251, 319591669, 731385621, 1676573854, 3849288924, 8850674950, 20378544752, 46982414535 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. In a semi-identity tree, the non-leaf branches of any given vertex are distinct. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1000 EXAMPLE The a(1) = 1 through a(7) = 11 trees: o (o) (oo) (ooo) (oooo) (ooooo) (oooooo) (o(o)) (o(oo)) (o(ooo)) (o(oooo)) (oo(o)) (oo(oo)) (oo(ooo)) (ooo(o)) (ooo(oo)) ((o)(oo)) (oooo(o)) (o(o(o))) ((o)(ooo)) (o(o)(oo)) (o(o(oo))) (o(oo(o))) (oo(o(o))) ((o)(o(o))) MATHEMATICA sssb[n_]:=Switch[n, 1, {{}}, 2, {{{}}}, _, Join@@Function[c, Select[Union[Sort/@Tuples[sssb/@c]], UnsameQ@@DeleteCases[#, {}]&]]/@Rest[IntegerPartitions[n-1]]]; Table[Length[sssb[n]], {n, 10}] PROG (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)} seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020 CROSSREFS Not requiring any lone-child-avoidance gives A306200. The locally disjoint case is A324969 (essentially A000045). Matula-Goebel numbers of these trees are A331994. Lone-child-avoiding rooted identity trees are A000007. Semi-lone-child-avoiding rooted trees are A331934. Semi-lone-child-avoiding rooted identity trees are A331964. Lone-child-avoiding rooted semi-identity trees are A331966. Cf. A001678, A004111, A300660, A316694, A331683, A331686, A331783, A331875, A331933, A331963, A331965. Sequence in context: A026418 A063895 A337090 * A027214 A192652 A132831 Adjacent sequences: A331990 A331991 A331992 * A331994 A331995 A331996 KEYWORD nonn AUTHOR Gus Wiseman, Feb 05 2020 EXTENSIONS Terms a(26) and beyond from Andrew Howroyd, Feb 09 2020 STATUS approved

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Last modified September 8 13:05 EDT 2024. Contains 375753 sequences. (Running on oeis4.)