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 A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex. 10
 1, 1, 1, 2, 4, 6, 12, 18, 33, 52, 90, 142, 242, 384, 639, 1028, 1688, 2716, 4445, 7161, 11665, 18839, 30595, 49434, 80199, 129637, 210079, 339750, 550228, 889978, 1440909, 2330887, 3772845, 6103823, 9878357, 15982196, 25863454, 41845650, 67713550, 109559443 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1000 David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014). FORMULA a(n) = 1 + Sum_{i=2..n-2} floor((n-1)/i)*a(i). - Andrew Howroyd, Feb 09 2020 EXAMPLE The a(1) = 1 through a(8) = 18 trees:   o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)                 (o(o))  (o(oo))   (o(ooo))   (o(oooo))                         (oo(o))   (oo(oo))   (oo(ooo))                         ((o)(o))  (ooo(o))   (ooo(oo))                                   (o(o)(o))  (oooo(o))                                   (o(o(o)))  ((oo)(oo))                                              (o(o(oo)))                                              (o(oo(o)))                                              (oo(o)(o))                                              (oo(o(o)))                                              ((o)(o)(o))                                              (o((o)(o))) MATHEMATICA sseo[n_]:=Switch[n, 1, {{}}, 2, {{{}}}, _, Join@@Function[c, Select[Union[Sort/@Tuples[sseo/@c]], Length[Union[DeleteCases[#, {}]]]<=1&]]/@Rest[IntegerPartitions[n-1]]]; Table[Length[sseo[n]], {n, 10}] PROG (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(i=2, n-2, ((n-1)\i)*v[i])); v} \\ Andrew Howroyd, Feb 09 2020 CROSSREFS Not requiring lone-child-avoidance gives A320222. The non-semi version is A320268. Matula-Goebel numbers of these trees are A331936. Achiral trees are A003238. Semi-identity trees are A306200. Numbers S with at most one distinct prime index in S are A331912. Semi-lone-child-avoiding rooted trees are A331934. Cf. A000081, A001678, A004111, A050381, A214577, A291636, A320230, A320269, A331784, A331872, A331914, A331935, A331966. Sequence in context: A259941 A007436 A052847 * A052823 A063516 A306315 Adjacent sequences:  A331930 A331931 A331932 * A331934 A331935 A331936 KEYWORD nonn AUTHOR Gus Wiseman, Feb 03 2020 EXTENSIONS Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020 STATUS approved

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Last modified April 8 14:29 EDT 2020. Contains 333314 sequences. (Running on oeis4.)