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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.

%I A331933 #9 Feb 09 2020 18:58:00
%S A331933 1,1,1,2,4,6,12,18,33,52,90,142,242,384,639,1028,1688,2716,4445,7161,
%T A331933 11665,18839,30595,49434,80199,129637,210079,339750,550228,889978,
%U A331933 1440909,2330887,3772845,6103823,9878357,15982196,25863454,41845650,67713550,109559443
%N A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.
%C A331933 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
%H A331933 Andrew Howroyd, <a href="/A331933/b331933.txt">Table of n, a(n) for n = 1..1000</a>
%H A331933 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A331933 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%H A331933 Gus Wiseman, <a href="/A331933/a331933.png">The a(11) = 90 semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.</a>
%F A331933 a(n) = 1 + Sum_{i=2..n-2} floor((n-1)/i)*a(i). - _Andrew Howroyd_, Feb 09 2020
%e A331933 The a(1) = 1 through a(8) = 18 trees:
%e A331933   o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)
%e A331933                 (o(o))  (o(oo))   (o(ooo))   (o(oooo))
%e A331933                         (oo(o))   (oo(oo))   (oo(ooo))
%e A331933                         ((o)(o))  (ooo(o))   (ooo(oo))
%e A331933                                   (o(o)(o))  (oooo(o))
%e A331933                                   (o(o(o)))  ((oo)(oo))
%e A331933                                              (o(o(oo)))
%e A331933                                              (o(oo(o)))
%e A331933                                              (oo(o)(o))
%e A331933                                              (oo(o(o)))
%e A331933                                              ((o)(o)(o))
%e A331933                                              (o((o)(o)))
%t A331933 sseo[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sseo/@c]],Length[Union[DeleteCases[#,{}]]]<=1&]]/@Rest[IntegerPartitions[n-1]]];
%t A331933 Table[Length[sseo[n]],{n,10}]
%o A331933 (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
%Y A331933 Not requiring lone-child-avoidance gives A320222.
%Y A331933 The non-semi version is A320268.
%Y A331933 Matula-Goebel numbers of these trees are A331936.
%Y A331933 Achiral trees are A003238.
%Y A331933 Semi-identity trees are A306200.
%Y A331933 Numbers S with at most one distinct prime index in S are A331912.
%Y A331933 Semi-lone-child-avoiding rooted trees are A331934.
%Y A331933 Cf. A000081, A001678, A004111, A050381, A214577, A291636, A320230, A320269, A331784, A331872, A331914, A331935, A331966.
%K A331933 nonn
%O A331933 1,4
%A A331933 _Gus Wiseman_, Feb 03 2020
%E A331933 Terms a(31) and beyond from _Andrew Howroyd_, Feb 09 2020