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A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.

%I A331964 #8 Feb 09 2020 18:12:09
%S A331964 1,1,0,1,0,1,1,2,2,4,6,10,16,27,44,74,123,209,353,602,1026,1760,3019,
%T A331964 5203,8977,15538,26930,46792,81415,141939,247795,433307,758672,
%U A331964 1330219,2335086,4104064,7220937,12718694,22424283,39574443,69903759,123584852,218668323
%N A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.
%C A331964 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. It is an identity tree if the branches of any given vertex are all distinct.
%H A331964 Andrew Howroyd, <a href="/A331964/b331964.txt">Table of n, a(n) for n = 1..1000</a>
%H A331964 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 A331964 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 A331964 Gus Wiseman, <a href="/A331964/a331964.png">The semi-lone-child-avoiding rooted identity trees with up to 13 vertices.</a>
%e A331964 The a(9) = 2 through a(12) = 10 semi-lone-child-avoiding rooted identity trees:
%e A331964   ((o)(o(o(o))))  (o(o)(o(o(o))))   ((o)(o(o)(o(o))))  (o(o)(o(o)(o(o))))
%e A331964   (o((o)(o(o))))  (o(o(o)(o(o))))   ((o)(o(o(o(o)))))  (o(o)(o(o(o(o)))))
%e A331964                   (o(o(o(o(o)))))   ((o(o))(o(o(o))))  (o(o(o))(o(o(o))))
%e A331964                   ((o)((o)(o(o))))  (o((o)(o(o(o)))))  (o(o(o)(o(o(o)))))
%e A331964                                     (o(o)((o)(o(o))))  (o(o(o(o)(o(o)))))
%e A331964                                     (o(o((o)(o(o)))))  (o(o(o(o(o(o))))))
%e A331964                                                        ((o)((o)(o(o(o)))))
%e A331964                                                        ((o)(o((o)(o(o)))))
%e A331964                                                        ((o(o))((o)(o(o))))
%e A331964                                                        (o((o)((o)(o(o)))))
%t A331964 ssei[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[ssei/@c]],UnsameQ@@#&]]/@Rest[IntegerPartitions[n-1]]];
%t A331964 Table[Length[ssei[n]],{n,15}]
%o A331964 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A331964 seq(n)={my(v=[1,1]); for(n=2, n-1, v=concat(v, WeighT(v)[n] - v[n])); v} \\ _Andrew Howroyd_, Feb 09 2020
%Y A331964 The non-semi version is A000007.
%Y A331964 Matula-Goebel numbers of these trees are A331963.
%Y A331964 Rooted identity trees are A004111.
%Y A331964 Semi-lone-child-avoiding rooted trees are A331934.
%Y A331964 Cf. A000081, A001678, A050381, A276625, A300660, A306200, A316694, A331872, A331875, A331933, A331935, A331936, A331937, A331966.
%K A331964 nonn
%O A331964 1,8
%A A331964 _Gus Wiseman_, Feb 04 2020
%E A331964 Terms a(36) and beyond from _Andrew Howroyd_, Feb 09 2020