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A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.

%I A331993 #9 Feb 09 2020 18:58:57
%S A331993 1,1,1,2,3,6,11,22,43,90,185,393,835,1802,3904,8540,18756,41463,92022,
%T A331993 205179,459086,1030917,2321949,5245104,11878750,26967957,61359917,
%U A331993 139902251,319591669,731385621,1676573854,3849288924,8850674950,20378544752,46982414535
%N A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.
%C A331993 Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
%C A331993 In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
%H A331993 Andrew Howroyd, <a href="/A331993/b331993.txt">Table of n, a(n) for n = 1..1000</a>
%e A331993 The a(1) = 1 through a(7) = 11 trees:
%e A331993   o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
%e A331993                 (o(o))  (o(oo))  (o(ooo))   (o(oooo))
%e A331993                         (oo(o))  (oo(oo))   (oo(ooo))
%e A331993                                  (ooo(o))   (ooo(oo))
%e A331993                                  ((o)(oo))  (oooo(o))
%e A331993                                  (o(o(o)))  ((o)(ooo))
%e A331993                                             (o(o)(oo))
%e A331993                                             (o(o(oo)))
%e A331993                                             (o(oo(o)))
%e A331993                                             (oo(o(o)))
%e A331993                                             ((o)(o(o)))
%t A331993 sssb[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
%t A331993 Table[Length[sssb[n]],{n,10}]
%o A331993 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A331993 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
%Y A331993 Not requiring any lone-child-avoidance gives A306200.
%Y A331993 The locally disjoint case is A324969 (essentially A000045).
%Y A331993 Matula-Goebel numbers of these trees are A331994.
%Y A331993 Lone-child-avoiding rooted identity trees are A000007.
%Y A331993 Semi-lone-child-avoiding rooted trees are A331934.
%Y A331993 Semi-lone-child-avoiding rooted identity trees are A331964.
%Y A331993 Lone-child-avoiding rooted semi-identity trees are A331966.
%Y A331993 Cf. A001678, A004111, A300660, A316694, A331683, A331686, A331783, A331875, A331933, A331963, A331965.
%K A331993 nonn
%O A331993 1,4
%A A331993 _Gus Wiseman_, Feb 05 2020
%E A331993 Terms a(26) and beyond from _Andrew Howroyd_, Feb 09 2020