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A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

%I A331966 #9 Feb 09 2020 18:58:24
%S A331966 1,0,1,1,2,3,5,9,16,30,55,105,200,388,754,1483,2923,5807,11575,23190,
%T A331966 46608,94043,190287,386214,785831,1602952,3276845,6712905,13778079,
%U A331966 28330583,58350582,120370731,248676129,514459237,1065696295,2210302177,4589599429,9540623926
%N A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.
%C A331966 Lone-child-avoiding means there are no unary branchings.
%C A331966 In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
%H A331966 Andrew Howroyd, <a href="/A331966/b331966.txt">Table of n, a(n) for n = 1..1000</a>
%H A331966 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>
%e A331966 The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
%e A331966   o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
%e A331966                      (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
%e A331966                               (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
%e A331966                                         (ooo(oo))   (ooo(ooo))   (ooo(oooo))
%e A331966                                         (o(o(oo)))  (oooo(oo))   (oooo(ooo))
%e A331966                                                     ((oo)(ooo))  (ooooo(oo))
%e A331966                                                     (o(o(ooo)))  ((oo)(oooo))
%e A331966                                                     (o(oo(oo)))  (o(o(oooo)))
%e A331966                                                     (oo(o(oo)))  (o(oo)(ooo))
%e A331966                                                                  (o(oo(ooo)))
%e A331966                                                                  (o(ooo(oo)))
%e A331966                                                                  (oo(o(ooo)))
%e A331966                                                                  (oo(oo(oo)))
%e A331966                                                                  (ooo(o(oo)))
%e A331966                                                                  ((oo)(o(oo)))
%e A331966                                                                  (o(o(o(oo))))
%t A331966 ssb[n_]:=If[n==1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[ssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
%t A331966 Table[Length[ssb[n]],{n,10}]
%o A331966 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A331966 seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ _Andrew Howroyd_, Feb 09 2020
%Y A331966 The non-semi case is A000007.
%Y A331966 Lone-child-avoiding rooted trees are A001678.
%Y A331966 The locally disjoint case is A212804.
%Y A331966 Not requiring lone-child-avoidance gives A306200.
%Y A331966 Matula-Goebel numbers of these trees are A331965.
%Y A331966 The semi-lone-child-avoiding version is A331993.
%Y A331966 Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.
%K A331966 nonn
%O A331966 1,5
%A A331966 _Gus Wiseman_, Feb 05 2020
%E A331966 Terms a(31) and beyond from _Andrew Howroyd_, Feb 09 2020