This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331934 #9 Feb 09 2020 17:34:03
%S A331934 1,1,1,2,4,7,15,29,62,129,279,602,1326,2928,6544,14692,33233,75512,
%T A331934 172506,395633,911108,2105261,4880535,11346694,26451357,61813588,
%U A331934 144781303,339820852,799168292,1882845298,4443543279,10503486112,24864797324,58944602767,139918663784
%N A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.
%C A331934 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 A331934 Andrew Howroyd, <a href="/A331934/b331934.txt">Table of n, a(n) for n = 1..1000</a>
%H A331934 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 A331934 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>
%F A331934 Product_{k > 0} 1/(1 - x^k)^a(k) = A(x) + A(x)/x - x where A(x) = Sum_{k > 0} x^k a(k).
%F A331934 Euler transform is b(1) = 1, b(n > 1) = a(n) + a(n + 1).
%e A331934 The a(1) = 1 through a(7) = 15 trees:
%e A331934 o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
%e A331934 (o(o)) (o(oo)) (o(ooo)) (o(oooo))
%e A331934 (oo(o)) (oo(oo)) (oo(ooo))
%e A331934 ((o)(o)) (ooo(o)) (ooo(oo))
%e A331934 ((o)(oo)) (oooo(o))
%e A331934 (o(o)(o)) ((o)(ooo))
%e A331934 (o(o(o))) ((oo)(oo))
%e A331934 (o(o)(oo))
%e A331934 (o(o(oo)))
%e A331934 (o(oo(o)))
%e A331934 (oo(o)(o))
%e A331934 (oo(o(o)))
%e A331934 ((o)(o)(o))
%e A331934 ((o)(o(o)))
%e A331934 (o((o)(o)))
%t A331934 sse[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Union[Sort/@Tuples[sse/@c]]]/@Rest[IntegerPartitions[n-1]]];
%t A331934 Table[Length[sse[n]],{n,10}]
%o A331934 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A331934 seq(n)={my(v=[1,1]); for(n=2, n-1, v=concat(v, EulerT(v)[n] - v[n])); v} \\ _Andrew Howroyd_, Feb 09 2020
%Y A331934 The same trees counted by leaves are A050381.
%Y A331934 The locally disjoint version is A331872.
%Y A331934 Matula-Goebel numbers of these trees are A331935.
%Y A331934 Lone-child-avoiding rooted trees are A001678.
%Y A331934 Cf. A000081, A000669, A198518, A289501, A291636, A306200, A320268, A330465, A330951, A331873, A331874, A331933, A331966.
%K A331934 nonn
%O A331934 1,4
%A A331934 _Gus Wiseman_, Feb 03 2020
%E A331934 Terms a(25) and beyond from _Andrew Howroyd_, Feb 09 2020