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 A198518 #50 Feb 09 2020 04:04:46
%S A198518 1,1,1,2,3,5,9,16,29,54,102,194,375,730,1434,2837,5650,11311,22767,
%T A198518 46023,93422,190322,389037,797613,1639878,3380099,6983484,14459570,
%U A198518 29999618,62357426,129843590,270807835,565674584,1183301266,2478624060,5198504694,10916110768,22948299899
%N A198518 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^n)/(1+x^n) * x^n/n ).
%C A198518 For n>=1, a(n) is the number of rooted trees (see A000081) with n non-root nodes where non-root nodes cannot have out-degree 1, see the note by _David Callan_ and the example. Imposing the condition also for the root node gives A001678. - _Joerg Arndt_, Jun 28 2014
%C A198518 Compare definition to G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ), where G(x) is the g.f. of A000081, the number of rooted trees with n nodes.
%C A198518 Number of forests of lone-child-avoiding rooted trees with n unlabeled vertices. - _Gus Wiseman_, Feb 03 2020
%H A198518 Alois P. Heinz, <a href="/A198518/b198518.txt">Table of n, a(n) for n = 0..1000</a>
%H A198518 David Callan, <a href="/A198518/a198518.pdf">Rooted trees with no out-degree = 1</a>, (7-July-2014).
%H A198518 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 A198518 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 A198518 Euler transform of coefficients in A(x)/(1+x), where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%F A198518 a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711..., c = 1.3437262442171062526771597... . - _Vaclav Kotesovec_, Sep 03 2014
%F A198518 a(n) = A001678(n + 1) + A001678(n + 2). - _Gus Wiseman_, Jan 22 2020
%F A198518 Euler transform of A001678(n + 1). - _Gus Wiseman_, Feb 03 2020
%e A198518 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 16*x^7 + 29*x^8 +...
%e A198518 where
%e A198518 log(A(x)) = A(x)/(1+x)*x + A(x^2)/(1+x^2)*x^2/2 + A(x^3)/(1+x^3)*x^3/3 +...
%e A198518 The coefficients in A(x)/(1+x) begin:
%e A198518 [1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 67, 127, 248, 482, 952, 1885, 3765, ...]
%e A198518 (this is, up to offset, A001678),
%e A198518 from which g.f. A(x) may be generated by the Euler transform:
%e A198518 A(x) = 1/((1-x)^1*(1-x^2)^0*(1-x^3)^1*(1-x^4)^1*(1-x^5)^2*(1-x^6)^3*(1-x^7)^6*(1-x^8)^10*(1-x^9)^19*(1-x^10)^35*...).
%e A198518 From _Joerg Arndt_, Jun 28 2014: (Start)
%e A198518 The a(6) = 9 rooted trees with 6 non-root nodes as described in the comment are:
%e A198518 : level sequence out-degrees (dots for zeros)
%e A198518 : 1: [ 0 1 2 3 3 3 2 ] [ 1 2 3 . . . . ]
%e A198518 : O--o--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 2: [ 0 1 2 3 3 2 2 ] [ 1 3 2 . . . . ]
%e A198518 : O--o--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 3: [ 0 1 2 3 3 2 1 ] [ 2 2 2 . . . . ]
%e A198518 : O--o--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 4: [ 0 1 2 2 2 2 2 ] [ 1 5 . . . . . ]
%e A198518 : O--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 5: [ 0 1 2 2 2 2 1 ] [ 2 4 . . . . . ]
%e A198518 : O--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 6: [ 0 1 2 2 2 1 1 ] [ 3 3 . . . . . ]
%e A198518 : O--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 7: [ 0 1 2 2 1 2 2 ] [ 2 2 . . 2 . . ]
%e A198518 : O--o--o
%e A198518 : .--o
%e A198518 : .--o--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 8: [ 0 1 2 2 1 1 1 ] [ 4 2 . . . . . ]
%e A198518 : O--o--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 :
%e A198518 : 9: [ 0 1 1 1 1 1 1 ] [ 6 . . . . . . ]
%e A198518 : O--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 : .--o
%e A198518 (End)
%e A198518 From _Gus Wiseman_, Jan 22 2020: (Start)
%e A198518 The a(0) = 1 through a(6) = 9 rooted trees with n + 1 nodes where non-root vertices cannot have out-degree 1:
%e A198518 o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
%e A198518 ((oo)) ((ooo)) ((oooo)) ((ooooo))
%e A198518 (o(oo)) (o(ooo)) (o(oooo))
%e A198518 (oo(oo)) (oo(ooo))
%e A198518 ((o(oo))) (ooo(oo))
%e A198518 ((o(ooo)))
%e A198518 ((oo)(oo))
%e A198518 ((oo(oo)))
%e A198518 (o(o(oo)))
%e A198518 (End)
%p A198518 with(numtheory):
%p A198518 b:= proc(n) b(n):= `if`(n=0, 1, a(n)-b(n-1)) end:
%p A198518 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A198518 d*b(d-1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A198518 end:
%p A198518 seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 02 2014
%t A198518 b[n_] := b[n] = If[n==0, 1, a[n] - b[n-1]];
%t A198518 a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*b[d-1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
%t A198518 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Mar 21 2017, after _Alois P. Heinz_ *)
%t A198518 urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
%t A198518 Table[Length[Select[urt[n],FreeQ[Z@@#,{_}]&]],{n,10}] (* _Gus Wiseman_, Jan 22 2020 *)
%o A198518 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A/(1+x),x,x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)}
%Y A198518 Cf. A052855, A246403.
%Y A198518 The labeled version is A254382.
%Y A198518 Unlabeled rooted trees are A000081.
%Y A198518 Lone-child-avoiding rooted trees are A001678(n+1).
%Y A198518 Topologically series-reduced rooted trees are A001679.
%Y A198518 Labeled lone-child-avoiding rooted trees are A060356.
%Y A198518 Cf. A000669, A004111, A108919, A291636, A330951, A331488, A331934.
%K A198518 nonn
%O A198518 0,4
%A A198518 _Paul D. Hanna_, Oct 26 2011