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 A060356 #33 Apr 02 2020 03:20:32
%S A060356 0,1,0,3,4,65,306,4207,38424,573057,7753510,134046671,2353898196,
%T A060356 47602871329,1013794852266,23751106404495,590663769125296,
%U A060356 15806094859299329,448284980183376078,13515502344669830287
%N A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).
%C A060356 Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - _Gus Wiseman_, Jan 20 2020
%H A060356 Harry J. Smith, <a href="/A060356/b060356.txt">Table of n, a(n) for n = 0..100</a>
%H A060356 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 A060356 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 A060356 a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - _Vladeta Jovovic_, Sep 17 2003
%F A060356 a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - _Vaclav Kotesovec_, Nov 27 2012
%F A060356 a(n) = n * A108919(n). - _Gus Wiseman_, Dec 31 2019
%e A060356 From _Gus Wiseman_, Dec 31 2019: (Start)
%e A060356 Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
%e A060356 1[2,3[4,5[6,7]]]
%e A060356 1[2,3[4,5,6,7]]
%e A060356 1[2[3,4],5[6,7]]
%e A060356 1[2,3,4[5,6,7]]
%e A060356 1[2,3,4,5[6,7]]
%e A060356 1[2,3,4,5,6,7]
%e A060356 (End)
%p A060356 seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # _G. C. Greubel_, Mar 16 2020
%t A060356 CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%t A060356 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A060356 a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
%t A060356 Array[a,10] (* _Gus Wiseman_, Dec 31 2019 *)
%o A060356 (PARI) { for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jul 04 2009
%o A060356 (PARI) my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ _G. C. Greubel_, Feb 19 2018
%o A060356 (GAP) List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # _Muniru A Asiru_, Feb 19 2018
%Y A060356 Cf. A052871, A060313.
%Y A060356 Cf. A008297.
%Y A060356 Column k=0 of A231602.
%Y A060356 The unlabeled version is A001678(n + 1).
%Y A060356 The case where the root is fixed is A108919.
%Y A060356 Unlabeled rooted trees are counted by A000081.
%Y A060356 Lone-child-avoiding rooted trees with labeled leaves are A000311.
%Y A060356 Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y A060356 Singleton-reduced rooted trees are counted by A330951.
%Y A060356 Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.
%K A060356 easy,nonn
%O A060356 0,4
%A A060356 _Vladeta Jovovic_, Apr 01 2001