%I #33 Apr 02 2020 03:20:32

%S 0,1,0,3,4,65,306,4207,38424,573057,7753510,134046671,2353898196,

%T 47602871329,1013794852266,23751106404495,590663769125296,

%U 15806094859299329,448284980183376078,13515502344669830287

%N Expansion of e.g.f.: -LambertW(-x/(1+x)).

%C 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 Harry J. Smith, <a href="/A060356/b060356.txt">Table of n, a(n) for n = 0..100</a>

%H 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 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 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 a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - _Vaclav Kotesovec_, Nov 27 2012

%F a(n) = n * A108919(n). - _Gus Wiseman_, Dec 31 2019

%e From _Gus Wiseman_, Dec 31 2019: (Start)

%e Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:

%e 1[2,3[4,5[6,7]]]

%e 1[2,3[4,5,6,7]]

%e 1[2[3,4],5[6,7]]

%e 1[2,3,4[5,6,7]]

%e 1[2,3,4,5[6,7]]

%e 1[2,3,4,5,6,7]

%e (End)

%p seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # _G. C. Greubel_, Mar 16 2020

%t CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];

%t Array[a,10] (* _Gus Wiseman_, Dec 31 2019 *)

%o (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 (PARI) my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ _G. C. Greubel_, Feb 19 2018

%o (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 Cf. A052871, A060313.

%Y Cf. A008297.

%Y Column k=0 of A231602.

%Y The unlabeled version is A001678(n + 1).

%Y The case where the root is fixed is A108919.

%Y Unlabeled rooted trees are counted by A000081.

%Y Lone-child-avoiding rooted trees with labeled leaves are A000311.

%Y Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.

%Y Singleton-reduced rooted trees are counted by A330951.

%Y Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Apr 01 2001