%I #36 Sep 25 2022 09:32:49

%S 1,0,1,1,13,51,601,4803,63673,775351,12186061,196158183,3661759333,

%T 72413918019,1583407093633,36916485570331,929770285841137,

%U 24904721121298671,711342228666833173,21502519995056598639,687345492498807434461,23135454269839313430715,818568166383797223246601,30357965273255025673685091

%N Number of series-reduced labeled trees with n nodes.

%C "Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.

%C Callan points out that A002792 is an incorrect version of this sequence. - _Joerg Arndt_, Jul 01 2014

%H Seiichi Manyama, <a href="/A108919/b108919.txt">Table of n, a(n) for n = 1..415</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], 2014.

%F a(n) = A060356(n)/n.

%F 1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.

%F E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - _Olivier Gérard_, Dec 31 2013 (edited by _Gus Wiseman_, Dec 31 2019)

%F E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - _Ilya Gutkovskiy_, Jul 01 2020

%t f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* _Robert G. Wilson v_, Jul 21 2005 *)

%o (PARI) a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n,k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ _Joerg Arndt_, Aug 28 2014

%Y The rooted version is A060356.

%Y Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.

%K easy,nonn

%O 1,5

%A _Vladeta Jovovic_, Jul 20 2005

%E More terms from _Robert G. Wilson v_, Jul 21 2005

%E New name (from A002792) by _Joerg Arndt_, Aug 28 2014

%E Offset corrected by _Gus Wiseman_, Dec 31 2019