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A108919 Number of series-reduced labeled trees with n nodes.

%I A108919 #36 Sep 25 2022 09:32:49
%S A108919 1,0,1,1,13,51,601,4803,63673,775351,12186061,196158183,3661759333,
%T A108919 72413918019,1583407093633,36916485570331,929770285841137,
%U A108919 24904721121298671,711342228666833173,21502519995056598639,687345492498807434461,23135454269839313430715,818568166383797223246601,30357965273255025673685091
%N A108919 Number of series-reduced labeled trees with n nodes.
%C A108919 "Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.
%C A108919 Callan points out that A002792 is an incorrect version of this sequence. - _Joerg Arndt_, Jul 01 2014
%H A108919 Seiichi Manyama, <a href="/A108919/b108919.txt">Table of n, a(n) for n = 1..415</a>
%H A108919 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 A108919 a(n) = A060356(n)/n.
%F A108919 1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
%F A108919 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 A108919 E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - _Ilya Gutkovskiy_, Jul 01 2020
%t A108919 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 A108919 (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 A108919 The rooted version is A060356.
%Y A108919 Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.
%K A108919 easy,nonn
%O A108919 1,5
%A A108919 _Vladeta Jovovic_, Jul 20 2005
%E A108919 More terms from _Robert G. Wilson v_, Jul 21 2005
%E A108919 New name (from A002792) by _Joerg Arndt_, Aug 28 2014
%E A108919 Offset corrected by _Gus Wiseman_, Dec 31 2019