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A052318 Number of labeled rooted trimmed trees with n nodes.

%I A052318 #38 Jun 02 2024 11:08:31
%S A052318 1,2,3,16,145,1536,19579,290816,4942305,94689280,2020278931,
%T A052318 47523053568,1222147737265,34117226135552,1027550555918475,
%U A052318 33213871550365696,1146891651823112641,42135941698113503232,1641164216596258397347,67550839668807638712320
%N A052318 Number of labeled rooted trimmed trees with n nodes.
%C A052318 A rooted trimmed tree is a tree with a forbidden limb of length 2.
%C A052318 A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.
%H A052318 Alois P. Heinz, <a href="/A052318/b052318.txt">Table of n, a(n) for n = 1..400</a>
%H A052318 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052318 E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
%F A052318 E.g.f.: -LambertW(-x/exp(x^2)). - _Vaclav Kotesovec_, Jan 08 2014
%F A052318 a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - _Vaclav Kotesovec_, Jan 08 2014
%p A052318 A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # _Alois P. Heinz_, Aug 23 2008
%t A052318 a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, Sep 10 2012, after _Vladimir Kruchinin_ *)
%t A052318 Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o A052318 (Maxima) a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* _Vladimir Kruchinin_, Aug 07 2012 */
%Y A052318 Cf. A002955, A002988-A002992, A052319-A052329.
%K A052318 nonn,eigen
%O A052318 1,2
%A A052318 _Christian G. Bower_, Dec 15 1999