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A001862 Number of forests of least height with n nodes.

%I A001862 M1773 N0702 #38 May 19 2024 14:02:59
%S A001862 1,1,2,7,26,111,562,3151,19252,128449,925226,7125009,58399156,
%T A001862 507222535,4647051970,44747776651,451520086208,4761032807937,
%U A001862 52332895618066,598351410294193,7102331902602676,87365859333294151,1111941946738682522,14621347433458883187
%N A001862 Number of forests of least height with n nodes.
%C A001862 From _Gus Wiseman_, Feb 14 2024: (Start)
%C A001862 Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):
%C A001862   {}  {1}  {12}   {1-23}
%C A001862            {1-2}  {2-13}
%C A001862                   {3-12}
%C A001862                   {12-13}
%C A001862                   {12-23}
%C A001862                   {13-23}
%C A001862                   {1-2-3}
%C A001862 (End)
%D A001862 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.
%D A001862 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001862 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001862 T. D. Noe, <a href="/A001862/b001862.txt">Table of n, a(n) for n = 0..100</a>
%H A001862 John Riordan, <a href="https://doi.org/10.1016/S0021-9800(68)80033-X">Forests of labeled trees</a>, J. Combin. Theory, 5 (1968), 90-103.
%H A001862 John Riordan, <a href="/A001861/a001861.pdf">Letter to N. J. A. Sloane, Oct. 1970</a>
%H A001862 John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>
%F A001862 E.g.f.: exp(x*(exp(x)-x/2)).
%F A001862 Binomial transform of A053530. - _Gus Wiseman_, Feb 14 2024
%t A001862 Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* _Geoffrey Critzer_, Mar 13 2011 *)
%t A001862 fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
%t A001862 Table[Length@fasmin[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&]],{n,0,4}] (* _Gus Wiseman_, Feb 14 2024 *)
%Y A001862 The connected case is A000272.
%Y A001862 Without loops we have A053530, minimal case of A369191.
%Y A001862 This is the minimal case of A322661.
%Y A001862 A000666 counts unlabeled loop-graphs, covering A322700.
%Y A001862 A006125 counts simple graphs; also loop-graphs if shifted left.
%Y A001862 A006129 counts covering graphs, unlabeled A002494.
%Y A001862 A054548 counts graphs covering n vertices with k edges, with loops A369199.
%Y A001862 Cf. A000085, A000169, A003465, A062740, A066383, A133686, A368597.
%K A001862 nonn
%O A001862 0,3
%A A001862 _N. J. A. Sloane_
%E A001862 Formula and more terms from _Vladeta Jovovic_, Mar 27 2001