Search: id:A057817
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%I A057817
%S A057817 1,0,1,6,51,560,7575,122052,2285353,48803904,1171278945,31220505800,
%T A057817 915350812299,29281681800384,1015074250155511,37909738774479600,
%U A057817 1517587042234033425,64830903253553212928,2944016994706445303937
%N A057817 Moebius invariant of cographic hyperplane arrangement for complete graph 
               K_n. Also value of Tutte dichromatic polynomial T_G(0,1) for G=K_n. 
               Also alternating sum F_{n,1} - F_{n,2} + F_{n,3} - ..., where F_{n,
               k} is the number of labeled forests on n nodes with k connected components.
%C A057817 The rank of reduced homology groups for the matroid complex of acyclic 
               subgraphs in complete graph K_n (n>1). It is also the number of labeled 
               edge-rooted forests on n-1 nodes where each connected component contains 
               at least one edge.
%C A057817 The description of this sequence as the number of labeled edge-rooted 
               forests on n-1 nodes appeared in W. Kook's Ph.D. thesis (G. Carlsson, 
               advisor), Categories of acyclic graphs and automorphisms of free 
               groups, Stanford University, 1996.
%D A057817 W. Kook, Categories of acyclic graphs and automorphisms of free groups, 
               Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996
%H A057817 I. Novik, A. Postnikov and B. Sturmfels, <a href="http://arXiv.org/abs/
               math.CO/0009241">Syzygies of oriented matroids</a>
%H A057817 A. Postnikov, <a href="http://www.math.berkeley.edu/~apost/papers/">Source</
               a>
%F A057817 E.g.f.: exp(1/2*LambertW(-x)^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 10 2001
%F A057817 Exponential generating function: \int \exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/
               m!) dx
%F A057817 (n-1) Sum_{k=0}^{[(n-2)/2]} {(n-2)! \over 2^k k! (n-2-2k)!} n^{n-2-2k}.
%F A057817 E.g.f.: \exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/m!).
%F A057817 E.g.f.: \int(exp(1/2*LambertW(-x)^2)dx). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 10 2001
%e A057817 For n=4, the number of labeled edge-rooted forests on three (= n-1) nodes 
               is 6: There are 3 labeled trees on three nodes. These are the only 
               forests with at least one edge in each connected component. Each 
               tree has 2 edges and each of the two may be marked as the root.
%p A057817 for n from 1 to 50 do printf(`%d,`, (n-1)*sum((n-2)!/(2^k*k!*(n-2-2*k)!)*n^(n-2-2*k), 
               k=0..floor((n-2)/2))) od:
%t A057817 s=20;(*generates first s terms starting from n=2*) K := Exp[Sum[(m-1)*(m^(m-2))*(x^m)/
               m!, {m, 2, 2s}]]; S := Series[K, {x, 0, s}]; h[i_] := SeriesCoefficient[S, 
               i-1]*(i-1)!; Table[h[n+1], {n, s}]
%o A057817 (PARI) a(n)=if(n<1,0,(n-1)!*polcoeff(exp(sum(k=1,n-1,k^(k-1)*x^k/k!,O(x^n))^2/
               2),n-1))
%o A057817 (PARI) a(n)=if(n<2,n==1,sum(k=0,(n-3)\2,(n-1)!/(2^k*k!*(n-3-2*k)!)*(n-1)^(n-4-2*k)))
%Y A057817 Cf. A053506, A060917, A060918.
%Y A057817 Sequence in context: A002295 A027393 A124565 this_sequence A000405 A113352 
               A063169
%Y A057817 Adjacent sequences: A057814 A057815 A057816 this_sequence A057818 A057819 
               A057820
%K A057817 nonn,nice,easy
%O A057817 1,4
%A A057817 Alex Postnikov (apost(AT)math.berkeley.edu), Nov 06 2000
%E A057817 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Nov 08 2000
%E A057817 Additional comments from Woong Kook (andrewk(AT)math.uri.edu), Feb 12 
               2002
%E A057817 Further comments from Michael Somos, Sep 18 2002

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