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%I #28 Feb 07 2020 20:49:52
%S 1,0,1,6,51,560,7575,122052,2285353,48803904,1171278945,31220505800,
%T 915350812299,29281681800384,1015074250155511,37909738774479600,
%U 1517587042234033425,64830903253553212928,2944016994706445303937
%N 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 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 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 W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996
%H Vincenzo Librandi, <a href="/A057817/b057817.txt">Table of n, a(n) for n = 1..200</a>
%H I. Novik, A. Postnikov and B. Sturmfels, <a href="https://arxiv.org/abs/math/0009241">Syzygies of oriented matroids</a>, arXiv:math/0009241 [math.CO], 2000.
%H A. Postnikov, <a href="http://math.mit.edu/~apost/papers/">Papers</a>
%F E.g.f.: exp(1/2*LambertW(-x)^2). - _Vladeta Jovovic_, Apr 10 2001
%F E.g.f.: integral exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/m!) dx (n-1) Sum_{k=0}^{[(n-2)/2]} binomial((n-2)! , 2^k k! (n-2-2k)!) n^{n-2-2k}.
%F E.g.f.: exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/m!).
%F E.g.f.: integral(exp(1/2*LambertW(-x)^2)dx). - _Vladeta Jovovic_, Apr 10 2001
%F a(n) ~ exp(-1/2)*n^(n-2). - _Vaclav Kotesovec_, Dec 12 2012
%F a(n) = n^(n-2) - Sum_{k=1..n-1} binomial(n-1,k-1) * k^(k-2) * a(n-k). - _Ilya Gutkovskiy_, Feb 07 2020
%e 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 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 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}]
%t a[n_] := (n-2)*Sum[ (n-1)^(n-2k-3)*(n-3)! / (2^k*k!*(n-2k-3)!), {k, 0, Floor[ (n-3)/ 2 ]}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Dec 11 2012, after Maple *)
%o (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 (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)))
%o (PARI)
%o df(n)=(2*n)!/(n!*2^n); \\ A001147
%o he(n,x)=x^n+sum(k=1, n\2, binomial(n,2*k) * df(k) * x^(n-2*k) );
%o a(n)=if( n<3, n==1, (n-2)*he(n-3, n-1) );
%o /* _Joerg Arndt_, May 06 2013 */
%Y Cf. A053506, A060917, A060918.
%Y Cf. column k=2 of A243098.
%K nonn,nice,easy
%O 1,4
%A Alex Postnikov (apost(AT)math.mit.edu), Nov 06 2000
%E More terms from _James A. Sellers_, Nov 08 2000
%E Additional comments from Woong Kook (andrewk(AT)math.uri.edu), Feb 12 2002
%E Further comments from _Michael Somos_, Sep 18 2002
%E Updated author's URL and e-mail address _R. J. Mathar_, May 23 2010
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