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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. 6
1, 0, 1, 6, 51, 560, 7575, 122052, 2285353, 48803904, 1171278945, 31220505800, 915350812299, 29281681800384, 1015074250155511, 37909738774479600, 1517587042234033425, 64830903253553212928, 2944016994706445303937 (list; graph; refs; listen; history; text; internal format)



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.

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.


W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996


Vincenzo Librandi, Table of n, a(n) for n = 1..200

I. Novik, A. Postnikov and B. Sturmfels, Syzygies of oriented matroids, arXiv:math/0009241 [math.CO], 2000.

A. Postnikov, Papers


E.g.f.: exp(1/2*LambertW(-x)^2). - Vladeta Jovovic, Apr 10 2001

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}.

E.g.f.: exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/m!).

E.g.f.: integral(exp(1/2*LambertW(-x)^2)dx). - Vladeta Jovovic, Apr 10 2001

a(n) ~ exp(-1/2)*n^(n-2). - Vaclav Kotesovec, Dec 12 2012

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


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.


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:


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}]

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-Fran├žois Alcover, Dec 11 2012, after Maple *)


(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))

(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)))


df(n)=(2*n)!/(n!*2^n);  \\ A001147

he(n, x)=x^n+sum(k=1, n\2, binomial(n, 2*k) * df(k) * x^(n-2*k) );

a(n)=if( n<3, n==1, (n-2)*he(n-3, n-1) );

/* Joerg Arndt, May 06 2013 */


Cf. A053506, A060917, A060918.

Cf. column k=2 of A243098.

Sequence in context: A253097 A345259 A124565 * A000405 A113352 A063169

Adjacent sequences:  A057814 A057815 A057816 * A057818 A057819 A057820




Alex Postnikov (apost(AT)math.mit.edu), Nov 06 2000


More terms from James A. Sellers, Nov 08 2000

Additional comments from Woong Kook (andrewk(AT)math.uri.edu), Feb 12 2002

Further comments from Michael Somos, Sep 18 2002

Updated author's URL and e-mail address R. J. Mathar, May 23 2010



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Last modified September 19 08:05 EDT 2021. Contains 347556 sequences. (Running on oeis4.)