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.

1, 0, 1, 6, 51, 560, 7575, 122052, 2285353, 48803904, 1171278945, 31220505800, 915350812299, 29281681800384, 1015074250155511, 37909738774479600, 1517587042234033425, 64830903253553212928, 2944016994706445303937

Offset: 1

Author

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

Keywords

• nonn
• nice
• easy

Comments

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.

Examples

			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.
		

References

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

Links

• 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

Crossrefs

Cf. A053506, A060917, A060918.

Cf. column k=2 of A243098.

Programs

Maple

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:

Mathematica

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-François 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)))

PARI

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 */

Formula

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

Extensions

More terms from James 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