login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
OFFSET
1,4
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.
REFERENCES
W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996
LINKS
I. Novik, A. Postnikov and B. Sturmfels, Syzygies of oriented matroids, arXiv:math/0009241 [math.CO], 2000.
A. Postnikov, Papers
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
EXAMPLE
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.
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 *)
PROG
(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 */
CROSSREFS
Cf. column k=2 of A243098.
Sequence in context: A253097 A345259 A124565 * A000405 A113352 A063169
KEYWORD
nonn,nice,easy
AUTHOR
Alex Postnikov (apost(AT)math.mit.edu), Nov 06 2000
EXTENSIONS
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
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)