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Search: id:A057817
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| A057817 |
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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. |
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4
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| 1, 0, 1, 6, 51, 560, 7575, 122052, 2285353, 48803904, 1171278945, 31220505800, 915350812299, 29281681800384, 1015074250155511, 37909738774479600, 1517587042234033425, 64830903253553212928, 2944016994706445303937
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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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.
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REFERENCES
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W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996
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LINKS
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I. Novik, A. Postnikov and B. Sturmfels, Syzygies of oriented matroids
A. Postnikov, Source
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FORMULA
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E.g.f.: exp(1/2*LambertW(-x)^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 10 2001
Exponential generating function: \int \exp( Sum_{m>1}(m-1)*m^{m-2}*x^{m}/m!) dx
(n-1) Sum_{k=0}^{[(n-2)/2]} {(n-2)! \over 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.: \int(exp(1/2*LambertW(-x)^2)dx). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 10 2001
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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}]
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PROGRAM
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(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)))
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CROSSREFS
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Cf. A053506, A060917, A060918.
Sequence in context: A002295 A027393 A124565 this_sequence A000405 A113352 A063169
Adjacent sequences: A057814 A057815 A057816 this_sequence A057818 A057819 A057820
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Alex Postnikov (apost(AT)math.berkeley.edu), Nov 06 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Nov 08 2000
Additional comments from Woong Kook (andrewk(AT)math.uri.edu), Feb 12 2002
Further comments from Michael Somos, Sep 18 2002
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