1,4

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

I. Novik, A. Postnikov and B. Sturmfels, Syzygies of oriented matroids

A. Postnikov, Source

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

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}]

(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)))

Cf. A053506, A060917, A060918.

Adjacent sequences: A057814 A057815 A057816 this_sequence A057818 A057819 A057820

Sequence in context: A002295 A027393 A124565 this_sequence A000405 A113352 A063169

nonn,nice,easy

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

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