OFFSET
0,4
COMMENTS
Equivalently, this is the number of "hypertrees" on n unlabeled nodes, i.e., connected hypergraphs that have no cycles, assuming that each edge contains at least two vertices. - Don Knuth, Jan 26 2008. See A134955 for hyperforests.
Graphs where every block is a complete graph are also called block graphs or clique tree. They can be characterized as induced-diamond-free chordal graphs. - Falk Hüffner, Jul 25 2019
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.14).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Robert Hellmann and Eckard Bich, A systematic formulation of the virial expansion for nonadditive interaction potentials, J. Chem. Phys. 135, 084117 (2011); doi:10.1063/1.3626524 (7 pages).
R. Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO].
Eric Weisstein's World of Mathematics, Block Graph
Eric Weisstein's World of Mathematics, Connected Graph
Wikipedia, Block graph.
FORMULA
a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.245899549044224207821149415964395... . - Vaclav Kotesovec, Jul 26 2014
EXAMPLE
From Gus Wiseman, May 20 2018: (Start)
Non-isomorphic representatives of the a(5) = 9 hypertrees are the following:
{{1,2,3,4,5}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,4},{2,5},{3,4,5}}
{{1,5},{2,5},{3,4,5}}
{{1,3},{2,4},{3,5},{4,5}}
{{1,4},{2,5},{3,5},{4,5}}
{{1,5},{2,5},{3,5},{4,5}}
(End)
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(B): c:= etr(b): B:= n-> if n=0 then 0 else c(n-1) fi: C:= etr(B): a:= n-> B(n)+C(n) -add(B(k)*C(n-k), k=0..n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008
MATHEMATICA
ClearAll[etr, b, a]; etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[0]=0; b[n_] := b[n] = etr[etr[b]][n-1]; a[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}]; Table[ a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 09 2012, after Alois P. Heinz *)
PROG
(PARI) \\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, May 22 2018
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Christian G. Bower, Oct 15 1998
STATUS
approved