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 A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph. 36
 1, 1, 1, 2, 4, 9, 22, 59, 165, 496, 1540, 4960, 16390, 55408, 190572, 665699, 2354932, 8424025, 30424768, 110823984, 406734060, 1502876903, 5586976572, 20884546416, 78460794158, 296124542120, 1122346648913, 4270387848473 (list; graph; refs; listen; history; text; internal format)
 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]. Wikipedia, Block graph. FORMULA G.f.: A(x)=1+(C(x)-1)*(1-B(x)). B: G.f. for A007563. C: G.f. for A035052. a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.245899549044224207821149415964395... . - Vaclav Kotesovec, Jul 26 2014 a(n) = A304937(n) - A304937(n-1) for n>1, a(n) = 1 for n<2. - Gus Wiseman, May 22 2018 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; 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=); for(i=2, n, v=concat(, 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 Cf. A007549, A007563, A007716, A030019, A035051, A035052, A054921, A134957, A134959, A245566, A304867, A304887, A304937. Sequence in context: A092920 A177377 A321994 * A000571 A077003 A210726 Adjacent sequences:  A035050 A035051 A035052 * A035054 A035055 A035056 KEYWORD nonn,easy,nice AUTHOR Christian G. Bower, Oct 15 1998 STATUS approved

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