A035052 -id:A035052 - cp's OEIS frontend

A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

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

Offset: 0

Christian G. Bower, Oct 15 1998

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

			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)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.14).

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).
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Block Graph
Eric Weisstein's World of Mathematics, Connected Graph
Wikipedia, Block graph.

Cf. A007549, A007563, A007716, A030019, A035051, A035052, A054921, A134957, A134959, A245566, A304867, A304887, A304937.

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

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 *)

\\ 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

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

A007563 Number of rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

0, 1, 1, 3, 8, 25, 77, 258, 871, 3049, 10834, 39207, 143609, 532193, 1990163, 7503471, 28486071, 108809503, 417862340, 1612440612, 6248778642, 24309992576, 94905791606, 371691137827, 1459935388202, 5749666477454

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.13).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1600 (first 200 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 167
N. J. A. Sloane, Transforms

Cf. A007549, A030019, A035051, A035052, A035053.
Column k=2 of A144042.
Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): c:= etr(b): a:= n-> if n=0 then 0 else c(n-1) fi: seq(a(n), n=0..25); # Alois P. Heinz, Sep 06 2008

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]; a[0] = 0; a[n_] := etr[etr[a]][n-1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([0], v)} \\ Andrew Howroyd, May 20 2018

Shifts left when Euler transform is applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524044275... (see A245566), c = 0.1977574301782950818433893126632477845870281049591883888... . - Vaclav Kotesovec, Jul 26 2014

New description from Christian G. Bower, Oct 15 1998

A245566 Decimal expansion of a constant related to A007563.

Original entry on oeis.org

4, 1, 8, 9, 6, 1, 0, 9, 5, 8, 3, 9, 3, 8, 2, 6, 9, 6, 5, 5, 2, 7, 0, 3, 6, 4, 5, 4, 5, 2, 4, 0, 4, 4, 2, 7, 5, 9, 4, 2, 3, 8, 9, 9, 2, 5, 9, 1, 5, 9, 3, 6, 5, 9, 4, 1, 3, 2, 8, 5, 7, 7, 4, 2, 5, 9, 8, 9, 8, 7, 0, 6, 4, 9, 1, 2, 0, 6, 1, 9, 9, 0, 1, 7, 6, 0, 7, 4, 0, 6, 3, 9, 5, 8, 9, 6, 8, 5, 6, 3, 3, 8, 2, 5, 3

Offset: 1

Vaclav Kotesovec, Jul 26 2014

			4.18961095839382696552703645452404427594238992591593659413285774...

Cf. A007563, A035052, A035053, A134955.

Equals lim n -> infinity A007563(n)^(1/n).
Equals lim n -> infinity A035052(n)^(1/n).
Equals lim n -> infinity A035053(n)^(1/n).
Equals lim n -> infinity A134955(n)^(1/n).

A318607 Triangle read by rows: T(n,k) is the number of sets of rooted hypertrees on a total of n unlabeled nodes with a total of k edges, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 9, 1, 5, 20, 42, 46, 20, 1, 6, 30, 86, 145, 128, 48, 1, 7, 42, 153, 353, 483, 364, 115, 1, 8, 56, 248, 729, 1369, 1592, 1029, 286, 1, 9, 72, 376, 1345, 3236, 5150, 5151, 2930, 719, 1, 10, 90, 541, 2287, 6728, 13708, 18792, 16513, 8344, 1842

Offset: 1

Andrew Howroyd, Aug 30 2018

Equivalently, the number of sets of rooted connected graphs on a total of n unlabeled nodes with a total of k blocks where every block is a complete graph.

Bivariate Euler transform of triangle A318602.

			Triangle begins:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 6, 4;
  1, 4, 12, 16, 9;
  1, 5, 20, 42, 46, 20;
  1, 6, 30, 86, 145, 128, 48;
  1, 7, 42, 153, 353, 483, 364, 115;
  1, 8, 56, 248, 729, 1369, 1592, 1029, 286;
  ...
Case n=3: There are 5 sets of rooted graph which are illustrated below (an x marks a root node). These have 0, 1, 1, 2, 2 blocks so row 3 is 1, 2, 2.
      x        o        o        o        o
              /        / \        \      /
    x   x    x   x    x---o    x---o    x---o

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Rightmost diagonal is A000081 (rooted trees).
Row sums are A035052.
Cf. A318601, A318602.

\\ here EulerMT is Euler transform (bivariate version).
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
A(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- EulerMT(v)]}
{ my(T=A(10)); for(n=1, #T, print(T[n])) }

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A035053 Number of connected graphs on n unlabeled nodes where every block is a complete graph.

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A007563 Number of rooted connected graphs where every block is a complete graph.

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A245566 Decimal expansion of a constant related to A007563.

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A318607 Triangle read by rows: T(n,k) is the number of sets of rooted hypertrees on a total of n unlabeled nodes with a total of k edges, (0 <= k < n).

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