A035079 -id:A035079 - cp's OEIS frontend

A007561 Number of asymmetric rooted connected graphs where every block is a complete graph.

0, 1, 1, 1, 3, 6, 16, 43, 120, 339, 985, 2892, 8606, 25850, 78347, 239161, 734922, 2271085, 7054235, 22010418, 68958139, 216842102, 684164551, 2165240365, 6871792256, 21865189969, 69737972975, 222915760126, 714001019626, 2291298553660, 7366035776888

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1900
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 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]
N. J. A. Sloane, Transforms

Cf. A007563, A035079-A035081.

Column k=2 of A316101.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 0, b(n-1, n-1)):
seq(a(n), n=0..40); # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<1, 0, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 0, 40}] (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)

Shifts left when weigh-transform applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 3.382016466020272807429818743..., c = 0.161800727760188847021075748... . - Vaclav Kotesovec, Jul 26 2014

Additional comments from Christian G. Bower

A035081 Number of increasing asymmetric rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 1, 7, 27, 167, 1451, 12672, 133356, 1573608, 20731512, 299642958, 4732486932, 81201040470, 1500094187292, 29730606352920, 628968809015766, 14147458062941100, 337143091156288002, 8485143902146640124

Offset: 1

Christian G. Bower, Nov 15 1998

In an increasing rooted graph nodes are numbered and numbers increase as you move away from root.

Andrew Howroyd, Table of n, a(n) for n = 1..100
C. G. Bower, Transforms (2)

Cf. A007549, A007561, A035079, A035080.

EGJ(v)={Vec(serlaplace(prod(k=1, #v, (1 + x^k/k! + O(x*x^#v))^v[k]))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], EGJ(EGJ(v)))); v} \\ Andrew Howroyd, Sep 11 2018

Shifts left when EGJ transform applied twice.

A035080 Number of asymmetric connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 3, 7, 21, 60, 168, 472, 1344, 3843, 11104, 32305, 94734, 279708, 831401, 2485877, 7474667, 22589771, 68594611, 209198103, 640591332, 1968920180, 6072766832, 18791062733, 58321579888, 181524367875, 566488767763, 1772261945866, 5557515157647

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A007561, A035081.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b((n-1)$2)+g(n$2)-add(b((i-1)$2)*g((n-i)$2), i=0..n):
seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i < 1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n-1, n-1] + g[n, n] - Sum[b[i-1, i-1]*g[n-i, n-i], {i, 0, n}]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)

G.f.: A(x) = B(x) + C(x) - B(x)*C(x), where B and C are g.f.s of A007561 and A035079, respectively.

a(n) ~ c * d^n / n^(5/2), where d = 3.38201646602027280742981874... (same as for A007561), c = 0.12430588691278777480105... . - Vaclav Kotesovec, Sep 10 2014

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A007561 Number of asymmetric rooted connected graphs where every block is a complete graph.

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A035081 Number of increasing asymmetric rooted connected graphs where every block is a complete graph.

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A035080 Number of asymmetric connected graphs where every block is a complete graph.

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