cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A007563, A035079-A035081.
Column k=2 of A316101.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

Additional comments from Christian G. Bower

A035079 Weigh transform of A007561.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 26, 71, 197, 564, 1639, 4833, 14406, 43374, 131652, 402525, 1238419, 3831520, 11912913, 37204431, 116655147, 367100319, 1159026041, 3670339794, 11655070593, 37104257405, 118398974620, 378627600346, 1213247498254, 3894924465243
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A035080, A035081.

Programs

  • Maple
    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-> g(n, n):
seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013
  • Mathematica
    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_] := g[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 22 2017, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 3.382016466020272807429818743... (same as for A035080), c = 0.2780120087122189647675707... . - Vaclav Kotesovec, Sep 12 2014

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

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A007561, A035081.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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