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 1-6 of 6 results.

A035082 Number of rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 5, 13, 27, 67, 157, 390, 963, 2437, 6186, 15908, 41127, 107148, 280569, 738675, 1953054, 5185364, 13816018, 36934431, 99030038, 266254593, 717652816, 1938831589, 5249221790, 14240130827, 38702218134, 105367669062
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures.
  • F. Harary and E. M. Palmer, Graphical Enumeration, p. 71

Links

Crossrefs

Cf. A003080, A035083, A035084, A035085, A035086, A035087, A035088.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(p))} \\ Andrew Howroyd, Aug 30 2018

Formula

Shifts left under transform T where Ta = EULER(BIK(a)-a).

A035083 DIK(b)-DIK[ 2 ](b)-b where b is A035082.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 7, 14, 33, 74, 180, 438, 1090, 2741, 6994, 17966, 46565, 121440, 318597, 839953, 2224486, 5914248, 15780662, 42241422, 113402369, 305254039, 823690961, 2227640597, 6037142355, 16392945284, 44592703836
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A035082, A035084, A035085.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p,n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(DIK(p, n) - p - (p^2 + subst(p, x, x^2))/2, -(n+1))} \\ Andrew Howroyd, Aug 31 2018

A035084 BIK(b)-b where b is A035082.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 273, 676, 1694, 4296, 10991, 28350, 73614, 192327, 505093, 1332801, 3531598, 9393501, 25070735, 67121670, 180216260, 485133376, 1309101329, 3540394176, 9594562328, 26051397890, 70861839620
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Links

Crossrefs

Cf. A035082, A035083, A035085.

Programs

  • PARI
    BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(BIK(p)-1)-Vec(p))} \\ Andrew Howroyd, Aug 30 2018

A035088 Number of labeled polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 3, 27, 240, 2985, 42840, 731745, 14243040, 313570845, 7683984000, 207685374435, 6135743053440, 196754537704725, 6805907485977600, 252620143716765825, 10015402456976716800, 422410127508300756825, 18884777200534941696000
Offset: 0

Views

Author

Christian G. Bower, Nov 15 1998

Keywords

Comments

A Husimi tree is a connected graph in which no line lies on more than one cycle [Harary, 1953]. - Jonathan Vos Post, Mar 12 2010

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
  • F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141.
  • F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
  • F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39. pp. 315-322, 1953.
  • F. Harary, G. Uhlenbeck (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences 39: 315-322. - From Jonathan Vos Post, Mar 12 2010

Links

Crossrefs

Cf. A035082, A035083, A035084, A035085, A035086, A035087.

Programs

  • Mathematica
    max = 20; s = 1+InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, max}], x]; a[n_] := SeriesCoefficient[s, n]*(n-1)!; a[0]=1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 27 2016, after Vaclav Kotesovec at A035087 *)

Formula

a(n) = A035087(n)/n, n > 0.

A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.

Original entry on oeis.org

1, 1, 2, 4, 10, 30, 105, 400, 1654, 7229, 32944, 154749, 744973, 3655993, 18232812, 92162974, 471301437, 2434542190, 12687850499, 66646225443, 352548333438, 1876770716627, 10048289587337, 54079948967654, 292447643655469, 1588388448970674, 8661869330014601
Offset: 1

Views

Author

Andrew Howroyd, Feb 18 2020

Keywords

Examples

			a(3) = 2 because there are two cacti on 5 nodes which are a pentagon and 2 triangles joined at a node.

Links

Crossrefs

Cf. A000083, A035085, A091487, A332649, A332651.

Programs

  • PARI
    \\ Here UCacti gives number of unrooted cacti with restricted polygons.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
RCacti(u)={my(v=[1]); while(#v<#u, my(g=x*Ser(v), g2=subst(g,x,x^2) + O(x^2*x^#v), r=sum(k=1, #u-1, my(c=u[k+1]); if(c, c*(g^k + g^(k%2)*g2^(k\2))))/2 + O(x^#u)); v=concat([1], EulerT(Vec(r, 1-serprec(r, x))))); v}
UCacti(u)={my(p=x*Ser(RCacti(u))); my(g(d)=subst(p + O(x*x^(#u\d)), x, x^d)); Vec(g(1) + sum(k=1, #u, my(c=u[k]); if(c, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - (g(1)^k)/2 + if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)))}
seq(n)={my(v=UCacti(vector(2*n-1, i, i>2 && isprime(i)))); vector(n, i, v[2*i-1])}

A332651 Number of polygonal cacti on n unlabeled nodes with every polygon having an even number of edges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 4, 2, 7, 9, 14, 26, 48, 71, 154, 243, 478, 894, 1631, 3149, 6062, 11295, 22469, 42900, 83528, 164829, 321012, 632960, 1255613, 2472803, 4928140, 9808439, 19533534, 39134059, 78345317, 157177556, 316398963, 636790282, 1284910954
Offset: 0

Views

Author

Andrew Howroyd, Feb 18 2020

Keywords

Comments

Bridges are disallowed.

Examples

			a(6) = 1 corresponding with a hexagon.
a(7) = 1 corresponding with two quadrilaterals joined at a node.

Links

Crossrefs

Cf. A000083, A035085, A091487, A332649, A332650.

Programs

  • PARI
    \\ See A332650 for UCacti.
seq(n)={concat([1], UCacti(vector(n, i, i>2&&i%2==0)))}
Showing 1-6 of 6 results.

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