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 1-7 of 7 results.

A361245 Number of noncrossing 2,3 cacti with n nodes.

Original entry on oeis.org

1, 1, 1, 4, 20, 115, 715, 4683, 31824, 222300, 1586310, 11514030, 84742320, 630946446, 4743789260, 35965715780, 274659794160, 2110810059795, 16312695488265, 126693445737170, 988340783454380, 7740875273884445, 60846920004855985, 479854293574853085
Offset: 0

Views

Author

Andrew Howroyd, Mar 08 2023

Keywords

Comments

A 2,3 cactus is a cactus composed of bridges and triangles.

Links

Crossrefs

Cf. A091481, A091485, A091486, A091487, A361242, A361244.

Programs

  • PARI
    seq(n) = Vec(1 + x/(1 - serreverse((sqrt(1 + 4*x + O(x^n)) - 1)*(1 - x)^2/2)))

A091485 Number of labeled 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 1, 4, 28, 290, 3996, 68992, 1434112, 34895772, 973450000, 30636233936, 1074020373504, 41510792057176, 1753764940408768, 80412829785000000, 3977094146761424896, 211058327532167398928, 11963018212810373415168, 721321146876339731628352
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Comments

As Alois P. Heinz has pointed out, the e.g.f in the Example section does not match the offset. However, the identity a(n) = A091481(n)/n holds with the present offset of 1. - N. J. A. Sloane, Jun 23 2017

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 290*x^4/4! + 3996*x^5/5! +...

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

Links

Crossrefs

Cf. A091481, A091487, A000085.

Programs

  • Mathematica
    CoefficientList[1/x InverseSeries[x/Exp[x+x^2/2]+O[x]^20], x] Range[0, 18]! (* Jean-François Alcover, Aug 06 2018 *)

Formula

a(n) = A091481(n)/n.
From Paul D. Hanna, Jun 01 2012: (Start)
E.g.f.: (1/x)*Series_Reversion( x/exp(x+x^2/2) ).
E.g.f. satisfies: A(x) = exp( x*A(x) + x^2*A(x)^2/2 ).
E.g.f. satisfies: A( x/exp(x+x^2/2) ) = exp(x+x^2/2).
(End)
a(n+1) = n! * Sum_{k=0..n} (1/2)^(n-k) * (n+1)^(k-1) * binomial(k,n-k)/k!. - Seiichi Manyama, Aug 19 2023

A091486 Number of unlabeled rooted 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 1, 3, 7, 21, 60, 190, 600, 1977, 6589, 22408, 77050, 268178, 941599, 3333585, 11882427, 42615480, 153653039, 556664752, 2025330509, 7397242875, 27111563026, 99681629658, 367563272278, 1358945378906, 5036549490009, 18708739990129, 69640873691941
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Comments

Also number of unlabeled involution rooted trees.

Links

Crossrefs

Cf. A091487, A091488.

Programs

  • PARI
    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(p + (p^2 + subst(p,x,x^2))/2))))); Vec(p)} \\ Andrew Howroyd, Aug 30 2018

Formula

Shifts left under transform T where Ta = EULER(E_1, 2(a)). E_1, 2(a) has g.f. A(x)+(A(x^2)+A(x)^2)/2.

A380634 Number of unlabeled 2,3 cacti (triangular cacti with bridges) with n triangles and every node contained in exactly one triangle.

Original entry on oeis.org

1, 1, 1, 2, 6, 18, 66, 265, 1140, 5186, 24588, 120062, 600884, 3066490, 15907266, 83665520, 445317808, 2394928214, 12997988041, 71116953074, 391931826699, 2174062325068, 12130745830640, 68049392678632, 383601371168527, 2172093593344465, 12349917974708867
Offset: 0

Views

Author

Andrew Howroyd, Feb 24 2025

Keywords

Comments

The number of vertices is 3*n and for n > 0, the number of bridges is n-1.

Examples

			The a(3) = 2 cactus graphs are:
    o       o       o        o   o---o   o
   / \     / \     / \      / \   \ /   / \
  o---o---o---o---o---o    o---o---o---o---o

Links

Crossrefs

Cf. A091487, A287891, A380631, A381467.

Programs

  • PARI
    \\ here R(n) gives A287891 as g.f.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
raise(p,d) = {my(n=serprec(p,x)-1); subst(p + O(x^(n\d+1)), x, x^d)}
R(n)={my(p=1+O(x)); for(n=1, n, p = 1 + x*Ser(EulerT(Vec(p*(p^2 + raise(p,2))/2)))); p}
seq(n)={ my(p=R(n-1), g=p*(p^2 + raise(p,2))/2); Vec(1 + x*(x*(raise(g,2) - g^2) + p*raise(p,2) + (p^3 + 2*raise(p,3))/3)/2) }

Formula

a(n) = A380631(3*n,n) = A381467(3*n,n).

A091489 Number of asymmetric 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 3, 7, 21, 57, 159, 435, 1217, 3400, 9633, 27413, 78733, 227489, 661984, 1937211, 5701733, 16865522, 50126650, 149627241, 448448400, 1349060262, 4072508115, 12333762442, 37466367898, 114133581762
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Links

Crossrefs

Cf. A091487, A091488.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
b(n)={my(p=O(x)); for(n=1, n, p=x+x^2*(Ser(WeighT(Vec(p + (p^2 - subst(p,x,x^2))/2))))); p}
seq(n)={my(p=b(n)); Vec(p - p^2/2 - p^3/3 - subst(p, x, x^2)/2 + subst(p, x, x^3)/3)} \\ Andrew Howroyd, Aug 30 2018

Formula

G.f.: A(x) = B(x)-B(x)^2/2-B(x^2)/2+B(x^3)/3-B(x)^3/3 where B is g.f. of A091488

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 1-7 of 7 results.

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

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