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.

A287892 Number of unrooted unlabeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 1, 1, 3, 7, 25, 88, 366, 1583, 7336, 34982, 172384, 867638, 4452029, 23194392, 122462546, 653957197, 3527218134, 19192275883, 105248481503, 581223149532, 3230039198628, 18053111982952, 101426901301489, 572554846192811, 3246191706162233, 18478844801342495
Offset: 0

Views

Author

N. J. A. Sloane, Jun 21 2017

Keywords

Links

Crossrefs

Column k=4 of A332649.
Cf. A003081, A287889, A287890, A287891.

Programs

  • PARI
    \\ Here G(n) is A287891 as vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
G(n)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec(g*(g^2 + subst(g, x, x^2))/2))); concat([1], v)}
seq(n)={my(p=Ser(G(n))); my(g(d)=subst(p,x,x^d)); Vec(g(1) + x*(2*g(4) + 3*g(2)^2 - 2*g(1)^2*g(2) - 3*g(1)^4)/8)} \\ Andrew Howroyd, Feb 18 2020

Formula

G.f.: g(x) + x*(2*g(x^4) + 3*g(x^2)^2 - 2*g(x)^2*g(x^2) - 3*g(x)^4)/8 where g(x) is the g.f. of A287891.

Extensions

a(0) changed and terms a(12) and beyond from Andrew Howroyd, Feb 18 2020

A287889 Number of rooted labeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 12, 4410, 7560000, 35626991400, 357082280755200, 6536573599765809600, 197543239414923257856000, 9172025443146972656250000000, 619972004905097945232074342400000, 58507834434071888178873434004530400000, 7455351156359319047773396236777475276800000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 21 2017

Keywords

Links

Crossrefs

Cf. A034940, A287890, A287891, A287892.

Programs

  • Magma
    [(3*n+1)^n*Factorial(3*n)/(2^n*Factorial(n)): n in [0..12]]; // Vincenzo Librandi, Feb 19 2020
  • Mathematica
    Table[(3 n + 1)^n (3 n)! / (2^n n!), {n, 0, 15}] (* Vincenzo Librandi, Feb 19 2020 *)
  • PARI
    seq(n)={my(p=serlaplace(serreverse(x*exp(-x^3/2 + O(x^(3*n+1)))))); vector(n+1, k, polcoef(p, 3*k-2))} \\ Andrew Howroyd, Feb 17 2020

Formula

a(n) = (3*n+1)^n*(3*n)!/(2^n*n!). - Andrew Howroyd, Feb 17 2020

Extensions

a(0) changed and terms a(7) and beyond from Andrew Howroyd, Feb 17 2020

A287890 Number of unrooted labeled 4-cactus graphs on 3n+1 nodes.

Original entry on oeis.org

1, 3, 630, 756000, 2740537800, 22317642547200, 344030189461358400, 8979238155223784448000, 366881017725878906250000000, 22141857318039212329716940800000, 1887349497873286715447530129178400000, 219275034010568207287452830493455155200000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 21 2017

Keywords

Links

Crossrefs

Cf. A034941, A287889, A287891, A287892.

Programs

  • Magma
    [(3*n+1)^(n-1)*Factorial(3*n)/(2^n*Factorial(n)): n in [0..12]]; // Vincenzo Librandi, Feb 19 2020
  • Mathematica
    Table[(3 n + 1)^(n-1) (3 n)! / (2^n n!), {n, 0, 15}] (* Vincenzo Librandi, Feb 19 2020 *)
  • PARI
    seq(n)={my(p=serlaplace(serreverse(x*exp(-x^3/2 + O(x^(3*n+1))))/x)); vector(n+1, k, polcoef(p, 3*k-3))} \\ Andrew Howroyd, Feb 17 2020

Formula

a(n) = (3*n+1)^(n-1)*(3*n)!/(2^n*n!). - Andrew Howroyd, Feb 17 2020

Extensions

a(0) changed and terms a(7) and beyond from Andrew Howroyd, Feb 17 2020

A332648 Array read by antidiagonals: T(n,k) is the number of rooted unlabeled k-gonal cacti having n polygons.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 3, 5, 9, 1, 1, 1, 3, 11, 13, 20, 1, 1, 1, 4, 13, 46, 37, 48, 1, 1, 1, 4, 22, 62, 208, 111, 115, 1, 1, 1, 5, 25, 140, 333, 1002, 345, 286, 1, 1, 1, 5, 37, 176, 985, 1894, 5012, 1105, 719, 1, 1, 1, 6, 41, 319, 1397, 7374, 11258, 25863, 3624, 1842, 1
Offset: 0

Views

Author

Andrew Howroyd, Feb 18 2020

Keywords

Comments

The number of nodes will be n*(k-1) + 1.

Examples

			Array begins:
======================================================
n\k | 1   2    3     4     5      6      7       8
----+-------------------------------------------------
  0 | 1   1    1     1     1      1      1       1 ...
  1 | 1   1    1     1     1      1      1       1 ...
  2 | 1   2    2     3     3      4      4       5 ...
  3 | 1   4    5    11    13     22     25      37 ...
  4 | 1   9   13    46    62    140    176     319 ...
  5 | 1  20   37   208   333    985   1397    3059 ...
  6 | 1  48  111  1002  1894   7374  11757   31195 ...
  7 | 1 115  345  5012 11258  57577 103376  331991 ...
  8 | 1 286 1105 25863 68990 463670 937179 3643790 ...
  ...

Links

Crossrefs

Columns k=1..4 are A000012, A000081(n+1), A003080, A287891.
Cf. A303694, A332649.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[]); for(n=1, n, my(g=1+x*Ser(v)); v=EulerT(Vec((g^k + g^(k%2)*subst(g^(k\2), x, x^2))/2))); concat([1], v)}
T(n)={Mat(concat([vectorv(n+1,i,1)], vector(n,k,Col(R(n,k)))))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) }

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).

A381469 Number of unlabeled 2,3 cacti (triangular cacti with bridges) rooted at a triangle with n triangles and every node contained in exactly one triangle.

Original entry on oeis.org

0, 1, 1, 4, 15, 66, 304, 1503, 7622, 39856, 212447, 1151614, 6324924, 35127396, 196917025, 1112776860, 6332114208, 36252066562, 208665030299, 1206819559836, 7009605269315, 40871341270810, 239144296550695, 1403719120877546, 8263431521645830, 48774908707685849
Offset: 0

Views

Author

Andrew Howroyd, Feb 25 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A287891, A380634 (unrooted).

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)); Vec(x*(p^3 + 3*p*raise(p,2) + 2*raise(p,3))/6 + O(x*x^n), -n-1) }

Formula

G.f.: x*(B(x)^3 + 3*B(x)*B(x^2) + 2*B(x^3))/6 where B(x) is the g.f. of A287891.
Showing 1-6 of 6 results.

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

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