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 1-4 of 4 results.

A381467 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k cycles and no node a member of more than one cycle, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 6, 13, 1, 11, 33, 4, 23, 89, 21, 47, 240, 85, 2, 106, 657, 345, 16, 235, 1806, 1289, 109, 551, 5026, 4713, 627, 6, 1301, 13999, 16622, 3259, 64, 3159, 39260, 57535, 15576, 598, 7741, 110381, 195212, 69983, 4394, 18, 19320, 311465, 653318, 299354, 28286, 295
Offset: 0

Views

Author

Andrew Howroyd, Feb 24 2025

Keywords

Comments

All such graphs are cactus graphs (with bridges allowed).

Examples

			Triangle begins:
     1;
     1;
     1;
     1,     1;
     2,     2;
     3,     5;
     6,    13,     1;
    11,    33,     4;
    23,    89,    21;
    47,   240,    85,     2;
   106,   657,   345,    16;
   235,  1806,  1289,   109;
   551,  5026,  4713,   627,   6;
  1301, 13999, 16622,  3259,  64;
  3159, 39260, 57535, 15576, 598;
  ...

Links

Crossrefs

Row sums are A381468.
Columns k=0..2 are A000055, A001429, A381470.
Cf. A380631, A380634.

Programs

  • PARI
    EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
R(n,y)={my(g=x+O(x^2)); for(n=2, n, my(p=x*EulerMTS(g), p2=raise(p,2)); g=p + p*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
G(n,y=1)={my(g=R(n,y), p = x*EulerMTS(g) + O(x*x^n));
  my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
  my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
  1 + p + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
{my(A=T(15)); for(i=1, #A, print(A[i]))}

Formula

T(3*n, n) = A380634(n).

A380632 Number of simple connected graphs on n unlabeled nodes with each node a member of exactly one cycle.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 5, 9, 14, 28, 49, 95, 182, 369, 733, 1509, 3103, 6504, 13627, 28949, 61701, 132457, 285454, 618863, 1346022, 2940287, 6444364, 14172744, 31257883, 69142445, 153333476, 340880766, 759549740, 1696122213, 3795178540, 8508326129, 19109193805, 42991993545, 96881110654
Offset: 0

Views

Author

Andrew Howroyd, Feb 24 2025

Keywords

Comments

All such graphs are cactus graphs (with bridges allowed).

Links

Crossrefs

Row sums of A380631.
Cf. A000083, A380805.

Programs

  • PARI
    Vec(G(40)) \\ G() defined in A380631.

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).

A380633 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes of degree at most 3 with k cycles and each node a member of exactly one cycle, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 3, 0, 1, 3, 6, 0, 1, 4, 11, 2, 0, 1, 4, 17, 5, 0, 1, 5, 26, 17, 0, 1, 5, 36, 37, 2, 0, 1, 6, 50, 78, 12, 0, 1, 6, 65, 140, 44, 0, 1, 7, 85, 248, 131, 4, 0, 1, 7, 106, 396, 325, 23
Offset: 0

Views

Author

Andrew Howroyd, Feb 24 2025

Keywords

Comments

All such graphs are cactus graphs (with bridges allowed).

Examples

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  3;
  0, 1, 3,  6;
  0, 1, 4, 11,  2;
  0, 1, 4, 17,  5;
  0, 1, 5, 26, 17;
  0, 1, 5, 36, 37, 2;
  ...

Links

Crossrefs

Columns 0..3 are A000007, A000012(n+3), A004526(n+4), A003453(n+4).
Row sums are A380805.
Cf. A000672, A380631 (with vertices of any degree).

Programs

  • PARI
    raise(p,d) = {my(n=serprec(p,x)-1); substvec(p + O(x^(n\d+1)), [x, y], [x^d,y^d])}
R(n,y)={my(g=O(x^3)); for(n=1, (n-1)\2, my(p=x*(1 + g), p2=raise(p,2)); g=x*y*(p^2/(1 - p) + (1 + p)*p2/(1 - p2))/2); g}
G(n,y=1)={my(g=R(n,y), p = x*(1+g) + O(x*x^n));
  my( r=((1 + p)^2/(1 - raise(p,2)) - 1)/2 );
  my( c=-sum(d=1, n, eulerphi(d)/d*log(raise(1-p,d))) );
  1 + (raise(g,2) - g^2 + y*(r + c - 2*p - p^2 - raise(p,2)))/2 }
T(n)={[Vecrev(p) | p<-Vec(G(n,y))]}
{my(A=T(15)); for(i=1, #A, print(A[i]))}

Formula

T(3*n,n) = A000672(n).
Showing 1-4 of 4 results.

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