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.

A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 2, 11, 8, 1, 0, 0, 0, 0, 1, 15, 31, 12, 1, 0, 0, 0, 0, 1, 12, 63, 71, 16, 1, 0, 0, 0, 0, 0, 8, 89, 231, 144, 21, 1, 0, 0, 0, 0, 0, 5, 97, 513, 707, 274, 27, 1
Offset: 1

Views

Author

Hugo Pfoertner, May 21 2021

Keywords

Comments

The number of nonzero terms in the n-th row is A083920(n-1). - Hugo Pfoertner, Feb 01 2024

Examples

			The triangle begins
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  1;
  0, 0, 0, 0, 3,  5,  1;
  0, 0, 0, 0, 2, 11,  8,  1;
  0, 0, 0, 0, 1, 15, 31, 12,  1;
  0, 0, 0, 0, 1, 12, 63, 71, 16, 1;

Links

Crossrefs

Cf. A004108 (column sums), A342556 (row sums).
Cf. A083920, A369932 (not necessarily connected).

Programs

  • PARI
    \\ Needs G() defined in A369932.
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={my(r=Vec(InvEulerMTS(substvec(G(n),[x,y],[y,x])))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) }
{ my(A=T(12)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 07 2024

Formula

Bivariate inverse Euler transform of A369932. - Andrew Howroyd, Feb 07 2024

A369290 Number of unlabeled simple graphs without endpoints with n edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 6, 10, 25, 68, 182, 538, 1748, 5935, 21585, 82904, 334037, 1406934, 6167455, 28033776, 131770437, 638956188, 3189940453, 16369201031, 86214798929, 465480395911, 2573390342437, 14553415319929, 84118459655982, 496514424803358, 2990633679878654
Offset: 0

Views

Author

Andrew Howroyd, Jan 30 2024

Keywords

Links

Crossrefs

Row sums of A369932.
Cf. A004110 (n vertices), A307316 (multigraph), A342556 (connected).

Programs

  • PARI
    \\ See also A369932 for a more efficient program.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
seq(n)={my(s=0); forpart(p=2*n, s+=permcount(p)*prod(i=1, #p, 1-x^p[i])*edges(p, w->1+x^w + O(x*x^n))); Vec(s/(2*n)!)}

Formula

Euler transform of A342556.

A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 12, 0, 0, 0, 1, 85, 70, 0, 0, 0, 0, 100, 990, 465, 0, 0, 0, 0, 45, 2805, 11550, 3507, 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016, 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884, 0, 0, 0, 0, 0, 1335, 222516, 4710300, 22467312, 24556140, 3026655
Offset: 1

Views

Author

Andrew Howroyd, Feb 08 2024

Keywords

Comments

T(n,k) is the number of traceless symmetric binary matrices with 2n 1's and k rows and at least two 1's in every row.

Examples

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 3;
  0, 0, 0, 6,  12;
  0, 0, 0, 1,  85,   70;
  0, 0, 0, 0, 100,  990,    465;
  0, 0, 0, 0,  45, 2805,  11550,    3507;
  0, 0, 0, 0,  10, 3595,  59990,  140420,   30016;
  0, 0, 0, 0,   1, 2697, 147441, 1174670, 1802682, 286884;
  ...
The T(3,3) = 1 matrix is:
  [0 1 1]
  [1 0 1]
  [1 1 0]
The T(4,4) = 3 matrices are:
  [0 0 1 1]  [0 1 0 1]  [0 1 1 0]
  [0 0 1 1]  [1 0 1 0]  [1 0 0 1]
  [1 1 0 0]  [0 1 0 1]  [1 0 0 1]
  [1 1 0 0]  [1 0 1 0]  [0 1 1 0]

Links

Crossrefs

Row sums are A370059.
Column sums are A100743.
Main diagonal is A001205.
Cf. A369928, A369932 (unlabeled).

Programs

  • PARI
    G(n)={my(A=x/exp(x*y + O(x*x^n))); exp(y*x^2/2 - x + O(x*x^n)) * sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*A^k/k!)}
T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}

Formula

T(n,k) = k!*[x^k][y^n] exp(y*x^2/2 - x) * Sum_{j>=0} (1 + y)^binomial(j, 2)*(x/exp(y*x))^j/j!.

A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 0, 0, 1, 5, 14, 10, 4, 0, 0, 1, 6, 25, 33, 18, 4, 0, 0, 1, 8, 46, 96, 90, 31, 7, 0, 0, 1, 10, 75, 227, 330, 194, 52, 8, 0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12, 0, 0, 1, 14, 173, 982, 2847, 3908, 2591, 800, 128, 14
Offset: 0

Views

Author

Andrew Howroyd, Feb 08 2024

Keywords

Comments

An endpoint is a vertex that appears in only one edge. Equivalently, the degree of every vertex >= 2.

Examples

			Triangle begins:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  2,   2;
  0, 0, 1,  3,   5,   2;
  0, 0, 1,  5,  14,  10,    4;
  0, 0, 1,  6,  25,  33,   18,   4;
  0, 0, 1,  8,  46,  96,   90,  31,   7;
  0, 0, 1, 10,  75, 227,  330, 194,  52,  8;
  0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12;
  ...

Links

Crossrefs

Row sums are A307316.
Main diagonal is A002865.
Cf. A369932.

Programs

  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) / edges(p, w->1-y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}
T(n)={my(r=Vec(substvec(G(n), [x, y], [y, x]))); vector(#r, i, Vecrev(Pol(r[i]), i)) }
{ my(A=T(10)); for(i=1, #A, print(A[i])) }
Showing 1-4 of 4 results.

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

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