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 1-5 of 5 results.

A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 1, 7, 3, 1, 35, 16, 3, 1, 228, 77, 20, 3, 1, 2237, 485, 111, 21, 3, 1, 31559, 4013, 716, 127, 21, 3, 1, 1668997, 47648, 5623, 862, 131, 21, 3, 1, 3685886630, 1868157, 60201, 6739, 926, 132, 21, 3, 1
Offset: 1

Views

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

Comments

This is a two-dimensional Euler transform of A125699.

Examples

			The table starts:
     1;
     2,   1;
     7,   3,   1;
    35,  16,   3,  1;
   228,  77,  20,  3, 1;
  2237, 485, 111, 21, 3, 1;
  ...

Links

Crossrefs

Cf. A125696 (row sums), A058129 (column 1), A125699, A125701.

Formula

G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).
T(n,k) = A125701(n-k) when k >= (2/3)*n.
From Ben Spitz, Aug 30 2023: (Start)
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.
(End)

Extensions

a(23)-a(29) from Ben Spitz, Jul 17 2023
a(30)-a(36) from Ben Spitz, Aug 29 2023
a(37)-a(45) from Elijah Beregovsky, after the work of Cruttwell and Leblanc, May 20 2025

A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 7, 1, 35, 6, 228, 28, 2, 2237, 159, 11, 31559, 1075, 77, 3, 1668997, 9389, 497, 24
Offset: 1

Views

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

Comments

Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.

Examples

			The table starts:
        1;
        2;
        7,    1;
       35,    6;
      228,   28,   2;
     2237,  159,  11;
    31559, 1075,  77,  3;
  1668997, 9389, 497, 24;
  ...

Links

Crossrefs

Cf. A125696, A125698 (row sums), A110654 (row lengths), A058129 (column 1), A125700 (diagonal sums), A125702 (T(2n-1, n)).

Extensions

a(14)-a(20) from Ben Spitz, Sep 02 2023

A125700 Number of connected categories with n more morphisms than objects.

Original entry on oeis.org

1, 3, 15, 77, 494, 4068
Offset: 0

Views

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

Links

  • Eric Weisstein's World of Mathematics, Category

Crossrefs

Cf. A125696, A125699, A125701.

Formula

a(n) = Sum_{k=1..n+1} A125699(n+k,k).

Extensions

a(4)-a(5) from Ben Spitz, Aug 30 2023

A125701 Limiting values for category table A125697.

Original entry on oeis.org

1, 3, 21, 132, 950, 7698
Offset: 0

Views

Author

Christian G. Bower and Franklin T. Adams-Watters, Jan 05 2007

Keywords

Comments

Euler transform of A125700 (ignoring the initial 1).

Links

Crossrefs

Cf. A125697, A125700, A125696.

Formula

G.f.: Product_{i>=1} 1/(1-x^i)^A125700(i).
A125697(n,k) = a(n-k) if k >= (2/3)*n.

Extensions

a(4) from Ben Spitz, Jul 18 2023
a(5) from Ben Spitz, Aug 30 2023

A125698 Number of connected categories with n morphisms.

Original entry on oeis.org

1, 2, 8, 41, 258, 2407
Offset: 1

Views

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

Comments

Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.
Inverse Euler transform of A125696.

Examples

			The 8 categories with 3 morphisms consist of 7 3-element monoids and one 2-object category with a single morphism between the objects.

Links

  • Eric Weisstein's World of Mathematics, Category

Crossrefs

Cf. A125696, A125699, A125700.
Showing 1-5 of 5 results.

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

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