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

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

Author

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

Keywords

• tabl
• hard
• more
• nonn

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

• Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
• Ben Spitz, SmallCategories
• Eric Weisstein's World of Mathematics, Category

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