A125697 - OEIS

%I #29 Feb 16 2025 08:33:04

%S 1,2,1,7,3,1,35,16,3,1,228,77,20,3,1,2237,485,111,21,3,1,31559,4013,

%T 716,127,21,3,1,1668997,47648,5623,862,131,21,3,1

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

%C This is a two-dimensional Euler transform of A125699.

%H Ben Spitz, <a href="https://github.com/diracdeltafunk/SmallCategories#statistics">SmallCategories</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Category.html">Category</a>

%F G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).

%F T(n,k) = A125701(n-k) when k >= (2/3)*n.

%F From _Ben Spitz_, Aug 30 2023: (Start)

%F T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.

%F T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.

%F T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.

%F (End)

%e The table starts:

%e      1;

%e      2,   1;

%e      7,   3,   1;

%e     35,  16,   3,  1;

%e    228,  77,  20,  3, 1;

%e   2237, 485, 111, 21, 3, 1;

%e   ...

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

%K tabl,hard,more,nonn

%O 1,2

%A _Franklin T. Adams-Watters_ and _Christian G. Bower_, Jan 05 2007

%E a(23)-a(29) from _Ben Spitz_, Jul 17 2023

%E a(30)-a(36) from _Ben Spitz_, Aug 29 2023