cp's OEIS Frontend

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

%I A125697 #33 May 24 2025 23:52:42
%S A125697 1,2,1,7,3,1,35,16,3,1,228,77,20,3,1,2237,485,111,21,3,1,31559,4013,
%T A125697 716,127,21,3,1,1668997,47648,5623,862,131,21,3,1,3685886630,1868157,
%U A125697 60201,6739,926,132,21,3,1
%N A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.
%C A125697 This is a two-dimensional Euler transform of A125699.
%H A125697 Geoff Cruttwell, <a href="https://www.reluctantm.com/gcruttw/publications/ams2014CruttwellCountingFiniteCats.pdf">Counting Finite Categories</a>, presentation, (2018).
%H A125697 Ben Spitz, <a href="https://github.com/diracdeltafunk/SmallCategories#statistics">SmallCategories</a>
%H A125697 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Category.html">Category</a>
%F A125697 G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).
%F A125697 T(n,k) = A125701(n-k) when k >= (2/3)*n.
%F A125697 From _Ben Spitz_, Aug 30 2023: (Start)
%F A125697 T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
%F A125697 T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.
%F A125697 T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.
%F A125697 (End)
%e A125697 The table starts:
%e A125697      1;
%e A125697      2,   1;
%e A125697      7,   3,   1;
%e A125697     35,  16,   3,  1;
%e A125697    228,  77,  20,  3, 1;
%e A125697   2237, 485, 111, 21, 3, 1;
%e A125697   ...
%Y A125697 Cf. A125696 (row sums), A058129 (column 1), A125699, A125701.
%K A125697 tabl,hard,more,nonn
%O A125697 1,2
%A A125697 _Franklin T. Adams-Watters_ and _Christian G. Bower_, Jan 05 2007
%E A125697 a(23)-a(29) from _Ben Spitz_, Jul 17 2023
%E A125697 a(30)-a(36) from _Ben Spitz_, Aug 29 2023
%E A125697 a(37)-a(45) from _Elijah Beregovsky_, after the work of Cruttwell and Leblanc, May 20 2025