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%I #28 Aug 31 2023 02:53:20
%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="http://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
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