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%I #12 Jul 28 2020 11:32:00
%S 1,1,1,1,1,1,2,3,1,1,1,3,3,1,1,2,4,5,3,1,1,1,5,6,5,3,1,1,5,8,10,9,5,3,
%T 1,1,2,10,14,12,9,5,3,1,1,2,13,21,20,15,9,5,3,1,1,1,13,24,29,23,15,9,
%U 5,3,1,1,5,20,39,42,37,27,15,9,5,3,1,1,1,19,43,58,53,40,27,15,9,5,3,1,1,2
%N Table read by rows: T(n,k) is the number of groupoids (categories all of whose morphisms are invertible) with n morphisms and k objects.
%C The first column is T(n,1) = A000001(n) (number of groups of order n).
%C T(n,k) >= A136406(n,k).
%C The sum of the n^th row is A140189(n).
%C For 2k<=n, T(n,n-k) = A140190(k) does not depend on n.
%F T(n,k) is the sum over the quadratic bi-partitions (n_i,k_i) of (n,k) (see A136406) of the "product" of the A000001(n_i), where the "product" is the usual product except when (n_i1,k_i1)=...=(n_ip,k_ip), in which case a^p is replaced by binomial(a+p-1,p).
%Y Cf. A140185.
%K nonn,tabl
%O 1,7
%A _Benoit Jubin_, May 12 2008
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