%I A140188
%S A140188 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,
               1,
%T A140188 1,2,10,14,12,9,5,3,1,1,2,13,21,20,15,9,5,3,1,1,1,13,24,29,23,15,9,5,3,
%U A140188 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 A140188 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 A140188 The first column is T(n,1) = A00001(n) (number of groups of order n).
%C A140188 T(n,k) >= A136406(n,k).
%C A140188 The sum of the n^th row is A140189(n).
%C A140188 For 2k<=n, T(n,n-k) = A140190(k) does not depend on n.
%H A140188 B. Jubin, <a href="http://math.berkeley.edu/~jubin/oeis.html">Sequences 
               contributed to the OEIS</a>.
%F A140188 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 binom(a+p-1,p).
%Y A140188 Cf. A140185.
%Y A140188 Sequence in context: A139336 A100619 A094006 this_sequence A140737 A108756 
               A106178
%Y A140188 Adjacent sequences: A140185 A140186 A140187 this_sequence A140189 A140190 
               A140191
%K A140188 nonn,tabl
%O A140188 1,7
%A A140188 Benoit Jubin (benoit_jubin(AT)yahoo.fr), May 12 2008