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A227322 Triangle read by rows: T(n, m) for 0 <= m <= n is the number of bipartite connected labeled graphs with parts of size n and m.

%I A227322 #24 Oct 04 2021 09:02:14
%S A227322 1,1,1,0,1,5,0,1,19,205,0,1,65,1795,36317,0,1,211,14221,636331,
%T A227322 23679901,0,1,665,106819,10365005,805351531,56294206205,0,1,2059,
%U A227322 778765,162470155,26175881341,3735873535339,502757743028605
%N A227322 Triangle read by rows: T(n, m) for 0 <= m <= n is the number of bipartite connected labeled graphs with parts of size n and m.
%H A227322 F. Harary and R. W. Robinson, <a href="http://cms.math.ca/cjm/v31/cjm1979v31.0060-0068.pdf">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68.
%F A227322 T(n, m) = 2^(n*m) - sum for all (i, j) in ({1, 2, ..., n} X {1, 2, ..., m} UNION (1, 0)) \ (n, m) \ (1 - n, 0) of T(i, j)*C(n - 1, i - 1)*C(m, j)*2^((n - i)*(m - j)), where C(n, m) is the binomial coefficient (A007318). This relation can be obtained considering connected component which contains the first vertex of the largest part. (If the largest part has zero size we get T(0, 0) = 2^0 - 0 = 1 which is true.)
%e A227322 Triangle T(n, m) begins:
%e A227322 n\m 0 1    2      3         4           5             6               7
%e A227322 0   1
%e A227322 1   1 1
%e A227322 2   0 1    5
%e A227322 3   0 1   19    205
%e A227322 4   0 1   65   1795     36317
%e A227322 5   0 1  211  14221    636331    23679901
%e A227322 6   0 1  665 106819  10365005   805351531   56294206205
%e A227322 7   0 1 2059 778765 162470155 26175881341 3735873535339 502757743028605
%e A227322 ...
%e A227322 Consider labeled bipartite graph with parts of size 2 and 2. To make graph connected it is possible to use all four possible edges or omit any one of them. Thus T(2, 2) = 5.
%Y A227322 Cf. A226658, A123260.
%Y A227322 Main diagonal gives: A005333.
%Y A227322 Columns m=2, 3, 4 give: A001047, A002501, A002502.
%K A227322 easy,nonn,tabl
%O A227322 0,6
%A A227322 _Pavel Irzhavski_, Jul 06 2013