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Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n nodes with exactly k bipartite connected components, n >= 0, 0 <= k <= n.
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%I #18 Mar 26 2020 23:18:43

%S 1,0,1,0,1,1,1,1,1,1,3,4,2,1,1,16,9,5,2,1,1,96,37,13,6,2,1,1,812,162,

%T 46,14,6,2,1,1,10957,1120,194,50,15,6,2,1,1,260494,12675,1219,204,51,

%U 15,6,2,1,1,11713772,276758,13099,1254,208,52,15,6,2,1,1

%N Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n nodes with exactly k bipartite connected components, n >= 0, 0 <= k <= n.

%C T(n,k) is the number of graphs on n nodes with incidence matrix of rank n-k, where the incidence matrix is defined as in Godsil-Royle reference below.

%D C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 166.

%F G.f.: Product_{i>=1} (1/(1-x^i))^A157051(i)*(1/(1-y*x^i))^A005142(i).

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 1, 1, 1, 1;

%e 3, 4, 2, 1, 1;

%e 16, 9, 5, 2, 1, 1;

%e 96, 37, 13, 6, 2, 1, 1;

%e 812, 162, 46, 14, 6, 2, 1, 1;

%e ...

%t Needs["Combinatorica`"];

%t Table[Table[Count[Prepend[Flatten[Table[g = {n, k};b = GraphData[g,"IncidenceMatrix"]; {n - MatrixRank[b]}, {k,2, NumberOfGraphs[n]}]], n], i], {i, 0, n}], {n, 0,7}] // Grid

%Y Cf. A157051 (column k=0 for n>0), A000088 (row sums), A157015, A005142.

%K nonn,tabl

%O 0,11

%A _Geoffrey Critzer_, Mar 04 2020