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Triangular array T(n,k) giving number of 3-connected graphs with n labeled nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n(n-1)/2).
4

%I #12 May 23 2024 04:24:59

%S 1,15,10,1,70,492,690,395,105,15,1,5040,28595,58905,63990,42392,18732,

%T 5880,1330,210,21,1,16800,442680,2485920,6629056,10684723,11716068,

%U 9409806,5824980,2872317,1147576,373156,98112,20475,3276

%N Triangular array T(n,k) giving number of 3-connected graphs with n labeled nodes and k edges (n >= 4, ceiling(3*n/2) <= k <= n(n-1)/2).

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

%H R. W. Robinson, <a href="/A123542/b123542.txt">Rows 4 through 15, flattened</a> (row 15 is incomplete).

%H T. R. S. Walsh, <a href="https://doi.org/10.1016/0095-8956(82)90072-7">Counting labeled three-connected and homeomorphically irreducible two-connected graphs</a>, J. Combin. Theory Ser. B 32 (1982), no. 1, 1-11, Table 1.

%e Triangle begins:

%e n = 4

%e k = 6 : 1

%e Total( 4) = 1

%e n = 5

%e k = 8 : 15

%e k = 9 : 10

%e k = 10 : 1

%e Total( 5) = 26

%e n = 6

%e k = 9 : 70

%e k = 10 : 492

%e k = 11 : 690

%e k = 12 : 395

%e k = 13 : 105

%e k = 14 : 15

%e k = 15 : 1

%e Total( 6) = 1768

%e n = 7

%e k = 11 : 5040

%e k = 12 : 28595

%e k = 13 : 58905

%e k = 14 : 63990

%e k = 15 : 42392

%e k = 16 : 18732

%e k = 17 : 5880

%e k = 18 : 1330

%e k = 19 : 210

%e k = 20 : 21

%e k = 21 : 1

%e Total( 7) = 225096

%Y Row sums give A005644. Cf. A123527, A123534.

%K nonn,tabf

%O 4,2

%A _N. J. A. Sloane_, Nov 13 2006