A123474 Triangle read by rows: T(n,k) = number of labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. The classes are interchangeable if k = n-k. Here n >= 2, k=1..n-1.

1, 0, 0, 0, 3, 0, 0, 10, 10, 0, 0, 15, 340, 15, 0, 0, 21, 6965, 6965, 21, 0, 0, 28, 51296, 246295, 51296, 28, 0, 0, 36, 326676, 14750946, 14750946, 326676, 36, 0, 0, 45, 1917840, 322476210, 796058676, 322476210, 1917840, 45, 0, 0, 55, 10683255

Offset: 2

Author

N. J. A. Sloane, Nov 12 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0,  0;
  0,  3,     0;
  0, 10,    10,      0;
  0, 15,   340,     15,     0;
  0, 21,  6965,   6965,    21,  0;
  0, 28, 51296, 246295, 51296, 28, 0;
  ...
Formatted as an array:
==========================================================
m/n | 1  2       3        4            5             6
----+-----------------------------------------------------
  1 | 1  0      0         0            0             0 ...
  2 | 0  3     10        15           21            28 ...
  3 | 0 10    340      6965        51296        326676 ...
  4 | 0 15   6965    246295     14750946     322476210 ...
  5 | 0 21  51296  14750946    796058676  105725374062 ...
  6 | 0 28 326676 322476210 105725374062 9736032295374 ...
  ...
		

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows; first 24 rows from R. W. Robinson)
• F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Crossrefs

Central coefficients are A005335.

Cf. A004100, A123301, A262307.

Formula

From Andrew Howroyd, Jan 03 2021: (Start)

T(n,k) = f(n-2*k) * binomial(n,k) * A123301(n, k) where f(0) = 1/2 and 1 otherwise.

A004100(n) = Sum_{k=0..floor(n/2)} T(n,k). (End)