A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 21, 12, 1, 1, 0, 0, 2, 0, 31, 0, 0, 0, 0, 3, 103, 220, 7, 1, 1, 0, 0, 3, 0, 1606, 0, 1, 0, 0, 0, 5, 752, 16829, 388, 9, 1, 1, 0, 0, 5, 0, 193900, 0, 6, 0, 0, 0

Offset: 1

Author

Jason Kimberley, Feb 22 2013

Keywords

• nonn
• hard
• tabf

Comments

In the n-th row 0 <= 2k <= n.

Examples

			01: 0;
02: 0, 0;
03: 0, 0;
04: 0, 0, 1;
05: 0, 0, 0;
06: 0, 0, 0, 1;
07: 0, 0, 0, 0;
08: 0, 0, 1, 2, 1;
09: 0, 0, 1, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 1, 0, 2, 0;
12: 0, 0, 2, 21, 12, 1, 1;
13: 0, 0, 2, 0, 31, 0, 0;
14: 0, 0, 3, 103, 220, 7, 1, 1;
15: 0, 0, 3, 0, 1606, 0, 1, 0;
16: 0, 0, 5, 752, 16829, 388, 9, 1, 1;
17: 0, 0, 5, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 7, 7385, 2452820, 406824, 267, 8, 1, 1;
19: 0, 0, 8, 0, 32670331, 0, 3727, 0, 0, 0;
20: 0, 0, 11, 91939, 456028487, 1125022326, 483012, 741, 13, 1, 1;
21: 0, 0, 12, 0, 6636066126, 0, 69823723, 0, 1, 0, 0;
22: 0, 0, 16, 1345933, 100135577863, 3813549359275, 14836130862, 2887493, ?, 14, 1;
		

Links

• Jason Kimberley, Table of n, a(i)=E(n,k) for i = 1..139 (n = 1..22)
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Crossrefs

The sum of the n-th row of this sequence is A198314(n).

Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), this sequence (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

Formula

E(n,k) = A186734(n,k) + A210704(n,k), noting the differing row lengths.

E(n,k) = A185304(n,k) - A185305(n,k), noting the differing row lengths.