A186734 Triangular array C(n,k) counting 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, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 20, 12, 1, 1, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 101, 220, 7, 1, 1, 0, 0, 0, 0, 1606, 0, 1, 0, 0, 0, 0, 743, 16828, 388, 9, 1, 1, 0, 0, 0, 0, 193900, 0, 6, 0, 0, 0, 0, 0, 7350

Offset: 1

Author

Jason Kimberley, Mar 20 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, 0, 2, 1;
09: 0, 0, 0, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 0, 0, 2, 0;
12: 0, 0, 0, 20, 12, 1, 1;
13: 0, 0, 0, 0, 31, 0, 0;
14: 0, 0, 0, 101, 220, 7, 1, 1;
15: 0, 0, 0, 0, 1606, 0, 1, 0;
16: 0, 0, 0, 743, 16828, 388, 9, 1, 1;
17: 0, 0, 0, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 0, 7350, 2452818, 406824, 267, 8, 1, 1;
19: 0, 0, 0, 0, 32670329, 0, 3727, 0, 0, 0;
20: 0, 0, 0, 91763, 456028472, 1125022325, 483012, 741, 13, 1, 1;
21: 0, 0, 0, 0, 6636066091, 0, 69823723, 0, 1, 0, 0;
		

Links

• Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Crossrefs

The sum of the n-th row of this sequence is A186744(n).

Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *exactly* g: A186733 (g=3), this sequence (g=4).

Triangular arrays C(n,k) counting connected simple k-regular graphs on n vertices with girth *at least* g: A068934 (g=3), A186714 (g=4).

Formula

C(n,k) = A186714(n,k) - A186715(n,k), noting the differing row lengths.

E(n,k) = A185644(n,k) - A210704(n,k), noting the differing row lengths.