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

1,23

Comments

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

Links

Table of n, a(n) for n=1..93.

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

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.

Example

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;

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).

Sequence in context: A037860 A037878 A107652 * A196096 A249344 A067150

Adjacent sequences: A186731 A186732 A186733 * A186735 A186736 A186737

Keyword

nonn,hard,tabf

Author

Jason Kimberley, Mar 20 2013

Status

approved