A333166 Number of n-regular graphs on 2n unlabeled vertices with half-edges.

1, 2, 3, 12, 118, 9638, 10622074, 135037240786, 18621890255342234, 28688490385422625653266, 511030957184968000138445253202

Offset: 0

Author

Andrew Howroyd, Mar 12 2020

Keywords

• nonn
• more

Comments

A half-edge is like a loop except it only adds 1 to the degree of its vertex.

a(n) is the number of non-isomorphic 2n X 2n symmetric matrices with entries in {+1, -1} and all rows and columns summing to zero where isomorphism is up to simultaneous permutation of rows and columns. The case where rows and columns can be permuted independently is covered by A333165.

Examples

			The a(1) = 1 matrix is:
  [+ -]
  [- +]
.
The a(2) = 2 matrices are:
  [+ + - -]   [- - + +]   [+ + - -]
  [+ + - -]   [- - + +]   [+ - + -]
  [- - + +]   [+ + - -]   [- + - +]
  [- - + +]   [+ + - -]   [- - + +]
		

Crossrefs

Central coefficients of A333161.

Formula

a(n) = A333161(2*n, n).

Cf. A333164, A333165.