A275334 Number of simple labeled graphs on n vertices that have at least one vertex of odd degree and at least one vertex of even degree.

0, 0, 6, 48, 960, 30720, 2064384, 264241152, 68451041280, 35046933135360, 35993612646875136, 73714918700800278528, 302157667927362455470080, 2475275615660953235210895360, 40562343327224770087344704323584

Offset: 1

Author

Geoffrey Critzer, Jul 23 2016

Keywords

• nonn

Examples

			a(4)=48 because there are 64 simple labeled graphs on 4 vertices but the graph with no edges, the 3 labelings of the 4-cycle graph, the 4 labelings of the 3 cycle with an isolated node, and the complements of each of these graphs are not counted.
		

Crossrefs

Cf. A122743.

Programs

Maple

A275334 := proc(n)
    if type(n,'even') then
        2^binomial(n,2)-2*2^binomial(n-1,2) ;
    else
        2^binomial(n,2)-2^binomial(n-1,2) ;
    end if;
end proc:
seq(A275334(n),n=1..30) ; # R. J. Mathar, Jul 15 2017

Mathematica

Table[If[EvenQ[n], 2^Binomial[n, 2] - 2 2^Binomial[n - 1, 2], 2^Binomial[n, 2] - 2^Binomial[n - 1, 2]], {n, 1, 15}]

Formula

a(n) = 2^binomial(n,2) - 2*2^binomial(n-1,2) if n is even.

a(n) = 2^binomial(n,2) - 2^binomial(n-1,2) if n is odd.