cp's OEIS Frontend

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.

%I A275334 #8 Jul 15 2017 09:56:19
%S A275334 0,0,6,48,960,30720,2064384,264241152,68451041280,35046933135360,
%T A275334 35993612646875136,73714918700800278528,302157667927362455470080,
%U A275334 2475275615660953235210895360,40562343327224770087344704323584
%N 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.
%F A275334 a(n) = 2^binomial(n,2) - 2*2^binomial(n-1,2) if n is even.
%F A275334 a(n) = 2^binomial(n,2) - 2^binomial(n-1,2) if n is odd.
%e A275334 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.
%p A275334 A275334 := proc(n)
%p A275334     if type(n,'even') then
%p A275334         2^binomial(n,2)-2*2^binomial(n-1,2) ;
%p A275334     else
%p A275334         2^binomial(n,2)-2^binomial(n-1,2) ;
%p A275334     end if;
%p A275334 end proc:
%p A275334 seq(A275334(n),n=1..30) ; # _R. J. Mathar_, Jul 15 2017
%t A275334 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}]
%Y A275334 Cf. A122743.
%K A275334 nonn
%O A275334 1,3
%A A275334 _Geoffrey Critzer_, Jul 23 2016