A367143 - OEIS

%I #11 Jul 04 2024 23:47:03

%S 1,0,0,0,3,12,88,732,10258,249976,11455832,994987528,163053176864,

%T 50171849022768,28953151594499584,31368377658489837792,

%U 63938162732587949277392,245807862122123877567929920,1787085853417304634682510751296,24634234097674713300981911735051072

%N Number of simple graphs on n unlabeled vertices without isolated or universal vertices.

%C An isolated vertex has degree 0 and a universal vertex has degree n-1.

%H Chai Wah Wu, <a href="/A367143/b367143.txt">Table of n, a(n) for n = 0..87</a>

%F a(n) = A000088(n) - 2*A000088(n-1) for n >= 2.

%F G.f.: x + (1 - 2*x)*B(x) where B(x) is the g.f. of A000088.

%p b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)

%p       +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),

%p        add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))

%p     end:

%p a:= n-> `if`(n<2, 1-n, b(n$2, [])-2*b(n-1$2, [])):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 06 2023

%Y Cf. A000088, A002494, A367142.

%K nonn

%O 0,5

%A _Andrew Howroyd_, Nov 06 2023