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A367143 Number of simple graphs on n unlabeled vertices without isolated or universal vertices.

%I A367143 #15 Jul 02 2025 03:45:18
%S A367143 1,0,0,0,3,12,88,732,10258,249976,11455832,994987528,163053176864,
%T A367143 50171849022768,28953151594499584,31368377658489837792,
%U A367143 63938162732587949277392,245807862122123877567929920,1787085853417304634682510751296,24634234097674713300981911735051072
%N A367143 Number of simple graphs on n unlabeled vertices without isolated or universal vertices.
%C A367143 An isolated vertex has degree 0 and a universal vertex has degree n-1.
%H A367143 Chai Wah Wu, <a href="/A367143/b367143.txt">Table of n, a(n) for n = 0..87</a>
%F A367143 a(n) = A000088(n) - 2*A000088(n-1) for n >= 2.
%F A367143 G.f.: x + (1 - 2*x)*B(x) where B(x) is the g.f. of A000088.
%p A367143 b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
%p A367143       +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
%p A367143        add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
%p A367143     end:
%p A367143 a:= n-> `if`(n<2, 1-n, b(n$2, [])-2*b(n-1$2, [])):
%p A367143 seq(a(n), n=0..20);  # _Alois P. Heinz_, Nov 06 2023
%t A367143 b[n_, i_, l_] := If[n == 0 || i == 1, 1/n!*2^(Function[p, Sum[Ceiling[(p[[j]]-1)/2] + Sum[GCD[p[[k]], p[[j]]], {k, 1, j-1}], {j, 1, Length[p]}]][Join[l, Table[1, {n}]]]), Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]];
%t A367143 a[n_] := If[n < 2, 1-n, b[n, n, {}] - 2*b[n-1, n-1, {}]];
%t A367143 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 02 2025, after _Alois P. Heinz_ *)
%Y A367143 Cf. A000088, A002494, A367142.
%K A367143 nonn
%O A367143 0,5
%A A367143 _Andrew Howroyd_, Nov 06 2023