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A367143
Number of simple graphs on n unlabeled vertices without isolated or universal vertices.
2
1, 0, 0, 0, 3, 12, 88, 732, 10258, 249976, 11455832, 994987528, 163053176864, 50171849022768, 28953151594499584, 31368377658489837792, 63938162732587949277392, 245807862122123877567929920, 1787085853417304634682510751296, 24634234097674713300981911735051072
OFFSET
0,5
COMMENTS
An isolated vertex has degree 0 and a universal vertex has degree n-1.
LINKS
FORMULA
a(n) = A000088(n) - 2*A000088(n-1) for n >= 2.
G.f.: x + (1 - 2*x)*B(x) where B(x) is the g.f. of A000088.
MAPLE
b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
+add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= n-> `if`(n<2, 1-n, b(n$2, [])-2*b(n-1$2, [])):
seq(a(n), n=0..20); # Alois P. Heinz, Nov 06 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 06 2023
STATUS
approved