OFFSET
0,2
COMMENTS
a(n) is the number of isolated points over all simple unlabeled graphs with (n+1) nodes. - Geoffrey Critzer, Apr 14 2012
REFERENCES
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..87
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 13.
FORMULA
O.g.f.: A(x)/(1-x), where A(x) is o.g.f. for A000088. - Geoffrey Critzer, Apr 12 2012
a(n) = Sum_{k=0..n} A000088(k). - Petros Hadjicostas, Apr 19 2020
EXAMPLE
a(2) = 4 includes the null graph G1 = [], G2 = [o], G3 = [o o], and G4 = [o-o].
a(3) = 8 includes the null graph G1 = [], G2 = [o], G3 = [o o], G4 = [o-o], G5 = [o o o], G6 = [o-o o], G7 = [o-o-o], and G8 = [triangle with three unlabeled nodes]. - Petros Hadjicostas, Apr 10 2020
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:= proc(n) option remember; b(n$2, [])+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 14 2019
MATHEMATICA
nn = 15; g = Sum[NumberOfGraphs[n] x^n, {n, 0, nn}]; CoefficientList[Series[g/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Apr 12 2012 *)
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
Name edited by Petros Hadjicostas, Apr 08 2020
STATUS
approved