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(Python)
from math import comb
def A369192(n): return sum(comb(comb(n, 2), k) for k in range(n+1)) # Chai Wah Wu, Jul 14 2024
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The version for loop-graphs is A066383, covering A369194.
The version for loop-graphs is A369195, covering A369194.
A001187 counts connected graphs, unlabeled A001349.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A057500 counts connected graphs, n vertices and n edges, unlabeled A001429.
A143543 counts simple labeled graphs by number of connected components.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, `A000666, ~A003465, `A001187, A006649, ~A140638, ~A367916A057500, A143543.
nonn,more,changed
1, 1, 2, 8, 57, 638, 9949, 198440, 4791323, 135142796, 4346814276, 156713948672, 6251579884084, 273172369790743, 12969420360339724, 664551587744173992, 36543412829258260135, 2146170890448154922648, 134053014635659737513358, 8872652968135849629240560
a(n) = Sum_{k=0..n} binomial(binomial(n,2),k).
allocated for Gus WisemanNumber of labeled simple graphs with n vertices and at most n edges (not necessarily covering).
1, 1, 2, 8, 57, 638, 9949, 198440
0,3
The a(0) = 1 through a(3) = 8 graphs:
{} {} {} {}
{{1,2}} {{1,2}}
{{1,3}}
{{2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[#]<=n&]], {n, 0, 5}]
The case of equality is A116508, covering A367863, also A367862.
The connected case is A129271, unlabeled A005703.
The covering case is A369191, minimal case A053530.
Counting only covered vertices gives A369193.
The version for loop-graphs is A369195, covering A369194.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges.
A057500 counts connected graphs, n vertices and n edges, unlabeled A001429.
A133686 counts choosable graphs, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, `A000666, ~A003465, `A006649, ~A140638, ~A367916.
allocated
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Gus Wiseman, Jan 17 2024
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