login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A279045 Number of pairs of vertices that share no common neighbor summed over all simple labeled graphs on n nodes. 1
0, 2, 18, 216, 4320, 155520, 10450944, 1337720832, 330225942528, 158508452413440, 148786600665415680, 274243462346494181376, 995653355660871966130176, 7136843253377130253221101568, 101189457574036357559516418539520 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n)/2^binomial(n,2) is the expected number of pairs of vertices in a simple labeled graph on n nodes that share no common neighbor. This expectation approaches 0 as n gets big. Hence almost all graphs have diameter 2.

REFERENCES

D. B. West, Introduction to Graph Theory, Pearson, 2015, page 432.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..45

FORMULA

a(n) = 2^binomial(n, 2)*binomial(n, 2)*(1 - (1/2)^2)^(n - 2).

EXAMPLE

a(3)=18. There are 3 such pairs of vertices in the empty graph. There are 3 pairs in each of the 3 labelings of the graph with one edge. There are 2 pairs in each of the 3 labelings of the path of length two. 3 + 3*3 + 2*3 = 18.

MATHEMATICA

Table[2^Binomial[n, 2] Binomial[n, 2] (1 - (1/2)^2)^(n - 2), {n, 1, 15}]

PROG

(MAGMA) [2^Binomial(n, 2)*Binomial(n, 2)*(1-(1/2)^2)^(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 08 2016

(PARI) a(n) = 2^binomial(n, 2)*binomial(n, 2)*(1-(1/2)^2)^(n-2) \\ Indranil Ghosh, Feb 25 2017

CROSSREFS

Sequence in context: A153647 A052726 A217239 * A155666 A227934 A349652

Adjacent sequences:  A279042 A279043 A279044 * A279046 A279047 A279048

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Dec 04 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 19:40 EST 2021. Contains 349526 sequences. (Running on oeis4.)