OFFSET
1,2
COMMENTS
Includes disconnected graphs. - Felix A. Pahl, Jul 15 2018
Loops and parallel edges are not allowed, but 2-cycles (in other words, edges A --> B and B --> A existing simultaneously) are allowed. - Mikhail Lavrov, Mar 04 2025
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Matteo Beccaria, Thermal properties of a string bit model at large N, arXiv:1709.01801 [hep-th], 2017.
Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.
Rüdiger Jehn, Kester Habermann, and Misha Lavrov, Number of ways that a football league can complete with all teams having the same number of points, arXiv:2503.14509 [math.GM], 2025.
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
B. D. McKay, The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs, Combinatorica 10 (1990), 367-377.
FORMULA
a(n) ~ e^(-1/4)*sqrt(n)(2^n/sqrt(Pi*n))^(n-1)*(1+O(1/sqrt(n))) [B. D. McKay, 1990]. - Thomas Curtright, Apr 11 2017
EXAMPLE
For n=3, the a(n) = 10 solutions are: (A . B . C), (A <--> B . C), (A <--> C . B), (B <--> C . A), (A --> B --> C --> A), (A --> C --> B --> A), (A <--> B <--> C), (A <--> C <--> B), (B <--> A <--> C), and (A <--> B <--> C <--> A). - Mikhail Lavrov, Mar 04 2025
MATHEMATICA
a[n_]:=Coefficient[Expand[Product[Product[x[i]+x[j], {j, 1, n}], {i, 1, n}]], Product[x[k]^n, {k, 1, n}]]/2^n (* practically unusable for n>7 *)
a[n_]:=N[(Sqrt[n]/E^(1/4))*(2^n/Sqrt[n*Pi])^(n-1)*(1+3/(16*n)+1/(7*n^2)+3/(20*n^3))]
(* four digit accuracy for n>7 *) (* Thomas Curtright, Apr 12 2017 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Terms a(12) and beyond from McKay (1983), added by Thomas Curtright, Apr 12 2017
STATUS
approved
