A007080 Number of labeled Eulerian digraphs with n nodes. (Formerly M1989)

1, 2, 10, 152, 7736, 1375952, 877901648, 2046320373120, 17658221702361472, 569773219836965265152, 69280070663388783890248448, 31941407692847758201303724506112, 56121720938871110502272391300032261120, 377362438996731353329256282026362716827887616, 9744754031799754169218003376206941877943086188308480, 969342741943194323476512925742876053501022995325734477987840

Offset

1,2

Comments

Includes disconnected graphs. - Felix A. Pahl, Jul 15 2018

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..16.

Matteo Beccaria, Thermal properties of a string bit model at large N, arXiv:1709.01801 [hep-th], 2017.

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

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

Sequence in context: A294373 A194026 A165940 * A231517 A134088 A076659

Adjacent sequences: A007077 A007078 A007079 * A007081 A007082 A007083

Keyword

nonn,nice

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Terms a(12) and beyond from McKay (1983), added by Thomas Curtright, Apr 12 2017

Status

approved