A102692 a(n) = number of digraphs (allowing loops) with vertices 1,2,...,n that have a unique Eulerian tour (up to cyclic shift).

1, 2, 4, 28, 336, 5808, 132000, 3731040, 126362880, 4993309440, 225677975040, 11487263961600, 650467886745600, 40565803419187200, 2763133948128153600, 204127536266119065600, 16257504491853520896000, 1388699783389209747456000, 126649067639795642253312000

Offset

0,2

References

R. P. Stanley, unpublished work.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..349

Formula

a(n) = (n-1)! * (b(n-1) + b(n)), where b(n) is a little Schroeder number (A001003).

a(n) ~ 2^(1/4) * (1 + sqrt(2))^(2*n) * n^(n-2) / exp(n). - Vaclav Kotesovec, May 22 2018

Example

a(3) = 2!*(3+11) = 28. There are 16 such digraphs which are triangles with a possible loop at each vertex and 12 which consist of two 2-cycles with a common vertex and a possible loop at the other two vertices.

Maple

a:= proc(n) option remember; `if`(n<2, n+1,
      (n-1)*(6*(2*n^2-4*n+1)*a(n-1)-(n-2)*
      (n-3)*(2*n-1)*a(n-2))/((n+1)*(2*n-3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 21 2018

Mathematica

b[n_] := Hypergeometric2F1[-n, n+1, 2, -1]/2;
a[n_] := Switch[n, 0, 1, 1, 2, _, (n-1)! (b[n-1] + b[n])];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 18 2020 *)

Crossrefs

Cf. A001003.

Sequence in context: A117443 A095858 A062792 * A292184 A248872 A329102

Adjacent sequences: A102689 A102690 A102691 * A102693 A102694 A102695

Keyword

nonn

Author

Richard Stanley, Feb 04 2005

Extensions

a(0)=1 prepended and one typo corrected by Alois P. Heinz, May 21 2018

Status

approved