A102693 a(n) is the number of digraphs (not allowing loops) with vertices 1,2,...,n that have a unique Eulerian tour (up to cyclic shift).

1, 5, 42, 504, 7920, 154440, 3603600, 98017920, 3047466240, 106661318400, 4151586700800, 177925144320000, 8326896754176000, 422590010274432000, 23118159385601280000, 1356265350621941760000, 84945040381058457600000, 5657339689378493276160000

Offset: 2

Author

Richard Stanley, Feb 04 2005

Keywords

• nonn

Comments

It appears that a(n) can be obtained from the permanent of (2,3,4,...,n+2) as in A203470. - Clark Kimberling, Jan 02 2012

Examples

			a(3) = 5. There are two such digraphs that are triangles and three that consist of two 2-cycles with a common vertex.
		

References

• R. P. Stanley, unpublished work.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..367
• Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
• Luz Grisales, Antoine Labelle, Rodrigo Posada, and Stoyan Dimitrov, Digraphs with exactly one Eulerian tour, arXiv:2104.10734 [math.CO], 2021.

Crossrefs

Cf. A000108, A065866, A203470, A262034.

Programs

Maple

with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T), Prod(T,Z,Epsilon),Prod(T,T,Z))},labeled]: seq(count(ZL,size=i)/(2*i),i=2..18); # Zerinvary Lajos, Dec 16 2007
# alternative Maple program:
a:= proc(n) option remember; `if`(n<3, (n-1)*n/2,
       2*(n-1)*(2*n-1)*a(n-1)/(n+1))
    end:
seq(a(n), n=2..20);  # Alois P. Heinz, Nov 03 2017

Mathematica

a[n_] := a[n] = If[n<3, n(n-1)/2, 2(n-1)(2n-1) a[n-1]/(n+1)];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

Formula

a(n) = (1/2)*A065866(n-1).

a(n) = C_n(n-1)!/2 = (n+2)(n+3)...(2n-1), where C_n denotes a Catalan number.

E.g.f.: Integral_{x} 2/(1+sqrt(1-4*x))^2 dx. - Alois P. Heinz, Sep 09 2015

a(n) = RisingFactorial(4 + n, n) assuming offset 0. - Peter Luschny, Mar 22 2022.

Sum_{n>=2} 1/a(n) = (25*exp(1/4)*sqrt(Pi)*erf(1/2) - 10)/8, where erf is the error function. - Amiram Eldar, Dec 04 2022