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

2,2

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

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R. P. Stanley, unpublished work.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.

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

Example

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

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 *)

Crossrefs

Cf. A000108, A065866, A203470, A262034.

Sequence in context: A352069 A355786 A347015 * A052654 A108398 A370907

Adjacent sequences: A102690 A102691 A102692 * A102694 A102695 A102696

Keyword

nonn

Author

Richard Stanley, Feb 04 2005

Status

approved