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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). 5
1, 5, 42, 504, 7920, 154440, 3603600, 98017920, 3047466240, 106661318400, 4151586700800, 177925144320000, 8326896754176000, 422590010274432000, 23118159385601280000, 1356265350621941760000, 84945040381058457600000, 5657339689378493276160000 (list; graph; refs; listen; history; text; internal format)
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R. P. Stanley, unpublished work.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

FORMULA

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

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

with(finance):seq(mul(cashflows([n-1, k, 1], 0), k=2..n-1), n=2..22); # Zerinvary Lajos, Dec 22 2008

# 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-Fran├žois Alcover, Jun 10 2018, after Alois P. Heinz *)

CROSSREFS

Equals (1/2) A065866(n-1).

Cf. A262034.

Sequence in context: A217810 A217809 A317352 * A052654 A108398 A239997

Adjacent sequences:  A102690 A102691 A102692 * A102694 A102695 A102696

KEYWORD

nonn

AUTHOR

Richard Stanley, Feb 04 2005

STATUS

approved

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Last modified May 24 18:34 EDT 2019. Contains 323534 sequences. (Running on oeis4.)