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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). 7
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R. P. Stanley, unpublished work.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
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
Sequence in context: A352069 A355786 A347015 * A052654 A108398 A370907
KEYWORD
nonn
AUTHOR
Richard Stanley, Feb 04 2005
STATUS
approved

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Last modified April 21 09:27 EDT 2024. Contains 371851 sequences. (Running on oeis4.)