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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).

%I A102693 #50 Dec 04 2022 08:32:30
%S A102693 1,5,42,504,7920,154440,3603600,98017920,3047466240,106661318400,
%T A102693 4151586700800,177925144320000,8326896754176000,422590010274432000,
%U A102693 23118159385601280000,1356265350621941760000,84945040381058457600000,5657339689378493276160000
%N 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).
%C A102693 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
%D A102693 R. P. Stanley, unpublished work.
%H A102693 Alois P. Heinz, <a href="/A102693/b102693.txt">Table of n, a(n) for n = 2..367</a>
%H A102693 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H A102693 Luz Grisales, Antoine Labelle, Rodrigo Posada, and Stoyan Dimitrov, <a href="https://arxiv.org/abs/2104.10734">Digraphs with exactly one Eulerian tour</a>, arXiv:2104.10734 [math.CO], 2021.
%F A102693 a(n) = (1/2)*A065866(n-1).
%F A102693 a(n) = C_n(n-1)!/2 = (n+2)(n+3)...(2n-1), where C_n denotes a Catalan number.
%F A102693 E.g.f.: Integral_{x} 2/(1+sqrt(1-4*x))^2 dx. - _Alois P. Heinz_, Sep 09 2015
%F A102693 a(n) = RisingFactorial(4 + n, n) assuming offset 0. - _Peter Luschny_, Mar 22 2022.
%F A102693 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
%e A102693 a(3) = 5. There are two such digraphs that are triangles and three that consist of two 2-cycles with a common vertex.
%p A102693 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
%p A102693 # alternative Maple program:
%p A102693 a:= proc(n) option remember; `if`(n<3, (n-1)*n/2,
%p A102693        2*(n-1)*(2*n-1)*a(n-1)/(n+1))
%p A102693     end:
%p A102693 seq(a(n), n=2..20);  # _Alois P. Heinz_, Nov 03 2017
%t A102693 a[n_] := a[n] = If[n<3, n(n-1)/2, 2(n-1)(2n-1) a[n-1]/(n+1)];
%t A102693 Table[a[n], {n, 2, 20}] (* _Jean-François Alcover_, Jun 10 2018, after _Alois P. Heinz_ *)
%Y A102693 Cf. A000108, A065866, A203470, A262034.
%K A102693 nonn
%O A102693 2,2
%A A102693 _Richard Stanley_, Feb 04 2005