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A007082 Number of Eulerian circuits on the complete graph K_{2n+1}, divided by (n-1)!^(2n+1).

%I A007082 M2183 #59 Jul 19 2025 17:45:04
%S A007082 2,264,1015440,90449251200,169107043478365440,
%T A007082 6267416821165079203599360,4435711276305905572695127676467200,
%U A007082 58393052751308545653929138771580386824519680,14021772793551297695593332913856884153315254190271692800,60498832138791357698014788383803842810832836262245623803123983974400
%N A007082 Number of Eulerian circuits on the complete graph K_{2n+1}, divided by (n-1)!^(2n+1).
%D A007082 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, p. 745, Problem 107.
%D A007082 B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
%D A007082 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007082 Florian Ragwitz, <a href="/A007082/b007082.txt">Table of n, a(n) for n = 1..16</a>
%H A007082 Brendan D. McKay and Robert W. Robinson, <a href="http://users.cecs.anu.edu.au/~bdm/papers/euler.pdf">Asymptotic enumeration of Eulerian circuits in the complete graph</a>, Combinatorics, Probability and Computing, 7 (1998), 437-449. (gives terms up to n=10, i.e., up through K_{21})
%F A007082 a(n) = A135388(n) / (n-1)!^(2n+1) = A350028(2n+1) / (n-1)!^(2n+1) = A357887(2n+1,n(2n+1)) / (n-1)!^(2n+1). - _Max Alekseyev_, Oct 19 2022
%e A007082 From _Günter Rote_, Dec 09 2021: (Start)
%e A007082 For n=2, in the graph K5, if we fix the Euler tour to start with the edge 12, we get 132 Euler tours. Here are the first and the last few in lexicographic order.
%e A007082   12314253451
%e A007082   12314254351
%e A007082   12314352451
%e A007082   12314354251
%e A007082   12314524351
%e A007082   ...
%e A007082   12543153241
%e A007082   12543241351
%e A007082   12543241531
%e A007082   12543513241
%e A007082   12543514231.
%e A007082 To get all 264*1!^5 = 264 Euler tours, the number must be multiplied by 2 to include the reversed tours. (End)
%o A007082 (Python)
%o A007082 # for n <= 4
%o A007082 def A(n,w="12"):
%o A007082     if len(w) > (2*n+1)*n: return 2
%o A007082     return sum(A(n, w+t) for t in "123456789"[:2*n+1]
%o A007082         if t!=w[-1] and t+w[-1] not in w and w[-1]+t not in w)
%Y A007082 Cf. A135388, A232545, A350028, A356366, A357855, A357856, A357857, A357885, A357886, A357887.
%K A007082 nonn,nice,walk
%O A007082 1,1
%A A007082 _N. J. A. Sloane_, _Mira Bernstein_
%E A007082 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003