cp's OEIS Frontend

A350028 Number of Euler tours of the complete graph on n vertices, minus a matching if n is even.

%I A350028 #36 Jul 28 2025 20:18:13
%S A350028 1,1,2,2,264,744,129976320,1847500800,911520057021235200,
%T A350028 91507897551783002112,257326999238092967427785160130560,
%U A350028 234051620220909442615820736748584960,6705710151431658873046319662156165939200000000000000
%N A350028 Number of Euler tours of the complete graph on n vertices, minus a matching if n is even.
%C A350028 For even n, the graph is a cocktail party graph (cf. A297383). - _Max Alekseyev_, Jul 24 2025
%H A350028 Eric Weisstein, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>, MathWorld.
%F A350028 a(2n+1) = A135388(n) = A357887(2n+1,n(2n+1)) = A007082(n) * (n-1)!^(2*n+1); a(2n) = 2 * A297383(n) = A357887(2n,2n(n-1)) / (2n-1)!!. - _Max Alekseyev_, Oct 19 2022
%e A350028 For n=6, if the edges 12,34,56 are removed from the complete graph and we fix the tour to start with the edge 13, we get 372 Euler tours. Here are the first and the last few in lexicographic order.
%e A350028   1324152635461
%e A350028   1324152645361
%e A350028   1324153625461
%e A350028   1324153645261
%e A350028   1324154625361
%e A350028   1324154635261
%e A350028   1324162536451
%e A350028   ...
%e A350028   1364532516241
%e A350028   1364532614251
%e A350028   1364532615241.
%e A350028 This must be multiplied by 2 to account for the reversed tours, for a total of 744.
%o A350028 (Python)
%o A350028 # for 3 <= n <= 9
%o A350028 def A(n,w="13"):
%o A350028     if n%2==0 and len(w) > n*(n-1)//2 - n//2: return 2
%o A350028     if n%2==1 and len(w) > n*(n-1)//2: return 2
%o A350028     return sum(A(n, w+t) for t in "123456789"[:n]
%o A350028         if t!=w[-1] and t+w[-1] not in w and w[-1]+t not in w
%o A350028         and (n%2==1 or t+w[-1] not in "121 343 565 787"))
%Y A350028 Cf. A007082, A135388, A232545, A297383, A334554, A356366, A357855, A357856, A357857, A357885, A357886, A357887.
%K A350028 nonn,walk,hard
%O A350028 1,3
%A A350028 _Günter Rote_, Dec 08 2021
%E A350028 a(1)-a(2) prepended, a(10)-a(13) added by _Max Alekseyev_, Jul 15 2025