This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382362 #32 Mar 25 2025 19:51:09
%S A382362 1,6,768,3888000,1238347284480,36133511823360000000,
%T A382362 132525036775962102988800000000,
%U A382362 80290170669240213088301154828288000000000,10219925826442937385376011199621103616000000000000000000,338787616987540767092926393308400759448386388551011812769792000000000000
%N A382362 Number of oriented Eulerian circuits from a fixed start vertex in the complete digraph K_n, counting distinct first arcs.
%H A382362 Wikipedia, <a href="https://en.wikipedia.org/wiki/BEST_theorem">BEST Theorem</a>.
%F A382362 a(n) = (n-1) * n^(n-2) * ((n-2)!)^n.
%F A382362 a(n) = A023443(n) * A232545(n) = A023443(n) * A000272(n) * A000142(n-2)^n.
%e A382362 For n=3, the complete digraph K_3 with vertex set {1,2,3} has 6 distinct Eulerian circuits (counting rotations / distinct first arcs) starting and ending at vertex 1: (1,2,3,1,3,2,1), (1,3,2,1,2,3,1), (1,2,1,3,2,3,1), (1,3,2,3,1,2,1), (1,2,3,2,1,3,1), and (1,3,1,2,3,2,1).
%e A382362 The first two are equal to their reversals. The others form pairs that are equal in reversed form. Since this sequence distinguishes arc order and direction, a(3)=6 rather than 4.
%e A382362 In contrast, A232545 counts only 3 tours for n=3 because it does not distinguish Eulerian circuits that differ solely by rotating the cycle to a different first arc out of vertex 1.
%o A382362 (PARI) a(n) = (n-1) * n^(n-2) * factorial(n-2)^n;
%o A382362 (Haskell) a n = (n-1) * n^(n-2) * (product [1..(n-2)])^n
%Y A382362 Cf. A232545, A135388, A369820, A124355, A000272.
%K A382362 nonn,walk
%O A382362 2,2
%A A382362 _Florian Ragwitz_, Mar 23 2025