cp's OEIS Frontend

A357887 Triangle read by rows: T(n,k) = number of circuits of length k in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.

%I A357887 #29 Jul 20 2025 15:38:52
%S A357887 1,2,0,3,0,0,2,4,0,0,8,6,0,0,5,0,0,20,30,24,60,120,0,0,264,6,0,0,40,
%T A357887 90,144,480,1440,2340,3840,9504,15840,11160,0,0,0,7,0,0,70,210,504,
%U A357887 2100,8280,23940,68880,217224,594720,1339800,2983680,6482880,10190880,12136320,24192000,39621120,0,0,129976320
%N A357887 Triangle read by rows: T(n,k) = number of circuits of length k in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.
%C A357887 A circuit of length k is viewed as a sequence of k+1 vertices it visits modulo cyclic rotations. Hence T(n,0) = n enumerates individual vertices.
%H A357887 Max Alekseyev, <a href="/A357887/b357887.txt">Table of n, a(n) for n = 1..175</a> (rows n=1..10)
%F A357887 For k >= 1, T(n,k) = A357885(n,k) * n / k.
%F A357887 Last nonzero element in row n:
%F A357887  T(2n+1,n(2n+1)) = A135388(n) = A350028(2n+1) = A007082(n) * (n-1)!^(2*n+1);
%F A357887  T(2n,2n(n-1)) = A350028(2n) * (2n-1)!! = A297383(n) * 2 * (2n-1)!!.
%e A357887 Triangle T(n,k) starts with:
%e A357887  n=1: 1,
%e A357887  n=2: 2, 0,
%e A357887  n=3: 3, 0, 0, 2,
%e A357887  n=4: 4, 0, 0, 8, 6, 0, 0,
%e A357887  n=5: 5, 0, 0, 20, 30, 24, 60, 120, 0, 0, 264,
%e A357887  ...
%Y A357887 Cf. A007082, A135388, A232545, A297383, A350028, A356366 (row sums), A357855, A357856, A357857, A357885, A357886.
%K A357887 tabf,nonn,walk
%O A357887 1,2
%A A357887 _Max Alekseyev_, Oct 19 2022