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A357885 Triangle read by rows: T(n,k) = number of closed trails of length k starting and ending at a fixed vertex in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.

%I A357885 #18 Jul 20 2025 15:38:58
%S A357885 1,1,0,1,0,0,2,1,0,0,6,6,0,0,1,0,0,12,24,24,72,168,0,0,528,1,0,0,20,
%T A357885 60,120,480,1680,3120,5760,15840,29040,22320,0,0,0,1,0,0,30,120,360,
%U A357885 1800,8280,27360,88560,310320,934560,2296800,5541120,12965760,21837600,27740160,58752000,101882880,0,0,389928960
%N A357885 Triangle read by rows: T(n,k) = number of closed trails of length k starting and ending at a fixed vertex in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.
%H A357885 Max Alekseyev, <a href="/A357885/b357885.txt">Table of n, a(n) for n = 1..175</a> (rows n=1..10)
%F A357885 For k >= 1, T(n,k) = A357887(n,k) * k / n.
%F A357885 Last nonzero element in row n:
%F A357885  T(2n+1,n(2n+1)) = A135388(n) * n = A350028(2n+1) * n = A007082(n) * n * (n-1)!^(2*n+1);
%F A357885  T(2n,2n(n-1)) = A350028(2n) * (n-1) * (2n-1)!! = A297383(n) * (2n-2) * (2n-1)!!.
%e A357885 Triangle starts:
%e A357885  n=1: 1
%e A357885  n=2: 1, 0
%e A357885  n=3: 1, 0, 0, 2
%e A357885  n=4: 1, 0, 0, 6, 6, 0, 0
%e A357885  n=5: 1, 0, 0, 12, 24, 24, 72, 168, 0, 0, 528
%e A357885  ...
%Y A357885 Cf. A007082, A135388, A232545, A297383, A350028, A356366, A357855 (row sums), A357856, A357857, A357886, A357887.
%K A357885 tabf,nonn,walk
%O A357885 1,7
%A A357885 _Max Alekseyev_, Oct 18 2022