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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.
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%I #12 Oct 21 2022 14:30:40

%S 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 60,120,480,1680,3120,5760,15840,29040,22320,0,0,0,1,0,0,30,120,360,

%U 1800,8280,27360,88560,310320,934560,2296800,5541120,12965760,21837600,27740160,58752000,101882880,0,0,389928960

%N 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 Max Alekseyev, <a href="/A357885/b357885.txt">Table of m, a(m) for m = 1..129</a> (rows n = 1..9)

%F For k >= 1, T(n,k) = A357887(n,k) * k / n.

%e Triangle starts:

%e n=1: 1

%e n=2: 1, 0

%e n=3: 1, 0, 0, 2

%e n=4: 1, 0, 0, 6, 6, 0, 0

%e n=5: 1, 0, 0, 12, 24, 24, 72, 168, 0, 0, 528

%e ...

%Y Cf. A007082, A135388, A232545, A350028, A356366, A357855 (row sums), A357856, A357857, A357886, A357887.

%K tabf,nonn,walk

%O 1,7

%A _Max Alekseyev_, Oct 18 2022