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.

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, 60, 120, 480, 1680, 3120, 5760, 15840, 29040, 22320, 0, 0, 0, 1, 0, 0, 30, 120, 360, 1800, 8280, 27360, 88560, 310320, 934560, 2296800, 5541120, 12965760, 21837600, 27740160, 58752000, 101882880, 0, 0, 389928960

Offset: 1

Author

Max Alekseyev, Oct 18 2022

Keywords

• tabf
• nonn
• walk

Examples

			Triangle starts:
 n=1: 1
 n=2: 1, 0
 n=3: 1, 0, 0, 2
 n=4: 1, 0, 0, 6, 6, 0, 0
 n=5: 1, 0, 0, 12, 24, 24, 72, 168, 0, 0, 528
 ...
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..175 (rows n=1..10)

Crossrefs

Cf. A007082, A135388, A232545, A297383, A350028, A356366, A357855 (row sums), A357856, A357857, A357886, A357887.

Formula

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

Last nonzero element in row n:

T(2n+1,n(2n+1)) = A135388(n) * n = A350028(2n+1) * n = A007082(n) * n * (n-1)!^(2*n+1);

T(2n,2n(n-1)) = A350028(2n) * (n-1) * (2n-1)!! = A297383(n) * (2n-2) * (2n-1)!!.