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.

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, 90, 144, 480, 1440, 2340, 3840, 9504, 15840, 11160, 0, 0, 0, 7, 0, 0, 70, 210, 504, 2100, 8280, 23940, 68880, 217224, 594720, 1339800, 2983680, 6482880, 10190880, 12136320, 24192000, 39621120, 0, 0, 129976320

Offset: 1

Author

Max Alekseyev, Oct 19 2022

Keywords

• tabf
• nonn
• walk

Comments

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.

Examples

			Triangle T(n,k) starts with:
 n=1: 1,
 n=2: 2, 0,
 n=3: 3, 0, 0, 2,
 n=4: 4, 0, 0, 8, 6, 0, 0,
 n=5: 5, 0, 0, 20, 30, 24, 60, 120, 0, 0, 264,
 ...
		

Links

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

Crossrefs

Cf. A007082, A135388, A232545, A297383, A350028, A356366 (row sums), A357855, A357856, A357857, A357885, A357886.

Formula

For k >= 1, T(n,k) = A357885(n,k) * n / k.

Last nonzero element in row n:

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

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