A357855 Number of closed trails starting and ending at a fixed vertex in the complete undirected graph on n labeled vertices.

1, 1, 3, 13, 829, 78441, 622316671, 3001764349333, 5926347237626029593, 2616519370820267981798929

Offset: 1

Author

Max Alekseyev, Oct 16 2022

Keywords

• nonn
• more
• walk

Comments

Trails are directed and pass through each (undirected) edge at most once in either of the two directions.

Examples

			For n = 3, we have a(3) = 3 trails starting and ending at vertex 1: 1 (single vertex), 1->2->3->1, and 1->3->2->1.
		

Crossrefs

Cf. A007082, A135388, A232545, A350028, A356366, A357856, A357857, A357885, A357886, A357887.

Formula

a(n) = Sum_{k = 0..n(n-1)/2} A357885(n,k) = 1 + Sum_{k = 1..n(n-1)/2} A357887(n,k) * k / n.

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022

a(10) from Max Alekseyev, Jul 17 2025