A185634 Number of n-length cycles from any point in a complete graph on n nodes.

1, 2, 21, 204, 2605, 39990, 720601, 14913080, 348678441, 9090909090, 261535698061, 8230246567620, 281241170407093, 10371206370520814, 410525522232055665, 17361641481138401520, 781282469559318055057, 37275544492386193492506, 1879498672877297909667781

Offset

2,2

Comments

If M is the n X n matrix filled with ones, a(n) is the upper left element of (M-Id)^n.

This is the middle primitive Betti number of Dwork hypersurfaces, see Theorem 2.2 in Goutet reference with d=n. - F. Chapoton, Sep 12 2025

The formula implies that a(p+1) is divisible by p for every odd prime p, and the quotients are A056852. - F. Chapoton, Sep 13 2025

Links

Table of n, a(n) for n=2..20.

Philippe Goutet, Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces, arXiv:0912.2075 [math.NT], 2009.

Formula

a(n) = floor((n-1)^n/n) + ((-1)^n+1)/2.

a(n) = floor((n-1)^n/n)+1 for n odd, a(n) = floor((n-1)^n/n) for n even.

a(n) = ((n-1)^n+(-1)^n*(n-1)) / n. - F. Chapoton, Sep 12 2025

Example

In a complete graph in 5 nodes, there are 204 different cycles with a length of 5, from a point to itself.

Prog

(SageMath)
[((n-1)**n+(-1)**n*(n-1)) / n for n in range(2, 21)]

Crossrefs

Cf. A173499, A081216, A056852.

Sequence in context: A329553 A365061 A110253 * A077249 A068070 A085953

Adjacent sequences: A185631 A185632 A185633 * A185635 A185636 A185637

Keyword

easy,nonn

Author

Sébastien Dumortier, Dec 18 2012

Status

approved