A361284 Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are not allowed.

0, 0, 0, 0, 0, 15, 420, 7140, 95760, 1116990, 11891880, 118776900, 1132182480, 10415938533, 93207174060, 815777235000, 7011723045600, 59364660734172, 496238466573648, 4102968354298200, 33602671702168800, 272909132004479355, 2200084921469527092, 17618774018675345340, 140252152286127750000

Offset

1,6

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Ivaylo Kortezov, Sets of Paths between Vertices of a Polygon, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.

Formula

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1).

E.g.f.: x^3*exp(x)*(exp(2*x) - 1)^3/384. - Andrew Howroyd, Mar 07 2023

Example

a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.

Prog

(PARI) a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1)} \\ Andrew Howroyd, Mar 07 2023

Crossrefs

If there is only one path, we get A261064. If there is are two paths, we get A360716. If all n points need to be used, we get A359404.

Sequence in context: A184222 A069431 A133791 * A323781 A253447 A302112

Adjacent sequences: A361281 A361282 A361283 * A361285 A361286 A361287

Keyword

nonn,easy

Author

Ivaylo Kortezov, Mar 07 2023

Status

approved