A361284 - OEIS

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

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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.

Index entries for linear recurrences with constant coefficients, signature (64,-1880,33600,-408316, 3571520,-23217064,114182720, -428394406,1227199680,-2668273640, 4345353664,-5182504636,4366326720, -2447285400,814968000,-121550625).

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

From Andrew Howroyd, Nov 23 2025: (Start)

Binomial transform of A359404.

G.f.: 3*x^6*(1 - 4*x)*(5 - 160*x + 2180*x^2 - 16480*x^3 + 75510*x^4 - 214880*x^5 + 372340*x^6 - 362400*x^7 + 153101*x^8)/((1 - x)^4*(1 - 3*x)^4*(1 - 5*x)^4*(1 - 7*x)^4). (End)

EXAMPLE

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

MATHEMATICA

A361284[n_] := n*(n-1)*(n-2)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1)/384;

Array[A361284, 25] (* Paolo Xausa, Jan 23 2026 *)

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

Column k=3 of A390896.

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 * A376872 A323781 A253447

Adjacent sequences: A361281 A361282 A361283 * A361285 A361286 A361287