cp's OEIS Frontend

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.

%I A361284 #14 Apr 03 2023 21:47:43
%S A361284 0,0,0,0,0,15,420,7140,95760,1116990,11891880,118776900,1132182480,
%T A361284 10415938533,93207174060,815777235000,7011723045600,59364660734172,
%U A361284 496238466573648,4102968354298200,33602671702168800,272909132004479355,2200084921469527092,17618774018675345340,140252152286127750000
%N 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.
%C A361284 Although each path is self-avoiding, the different paths are allowed to intersect.
%H A361284 Andrew Howroyd, <a href="/A361284/b361284.txt">Table of n, a(n) for n = 1..500</a>
%H A361284 Ivaylo Kortezov, <a href="http://wfnmc.org/Journal%202022%202.pdf">Sets of Paths between Vertices of a Polygon</a>, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.
%F A361284 a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1).
%F A361284 E.g.f.: x^3*exp(x)*(exp(2*x) - 1)^3/384. - _Andrew Howroyd_, Mar 07 2023
%e A361284 a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.
%o A361284 (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
%Y A361284 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.
%K A361284 nonn,easy
%O A361284 1,6
%A A361284 _Ivaylo Kortezov_, Mar 07 2023