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%I #10 Mar 11 2023 09:39:48
%S 0,0,1,10,85,695,5600,45080,364854,2973270,24382875,200967250,
%T 1662197251,13772638789,114126098450,944285871200,7791140945180,
%U 64038240953196,523977421054245,4266101869823850,34554155058753505,278417272387723315,2231755184899383220,17799741659621513240
%N 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 allowed.
%C Although each path is self-avoiding, the different paths are allowed to intersect.
%H 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 a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) + 9*5^(n-3) + 3^n + 27).
%F E.g.f.: x^3*exp(x)*(exp(2*x) + 3)^3/384. - _Andrew Howroyd_, Mar 07 2023
%e a(4) = A360021(4) + 4*A360021(3) = 6 + 4 = 10 since either all the 4 points are used or one is not.
%o (PARI) a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) + 9*5^(n-3) + 3^n + 27)} \\ _Andrew Howroyd_, Mar 07 2023
%Y If there is only one path, we get A360715. If there is are two paths, we get A360717. If all n points need to be used, we get A360021.
%K nonn,easy
%O 1,4
%A _Ivaylo Kortezov_, Mar 07 2023
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