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%I #22 Mar 06 2023 13:49:13
%S 0,1,6,33,185,1050,6027,35014,205326,1209375,7119860,41744703,
%T 243218703,1406685280,8073640785,45991600860,260131208396,
%U 1461591509805,8162196518322,45327133739245,250431036147285,1377169337010390,7540979990097191,41130452834689218,223528009015333050,1210753768099880875,6537995998163877312
%N Number of unordered pairs 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://www.wfnmc.org/journal.html">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)*2^(-5)*(5^(n-2) + 6*3^(n-2) + 9).
%F E.g.f.: exp(x)*((x*exp(2*x) + 3*x)/4)^2/2. - _Andrew Howroyd_, Feb 19 2023
%e a(4) = A359405(4) + 4*A359405(3) + 4*3/2 = 15 + 12 + 6 = 33 with the three summands corresponding to the cases of 4, 3 and 2 used points.
%Y If there is only one path, we get A360715. If one-node paths are not allowed, we get A360716.
%K nonn
%O 1,3
%A _Ivaylo Kortezov_, Feb 18 2023