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%I #9 Feb 17 2023 20:14:09
%S 0,0,10,105,1015,9625,90972,861420,8191920,78309000,752317280,
%T 7257522272,70223986560,680703296000,6601793730560,63984047339520,
%U 619018056228864,5972223901440000,57415027394027520,549677356175073280,5238367168966328320,49678823782558924800,468783944069762252800
%N Number of unordered quadruples of self-avoiding paths with nodes that cover all vertices of a convex n-gon; one-node paths are allowed.
%C Although each path is self-avoiding, the different paths are allowed to intersect.
%H Ivaylo Kortezov, <a href="https://doi.org/10.53656/math2022-6-4-set">Sets of Non-self-intersecting Paths Connecting the Vertices of a Convex Polygon</a>, Mathematics and Informatics, Vol. 65, No. 6, 2022.
%F a(n) = (1/3)*n*(n-1)*(n-2)*(n-3)*2^(n-15)*(4^(n-4) + 12*3^(n-4) + 54*2^(n-4) + 108) for n != 4.
%e a(6) = 6!/(2!2!2!2!)+6!*3/(3!3!) = 45+60 = 105; the first summand corresponds to the case of 2 two-node paths and 2 one-node paths; the second to the case of 1 three-node path and 3 one-node paths.
%Y Cf. A001792, A359405 (unordered pairs of paths), A360021 (unordered triples of paths).
%K nonn
%O 3,3
%A _Ivaylo Kortezov_, Feb 01 2023
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