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%I #21 Dec 15 2019 11:29:59
%S 1,0,1,3,10,35,128,483,1866,7344,29342,118701,485249,2001467,8319019,
%T 34810084,146519286,619939204,2635257950,11248889770,48198305528,
%U 207222648334,893704746508,3865335575201,16761606193951,72860178774410,317418310631983,1385703968792040
%N Number of noncrossing path sets on n nodes with each path having at least two nodes.
%C Paths are constructed using noncrossing line segments between the vertices of a regular n-gon. Isolated vertices are not allowed.
%C A noncrossing path set is a noncrossing forest (A054727) where each tree is restricted to being a path.
%H Andrew Howroyd, <a href="/A303730/b303730.txt">Table of n, a(n) for n = 0..200</a>
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F G.f.: G(x)/x where G(x) is the reversion of x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3).
%e Case n=3: There are 3 possibilities:
%e .
%e o o o
%e / \ / \
%e o---o o---o o o
%e .
%e Case n=4: There are 10 possibilities:
%e .
%e o o o o o---o o---o o---o
%e | | | | | | | |
%e o o o---o o---o o o o---o
%e .
%e o---o o---o o---o o o o o
%e / \ | / | | \ |
%e o---o o---o o---o o o o o
%e .
%t InverseSeries[x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O[x]^30, x] // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Jul 03 2018, from PARI *)
%o (PARI) Vec(serreverse(x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O(x^30)))
%Y Cf. A054727, A303729.
%K nonn
%O 0,4
%A _Andrew Howroyd_, Apr 29 2018
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