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A303730 Number of noncrossing path sets on n nodes with each path having at least two nodes.

%I A303730 #21 Dec 15 2019 11:29:59
%S A303730 1,0,1,3,10,35,128,483,1866,7344,29342,118701,485249,2001467,8319019,
%T A303730 34810084,146519286,619939204,2635257950,11248889770,48198305528,
%U A303730 207222648334,893704746508,3865335575201,16761606193951,72860178774410,317418310631983,1385703968792040
%N A303730 Number of noncrossing path sets on n nodes with each path having at least two nodes.
%C A303730 Paths are constructed using noncrossing line segments between the vertices of a regular n-gon. Isolated vertices are not allowed.
%C A303730 A noncrossing path set is a noncrossing forest (A054727) where each tree is restricted to being a path.
%H A303730 Andrew Howroyd, <a href="/A303730/b303730.txt">Table of n, a(n) for n = 0..200</a>
%H A303730 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 A303730 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 A303730 Case n=3: There are 3 possibilities:
%e A303730 .
%e A303730      o       o       o
%e A303730     /         \     / \
%e A303730    o---o   o---o   o   o
%e A303730 .
%e A303730 Case n=4: There are 10 possibilities:
%e A303730 .
%e A303730    o   o   o   o   o---o   o---o   o---o
%e A303730    |   |   |   |   |       |   |       |
%e A303730    o   o   o---o   o---o   o   o   o---o
%e A303730 .
%e A303730    o---o   o---o   o---o   o   o   o   o
%e A303730              /       \     | / |   | \ |
%e A303730    o---o   o---o   o---o   o   o   o   o
%e A303730 .
%t A303730 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 A303730 (PARI) Vec(serreverse(x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O(x^30)))
%Y A303730 Cf. A054727, A303729.
%K A303730 nonn
%O A303730 0,4
%A A303730 _Andrew Howroyd_, Apr 29 2018