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A303730 Number of noncrossing path sets on n nodes with each path having at least two nodes. 4
1, 0, 1, 3, 10, 35, 128, 483, 1866, 7344, 29342, 118701, 485249, 2001467, 8319019, 34810084, 146519286, 619939204, 2635257950, 11248889770, 48198305528, 207222648334, 893704746508, 3865335575201, 16761606193951, 72860178774410, 317418310631983, 1385703968792040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Paths are constructed using noncrossing line segments between the vertices of a regular n-gon. Isolated vertices are not allowed.

A noncrossing path set is a noncrossing forest (A054727) where each tree is restricted to being a path.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

FORMULA

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).

EXAMPLE

Case n=3: There are 3 possibilities:

.

     o       o       o

    /         \     / \

   o---o   o---o   o   o

.

Case n=4: There are 10 possibilities:

.

   o   o   o   o   o---o   o---o   o---o

   |   |   |   |   |       |   |       |

   o   o   o---o   o---o   o   o   o---o

.

   o---o   o---o   o---o   o   o   o   o

             /       \     | / |   | \ |

   o---o   o---o   o---o   o   o   o   o

.

MATHEMATICA

InverseSeries[x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O[x]^30, x] // CoefficientList[#, x]& // Rest (* Jean-Fran├žois Alcover, Jul 03 2018, from PARI *)

PROG

(PARI) Vec(serreverse(x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O(x^30)))

CROSSREFS

Cf. A054727, A303729.

Sequence in context: A078789 A299443 A128736 * A149037 A228769 A296164

Adjacent sequences:  A303727 A303728 A303729 * A303731 A303732 A303733

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Apr 29 2018

STATUS

approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)