login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A308914 Number of unordered pairs of non-intersecting non-selfintersecting paths with nodes that cover all vertices of a convex n-gon, n > 3. 1
2, 15, 75, 308, 1120, 3744, 11760, 35200, 101376, 282880, 768768, 2042880, 5324800, 13647872, 34467840, 85917696, 211681280, 516096000, 1246429184, 2984509440, 7090470912, 16724787200, 39190528000, 91276443648, 211392921600, 487025803264, 1116607610880 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,1

COMMENTS

Paths must have at least two nodes.

The number of non-selfintersecting paths that cover all vertices of a convex n-gon is given by A001792(n-2).

LINKS

Table of n, a(n) for n=4..30.

Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

FORMULA

a(n) = (1/3)*n*(n-1)*(n-3)*(n+4)*2^(n-8).

a(n) = (n/2)*Sum_{k=2..n-2} A001792(k-2)*A001792(n-k-2).

From Stefano Spezia, Feb 12 2020: (Start)

O.g.f.: x^4*(-2 + 5*x - 5*x^2 + 2*x^3)/(-1 + 2*x)^5.

E.g.f.: x^2*(3 + exp(2*x)*(-3 + 6*x + 2*x^2))/96.

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5) for n > 8.

(End)

EXAMPLE

a(5) = 15 since one of the non-selfintersecting paths has to be a segment connecting two adjacent vertices (5 choices) and the other path will connect the remaining vertices in one of three ways.

MAPLE

gf := x^2*(3 + exp(2*x)*(-3 + 6*x + 2*x^2))/96: ser := series(gf, x, 36):

seq(n!*coeff(ser, x, n), n=4..30); # Peter Luschny, Mar 01 2020

MATHEMATICA

Array[(1/3) # (# - 1) (# - 3) (# + 4)*2^(# - 8) &, 27, 4] (* Michael De Vlieger, Feb 25 2020 *)

CROSSREFS

Cf. A001792, A332426.

Sequence in context: A268644 A178321 A007232 * A099743 A283842 A344215

Adjacent sequences: A308911 A308912 A308913 * A308915 A308916 A308917

KEYWORD

easy,nonn

AUTHOR

Ivaylo Kortezov, Feb 12 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 07:08 EST 2022. Contains 358673 sequences. (Running on oeis4.)