

A308914


Number of unordered pairs of nonintersecting nonselfintersecting paths with nodes that cover all vertices of a convex ngon, 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
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OFFSET

4,1


COMMENTS

Paths must have at least two nodes.
The number of nonselfintersecting paths that cover all vertices of a convex ngon is given by A001792(n2).


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*(n1)*(n3)*(n+4)*2^(n8).
a(n) = (n/2)*Sum_{k=2..n2} A001792(k2)*A001792(nk2).
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(n1)  40*a(n2) + 80*a(n3)  80*a(n4) + 32*a(n5) for n > 8.
(End)


EXAMPLE

a(5) = 15 since one of the nonselfintersecting 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



