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A308914 Number of unordered pairs of non-intersecting non-selfintersecting paths with nodes that cover all vertices of a convex n-gon, n > 3.

%I A308914 #44 Mar 01 2020 09:53:04
%S A308914 2,15,75,308,1120,3744,11760,35200,101376,282880,768768,2042880,
%T A308914 5324800,13647872,34467840,85917696,211681280,516096000,1246429184,
%U A308914 2984509440,7090470912,16724787200,39190528000,91276443648,211392921600,487025803264,1116607610880
%N A308914 Number of unordered pairs of non-intersecting non-selfintersecting paths with nodes that cover all vertices of a convex n-gon, n > 3.
%C A308914 Paths must have at least two nodes.
%C A308914 The number of non-selfintersecting paths that cover all vertices of a convex n-gon is given by A001792(n-2).
%H A308914 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F A308914 a(n) = (1/3)*n*(n-1)*(n-3)*(n+4)*2^(n-8).
%F A308914 a(n) = (n/2)*Sum_{k=2..n-2} A001792(k-2)*A001792(n-k-2).
%F A308914 From  _Stefano Spezia_, Feb 12 2020: (Start)
%F A308914 O.g.f.: x^4*(-2 + 5*x - 5*x^2 + 2*x^3)/(-1 + 2*x)^5.
%F A308914 E.g.f.:  x^2*(3 + exp(2*x)*(-3 + 6*x + 2*x^2))/96.
%F A308914 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.
%F A308914 (End)
%e A308914 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.
%p A308914 gf := x^2*(3 + exp(2*x)*(-3 + 6*x + 2*x^2))/96: ser := series(gf, x, 36):
%p A308914 seq(n!*coeff(ser, x, n), n=4..30); # _Peter Luschny_, Mar 01 2020
%t A308914 Array[(1/3) # (# - 1) (# - 3) (# + 4)*2^(# - 8) &, 27, 4] (* _Michael De Vlieger_, Feb 25 2020 *)
%Y A308914 Cf. A001792, A332426.
%K A308914 easy,nonn
%O A308914 4,1
%A A308914 _Ivaylo Kortezov_, Feb 12 2020