A332426 - OEIS

%I #27 Mar 01 2020 09:53:19

%S 0,3,30,210,1260,6944,36288,182880,897600,4316928,20427264,95373824,

%T 440294400,2013020160,9126248448,41069371392,183607050240,

%U 816037560320,3607758766080,15874168848384,69544044134400,303465064562688

%N Number of unordered pairs of non-selfintersecting paths with nodes that cover all vertices of a convex n-gon.

%C Although each path is non-selfintersecting, the two paths are allowed to intersect.

%C Paths must have at least two nodes.

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

%C Given a sequence of two or more different vertices of the n-gon, if we connect each vertex after the first one by a segment to the preceding vertex, then the union of these segments is a path (the direction does not matter).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (18,-132,504,-1056,1152,-512).

%F a(n) = n*(n-1)*2^(n-6)*(2^(n-3)-1).

%F a(n) = (1/2)*Sum_{k=2..n-2} binomial(n,k)*A001792(k-2)*A001792(n-k-2).

%F From _Stefano Spezia_, Feb 12 2020: (Start)

%F O.g.f.: x^4*(3 - 24*x + 66*x^2 - 72*x^3 + 32*x^4)/(1 - 6*x + 8*x^2)^3.

%F E.g.f.: (exp(2*x) - 1)^2*x^2/32.

%F a(n) = 18*a(n-1) - 132*a(n-2) + 504*a(n-3) - 1056*a(n-4) + 1152*a(n-5) - 512*a(n-6) for n > 8.

%F (End)

%e a(5)=30 since one of the paths has to be a segment connecting two vertices (10 choices) and the other path will connect the remaining vertices in one of three ways.

%p gf := ((exp(2*x)-1)*x)^2/32: ser := series(gf, x, 32):

%p seq(n!*coeff(ser, x, n), n=3..24); # _Peter Luschny_, Mar 01 2020

%Y Cf. A001792, A308914.

%K easy,nonn

%O 3,2

%A _Ivaylo Kortezov_, Feb 12 2020