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 A102556 Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle. 10

%I

%S 3,15,159,867,19239,107985,1222563,6965835,319153335,1835486085,

%T 21185534577,122622340677,2846090375067,16550504577861,

%U 192854402926251,1125503935556763,105252693980913879,615999836125850637,7219077361263238917,42347454581722163361,994637701798929524937

%N Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

%H Robert Israel, <a href="/A102556/b102556.txt">Table of n, a(n) for n = 1..928</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>

%F From _Robert Israel_, Sep 29 2016: (Start)

%F a(n) is the numerator of p(n) = Sum_{k=n..2n-1} binomial(2n-1,k) 3^(2n-k)/4^(2n-1).

%F -(6n+3)p(n)+(14n+11)p(n+1)-(8n+8)p(n+2)=0 for n >= 1.

%F G.f. of p(n): 3x(1-1/sqrt(4-3x))/(2-2x). (End)

%e 3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ...

%p p:= gfun:-rectoproc({(-6*n-3)*v(n)+(14*n+11)*v(n+1)+(-8*n-8)*v(n+2), v(0) = 0, v(1) = 3/4, v(2) = 15/32},v(n),remember):

%p seq(numer(p(n)),n=1..50); # _Robert Israel_, Sep 29 2016

%t a[n_] := (3^n/4^(2n-1)) Binomial[2n-1, n] Hypergeometric2F1[1, 1-n, 1+n, -1/3] // Numerator; Array[a, 20] (* _Jean-François Alcover_, Mar 22 2019 *)

%o (PARI) a(n) = numerator(sum(k=n, 2*n-1, binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1))); \\ _Michel Marcus_, Mar 23 2019

%Y Cf. A102557, A102558, A102559.

%K nonn,frac

%O 1,1

%A _Eric W. Weisstein_, Jan 14 2005

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Last modified June 13 15:33 EDT 2021. Contains 345008 sequences. (Running on oeis4.)