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A102557 Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

%I A102557 #31 Feb 16 2025 08:32:56
%S A102557 4,32,512,4096,131072,1048576,16777216,134217728,8589934592,
%T A102557 68719476736,1099511627776,8796093022208,281474976710656,
%U A102557 2251799813685248,36028797018963968,288230376151711744,36893488147419103232,295147905179352825856,4722366482869645213696,37778931862957161709568
%N A102557 Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
%C A102557 Presumably this is the same as A093581? - _Andrew S. Plewe_, Apr 18 2007
%C A102557 a(n) equals A093581(n) for n <= 55000. - _G. C. Greubel_, Oct 20 2024
%H A102557 Robert Israel, <a href="/A102557/b102557.txt">Table of n, a(n) for n = 1..830</a>
%H A102557 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>
%F A102557 From _Robert Israel_, Sep 29 2016: (Start)
%F A102557 a(n) is the denominator of p(n) = Sum_{k=n..2n-1} binomial(2n-1,k) 3^(2n-k)/4^(2n-1).
%F A102557 8*(n+1)*p(n+2) = (14*n+11)*p(n+1) - 3*(2*n+1)*p(n), for n >= 1, with p(0) = 0, p(1) = 3/4, and p(2) = 15/32.
%F A102557 G.f. of p(n): 3*x*(1 - 1/sqrt(4-3*x))/(2*(1-x)). (End)
%F A102557 Assuming offset 0: a(n) = numerator((4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!)). - _Peter Luschny_, Dec 05 2024
%e A102557 3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ...
%p A102557 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 A102557 seq(denom(p(n)),n=1..50); # _Robert Israel_, Sep 29 2016
%p A102557 # Alternative, assuming offset 0:
%p A102557 a := n -> numer((4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!)): # _Peter Luschny_, Dec 05 2024
%t A102557 a[n_]:= (3^n/4^(2n-1)) Binomial[2n-1, n] Hypergeometric2F1[1, 1-n, 1+n, -1/3] // Denominator; Array[a, 20] (* _Jean-François Alcover_, Mar 22 2019 *)
%o A102557 (PARI) a(n) = denominator(sum(k=n, 2*n-1, binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1))); \\ _Michel Marcus_, Mar 23 2019
%o A102557 (Magma)
%o A102557 A102557:= func< n | Power(2, 4*n-2-(&+Intseq(2*(n-1), 2))) >;
%o A102557 [A102557(n): n in [1..30]]; // _G. C. Greubel_, Oct 20 2024
%o A102557 (SageMath)
%o A102557 def A102557(n): return pow(2, 4*n-2 - sum((2*n-2).digits(2)))
%o A102557 [A102557(n) for n in range(1,31)] # _G. C. Greubel_, Oct 20 2024
%Y A102557 Cf. A093581, A102556, A102558, A102559, A378071.
%K A102557 nonn,frac
%O A102557 1,1
%A A102557 _Eric W. Weisstein_, Jan 14 2005