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

%I A102556 #26 Feb 16 2025 08:32:56
%S A102556 3,15,159,867,19239,107985,1222563,6965835,319153335,1835486085,
%T A102556 21185534577,122622340677,2846090375067,16550504577861,
%U A102556 192854402926251,1125503935556763,105252693980913879,615999836125850637,7219077361263238917,42347454581722163361,994637701798929524937
%N A102556 Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
%H A102556 Robert Israel, <a href="/A102556/b102556.txt">Table of n, a(n) for n = 1..928</a>
%H A102556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>
%F A102556 From _Robert Israel_, Sep 29 2016: (Start)
%F A102556 a(n) is the numerator of p(n) = Sum_{k=n..2*n-1} binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1).
%F A102556 8(n+1)*p(n+2) = (14*n+11)*p(n+1) - 3*(2*n+1)*p(n), for n >= 1.
%F A102556 G.f. of p(n):  3*x*(1 - 1/sqrt(4-3*x))/(2-2*x). (End)
%e A102556 p(n) = {3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ... }_{n >= 1}.
%p A102556 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 A102556 seq(numer(p(n)),n=1..50); # _Robert Israel_, Sep 29 2016
%t A102556 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 A102556 (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
%o A102556 (Magma)
%o A102556 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A102556 A102556:= func< n | Numerator( Coefficient(R!( 3*x*(1-1/Sqrt(4-3*x))/(2-2*x) ), n) ) >;
%o A102556 [A102556(n): n in [1..30]]; // _G. C. Greubel_, Jan 31 2025
%o A102556 (SageMath)
%o A102556 def A102556(n): return ( 3*(1-1/sqrt(4-3*x))/(2*(1-x)) ).series(x,n+1).list()[n].numerator()
%o A102556 print([A102556(n) for n in range(31)]) # _G. C. Greubel_, Jan 31 2025
%Y A102556 Cf. A102557 (denominators), A102558, A102559.
%K A102556 nonn,frac
%O A102556 1,1
%A A102556 _Eric W. Weisstein_, Jan 14 2005