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

3, 15, 159, 867, 19239, 107985, 1222563, 6965835, 319153335, 1835486085, 21185534577, 122622340677, 2846090375067, 16550504577861, 192854402926251, 1125503935556763, 105252693980913879, 615999836125850637, 7219077361263238917, 42347454581722163361, 994637701798929524937

Offset: 1

Author

Eric W. Weisstein, Jan 14 2005

Keywords

• nonn
• frac

Examples

			p(n) = {3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ... }_{n >= 1}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..928
• Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Crossrefs

Cf. A102557 (denominators), A102558, A102559.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
A102556:= func< n | Numerator( Coefficient(R!( 3*x*(1-1/Sqrt(4-3*x))/(2-2*x) ), n) ) >;
[A102556(n): n in [1..30]]; // G. C. Greubel, Jan 31 2025

Maple

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):
seq(numer(p(n)),n=1..50); # Robert Israel, Sep 29 2016

Mathematica

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 *)

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

SageMath

def A102556(n): return ( 3*(1-1/sqrt(4-3*x))/(2*(1-x)) ).series(x,n+1).list()[n].numerator()
print([A102556(n) for n in range(31)]) # G. C. Greubel, Jan 31 2025

Formula

From Robert Israel, Sep 29 2016: (Start)

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).

8(n+1)*p(n+2) = (14*n+11)*p(n+1) - 3*(2*n+1)*p(n), for n >= 1.

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