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A102556
Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
10
3, 15, 159, 867, 19239, 107985, 1222563, 6965835, 319153335, 1835486085, 21185534577, 122622340677, 2846090375067, 16550504577861, 192854402926251, 1125503935556763, 105252693980913879, 615999836125850637, 7219077361263238917, 42347454581722163361, 994637701798929524937
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking
FORMULA
From Robert Israel, Sep 29 2016: (Start)
a(n) is the numerator of p(n) = Sum_{k=n..2n-1} binomial(2n-1,k) 3^(2n-k)/4^(2n-1).
-(6n+3)p(n)+(14n+11)p(n+1)-(8n+8)p(n+2)=0 for n >= 1.
G.f. of p(n): 3x(1-1/sqrt(4-3x))/(2-2x). (End)
EXAMPLE
3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Jan 14 2005
STATUS
approved