OFFSET
1,1
LINKS
Robert Israel, Table of n, a(n) for n = 1..928
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..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)
EXAMPLE
p(n) = {3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ... }_{n >= 1}.
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
(Magma)
R<x>:=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
(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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Jan 14 2005
STATUS
approved