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Expansion of sqrt(1-4*x)/(1-3*x).
2

%I #23 Sep 26 2024 22:20:06

%S 1,1,1,-1,-13,-67,-285,-1119,-4215,-15505,-56239,-202309,-724499,

%T -2589521,-9254363,-33111969,-118725597,-426892131,-1539965973,

%U -5575175319,-20260052337,-73908397851,-270657727593,-994938310059

%N Expansion of sqrt(1-4*x)/(1-3*x).

%C Hankel transform is A120617. In general, sqrt(1-4*x)/(1-k*x) has Hankel transform with g.f. of (1-2*x)/(1+2*(k+2)*x+4*x^2).

%H G. C. Greubel, <a href="/A137720/b137720.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} 3^k*C(2*n-2*k,n-k)/(1-(2*n-2*k)).

%F D-finite with recurrence: n*a(n) + (6-7*n)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 16 2011

%F a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 31 2014

%F a(n) = (-1)^n * A157674(2*n+1). - _Vaclav Kotesovec_, Jul 31 2014

%t CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 31 2014 *)

%t FullSimplify[Table[I*3^(-1/2+n) + 2^(1+2*n)*Gamma[1/2+n] * Hypergeometric2F1Regularized[1, 1/2+n, 2+n, 4/3]/(3*Sqrt[Pi]), {n, 0, 20}]] (* _Vaclav Kotesovec_, Jul 31 2014 *)

%o (PARI) x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ _G. C. Greubel_, Mar 21 2017

%Y Cf. A126966, A106191, A157674.

%K easy,sign

%O 0,5

%A _Paul Barry_, Feb 08 2008