A137720 Expansion of sqrt(1-4*x)/(1-3*x).

1, 1, 1, -1, -13, -67, -285, -1119, -4215, -15505, -56239, -202309, -724499, -2589521, -9254363, -33111969, -118725597, -426892131, -1539965973, -5575175319, -20260052337, -73908397851, -270657727593, -994938310059

Offset: 0

Author

Paul Barry, Feb 08 2008

Keywords

• easy
• sign

Comments

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).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A126966, A106191, A157674.

Programs

Mathematica

CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2014 *)
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 *)

PARI

x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ G. C. Greubel, Mar 21 2017

Formula

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

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

a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2014

a(n) = (-1)^n * A157674(2*n+1). - Vaclav Kotesovec, Jul 31 2014