A126966 Expansion of sqrt(1 - 4*x)/(1 - 2*x).

1, 0, -2, -8, -26, -80, -244, -752, -2362, -7584, -24892, -83376, -284324, -984672, -3455144, -12259168, -43908026, -158531392, -576352364, -2107982128, -7750490636, -28629222112, -106190978264, -395347083808, -1476813394916, -5533435084480, -20790762971864, -78316232088032

Offset: 0

Author

N. J. A. Sloane, Mar 22 2007

Keywords

• sign

Comments

Hankel transform is 2^n*(-1)^binomial(n+1, 2) = A120617(n). - Paul Barry, Feb 08 2008

Links

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

Crossrefs

Cf. A000108, A028329, A082590, A126967.

Programs

GAP

List([0..30], n-> (-1)*Sum([0..n], j-> 2^j*Binomial(2*(n-j), n-j)/(2*(n-j) -1) )); # G. C. Greubel, Jan 29 2020

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-2*x) )); // G. C. Greubel, Jan 29 2020

Maple

a := n -> -add(2^j*binomial(2*n-2*j,n-j)/(2*n-2*j-1), j=0..n):
seq(a(n),n=0..30); # Emeric Deutsch, Mar 25 2007
# second Maple program:
CatalanNumber := n -> binomial(2*n, n)/(n+1):
a := n -> 2^n*I + CatalanNumber(n)*simplify(hypergeom([1, n + 1/2], [n + 2], 2)):
seq(a(n), n=0..26); # Peter Luschny, Aug 04 2020
# third program:
A126966 := n -> 2*binomial(2*n, n) - add(2^(n-k)*binomial(2*k,k), k=0..n):
seq(A126966(n), n = 0 .. 27); # Mélika Tebni, Mar 08 2024

Mathematica

CoefficientList[Series[Sqrt[1-4*x]/(1-2*x), {x,0,30}], x] (* G. C. Greubel, Jan 31 2017 *)

PARI

Vec(sqrt(1-4*x)/(1-2*x) + O(x^30)) \\ G. C. Greubel, Jan 31 2017

Sage

def A126966_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(1-4*x)/(1-2*x) ).list()
A126966_list(30) # G. C. Greubel, Jan 29 2020

Formula

a(n) = -Sum_{j=0..n} ( 2^j*binomial(2n-2j, n-j)/(2n-2j-1) ). - Emeric Deutsch, Mar 25 2007

D-finite with recurrence: n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011, corrected Feb 17 2020

a(n) ~ -4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

a(n) = 2^n*i + CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 2). - Peter Luschny, Aug 04 2020

a(n) = A028329(n) - A082590(n). - Mélika Tebni, Mar 08 2024